Recent Works on Acoustics and Electroacoustics1

F. Trendelenburg, Berlin

1. Methods of acoustic measurements. 2. Processes in the sound field. Questions of room acoustics. 3. Sound emitters. 4. Sound receivers. 5. Speech and hearing. 6. Conclusion.

In the present article I shall report on a number of new works in acoustics and electroacoustics (1).2 Alongside works of a theoretical character, a large place is occupied by works concerning the improvement and detailed development of methods of acoustic measurements; it is precisely the improvement of methods that has led to significant successes, especially in the field of applied acoustics. Quantitative measurements—for example, the determination of the frequency and amplitude characteristics of an acoustic receiver and emitter—have made it possible to investigate in detail the operation of these devices and to establish to what extent particular modifications of the technical construction lead to improvement of the apparatus.

I shall therefore first take the liberty of giving an outline of the most recent development of acoustic methods of measurement; subsequently I shall turn to the presentation of various works in other areas of acoustics: works devoted to the sound field and room acoustics, to sound emitters and receivers, and, in the border areas of acoustics and physiology, to speech and hearing.

§ 1. Methods of Acoustic Measurements

At the present time the greatest practical interest is presented by measurements of sound-field processes in air—

spirit. The properties of the sound field may be regarded as physically definite if, at every point of it, one of three acoustic quantities is known as a function of time: the displacement of an air particle from its position of rest, the velocity of the oscillating particle, or the pressure in the medium.

For the sake of simplification let us first consider a sinusoidal acoustic process:

\[ a = a_0 \sin \omega t; \tag{1} \]

the velocity of the particle will be

\[ v = \frac{da}{dt} = \omega a_0 \cos \omega t, \tag{2} \]

and the pressure oscillation

\[ p = -\omega \rho_0 a_0 u \cos \omega t = p_0 \cos \omega t, \tag{3} \]

where \(u\) is the speed of sound, and \(\rho_0\) is the mean density.

The indicated relations apply to progressive plane waves; we shall speak below about certain other cases.

Here only the relations between the amplitudes of displacement, velocity, and pressure are given, because a large part of acoustic measurements is based on the determination of precisely these quantities. In particular, the pressure amplitude is comparatively easily accessible to measurement. Below we shall set forth in detail the methods of measuring it. Let us note only that acoustic processes can be measured by one further method, namely by determining the amplitude of the temperature oscillation with the aid of a thermomicrophone; temperature oscillations, according to the laws of thermodynamics, are directly connected with pressure oscillations. A. Gippel worked on the thermomicrophone effect \((^2)\). Absolute and relative measurements of temperature in standing sound waves were carried out by Friese and Wetmann \((^3)\).

The greatest importance for the measurement of the principal quantities characterizing the sound field has been acquired by those methods in which electrical sound receivers are used for recording sound, chiefly pressure receivers. In comparison with the data presented in my previous review \((^1)\), little has changed in the design of these receivers.

has changed, so that this question can be set forth only briefly. In those cases where it is necessary to obtain sound reception and transmission with a high degree of accuracy, condenser microphones have most often been used as receivers for acoustic measurements—namely, the Wente condenser microphone (⁵) and the high-frequency condenser microphone of Riegger (⁶). For such measurements the ribbon microphone of E. Gerlach (⁷) is also used.

A number of works are devoted to the development of calibration methods. Thus E. Meyer (⁸) developed a method for measuring the pressure amplitude by means of a calibrated condenser microphone, making it possible to attain high accuracy.

Fig. 1. Measurement of the pressure amplitude with a condenser microphone.

This method is based on the principle of compensation, first applied for acoustic purposes by Gerlach. Gerlach used a ribbon microphone to measure the pressure amplitude: the forces acting on the ribbon are compensated by a current of the corresponding phase and amplitude passed through the ribbon; compensation of the forces caused by the sound field takes place owing to the electrodynamic interaction between the current flowing through the ribbon and the magnetic field of the microphone. The zero setting is made by ear or by means of an amplifying device. Meyer compensates the forces acting on the membrane of the condenser microphone by means of electrostatic forces; the motion of the membrane and the pri-

bringing it to rest under compensation is established by observing the change of capacitance in a high-frequency circuit. The arrangement of the device is shown in Fig. 1; the condenser microphone is placed in the anode circuit according to Huth–Kühn’s scheme. When circuits I and II are detuned, owing to the change in the capacitance of the condenser microphone \(KM\), the constant component of the anode current \(J_a\) changes. When the capacitance of the microphone changes, the current \(J_a\) varies along the resonance curve, as shown in the left-hand part of Fig. 1. If point \(A\) is chosen as the operating point, then motions of the microphone membrane will produce oscillations of the constant anode current, and these oscillations will act on the amplifier. To compensate the motion of the membrane to zero, an auxiliary voltage of the corresponding frequency is used, which is applied to the plates of the condenser microphone through a high-frequency choke filter. Zero compensation can be established by means of a telephone connected after the amplifier or, at low frequencies, by means of a vibration galvanometer connected there. A special advantage of this method is the possibility of easily calibrating the instrument in absolute units: namely, if a pressure of a definite magnitude (negative) acts on the membrane, which can be read from a Töpler manometer for small pressures, then the displacement of the membrane can be compensated by means of a direct current of suitable strength and thus the absolute sensitivity of the instrument can be determined. E. Meyer, by means of the method just described, investigated precisely to what extent measurements of the sound field made in this way agree with the values obtained by means of Rayleigh’s disk.

The significance of Rayleigh’s disk for measuring a sound field is so great that I consider it appropriate to give a brief exposition of the theory of the disk. If a disk is placed in a sound field, inclined to the direction of the sound rays, then the disk tends to set itself perpendicular to the direction of the sound wave. The torque \(M\) of the disk, caused by the action of the sound wave, is proportional to the square of the velocity. This relation (under the condition of a very

(of small thickness of the disk) can be expressed, according to König \((^9)\), by the formula:

\[ M=\frac{2}{3}\rho_0 v_0^2 r^3 \sin 2\vartheta, \tag{4} \]

where \(r\) denotes the radius of the disk, and \(\vartheta\) the angle between the plane of the disk and the direction of the sound.

The work of E. Meyer mentioned above may serve as a new and, moreover, quite independent confirmation, by older methods, of the correctness and absolute applicability of formula (4).

Table 1 is taken from the above-mentioned article by Meyer.

Table 1

Frequency	Condenser microphone	Rayleigh disk
240 hertz	45 bar	53 bar
330 »	93 »	101 »
400 »	90 »	97 »
555 »	116 »	129 »
780 »	214 »	214 »
1150 »	88 »	97 »

The table gives measurement data (converted into pressure amplitudes) for sound oscillations in a Kundt tube, obtained with the aid of a condenser microphone and a Rayleigh disk. The discrepancy between the values obtained by the two methods averages only about 8%.

It should also be mentioned here that, for determining the amplitude characteristic of sound emitters, Meyer’s electrostatic compensation method \((^{10})\) is also used. The sound emitter is excited by a sinusoidal electromotive force; in this process, first the fundamental tone corresponding to the excitation is successively compensated, and then the overtones caused by the various imperfections of the sound emitter. We shall return to the interesting

as a result of these investigations in the chapter on sound radiators and receivers.

The Rayleigh disk, whose operation we briefly described above, has been used repeatedly in acoustic measurements. Thus, for example, Trendelenburg (11) calibrated Riegger’s high-frequency condenser microphone by placing it in the sound field next to a Rayleigh disk, the sound field being excited by Riegger’s “Blatthaller.” Hartmann (12) used a similar arrangement for calibrating Gerlach’s ribbon microphone. We shall discuss the results of these measurements in the chapter on sound radiators and receivers when considering the corresponding instruments. The Rayleigh disk was also used directly for investigating sound radiators. E. Meyer, with the aid of the Rayleigh disk, studied in detail the properties of loudspeakers; we shall also return to the results of these investigations later.

Substantial advances have been achieved in methods of acoustic measurement serving to obtain the frequency characteristics of apparatus; they have been so simplified technically that these measurements can now be carried out without a large expenditure of time and, most importantly, without resorting to subjective observation. Such a method of automatic recording is described by Cogen, Altridge, and West (13). In Germany, an apparatus for recording sound was constructed and used by Grützmacher and Meyer (14) at the State Telephone Administration, and by E. Gerlach (15) in the laboratory of Siemens & Halske.

The design of all the above-mentioned recording instruments is based on the common idea of connecting a photographic recording on a drum with a mechanism that determines the pitch of the tone; this is done, for example, in such a way that during one complete revolution of the drum the tone changes over the whole scale of sounds from very low to the highest frequencies that may still be of interest, i.e., approximately up to 10,000 hertz.

To excite the sound field, a radiator operating by the beat method is used, connected through a suitable

…amplifier with a loudspeaker. The drum for photographic recording is connected to a variable capacitor, which determines the frequency of one of the two electrical circuits of the emitter; the circuits are selected so that

Fig. 2. Diagram of the apparatus for obtaining frequency characteristics.

when the capacitor is turned from 0° to 180° the beat tone passes through the entire scale of sounds. Fig. 2 schematically shows the arrangement of the instruments. The receiver of the sound is a condenser microphone, which, through an amplifier and a rectifier, acts on a mirror galvanometer; the deflection of the galvanometer is recorded on the drum. By means of an artificial method found by Konom, Altridge, and West, it is possible to choose the characteristic of the rectifying tube so that it is practically linear; the method of connecting the rectifier is shown in Fig. 3. Grützmacher and Meyer used a tube with an oxidized filament (type B. O. Siemens and Halske). The grid was connected to the anode, and for

Fig. 3. Linearly operating rectifier.

F. TRENDELENBURG

To obtain a rectilinear characteristic, a resistance of 100,000 Ω was inserted in the anode circuit. A small additional voltage is applied to the grid through a potentiometer from the filament batteries. The measured voltage is applied across the resistance of 100,000 Ω. Under these conditions a characteristic is obtained

a          b          c

Fig. 4. Characteristics of the rectifier according to the circuit of Fig. 3.

which, over wide limits, even for very small voltages, is linear; in Fig. 4 a—c the current and voltage characteristics are shown. Let us emphasize that the method described can often be successfully used also in measurements with a cathode voltmeter.

Fig. 5. Voltage at the output of the sound emitter (buzzer).

By introducing the corresponding corrective elements into the circuit of an emitter constructed by the beat method, and also into the circuit of the amplifier following the condenser microphone, Grützmacher and Meyer achieved complete independence of the operation of the whole device from frequency. In Fig. 5 the voltage at the output of such an emitter is shown; in Fig. 6, the frequency characteristic of the recording apparatus as a whole.

We shall return further on to the interesting results obtained with the aid of the indicated methods of automatic recording.

The devices described make it possible automatically to obtain the frequency characteristic of a sound emitter; thereafter

Fig. 6. Frequency characteristic of the recording apparatus.

the problem was posed—on analogous principles—to build apparatus for the automatic analysis of sound, or, more generally, for the analysis of a mixture of different frequencies.

Fig. 7. Schematic diagram of automatic sound analysis.

Similar apparatus was developed by Moore and Curtis (¹⁶), and also by Gerlach (¹⁷) and Grützmacher (¹⁸).

The frequencies being analyzed act on an electrical sound receiver and, after appropriate amplification, are fed

to the grid of the rectifier tube. In addition, a voltage from the oscillator is applied to the grid of the tube, with beats whose frequency can be varied from 60 to 10,000 hertz by simply turning a variable capacitor.

Beyond the rectifier is placed a filter that passes only frequencies from 0 to 20 hertz; after it is an amplifier, to which is connected an instrument for photographic recording (Fig. 7).

We shall explain the operation of this automatic sound analyzer by means of a practical example: suppose that a sound consisting of components with frequencies of 200, 400, and 800 hertz falls on the receiver, so that, correspondingly, alternating electric voltages with frequencies of 200, 400, and 800 hertz are applied to the grid of the rectifier. These oscillations are combined with the oscillations coming from the oscillator. If the operating characteristic of the rectifier is quadratic, then after the rectifier there are also obtained combination tones between the oscillator oscillations and the sound frequencies. Of all these combination tones, the sound filter, as already mentioned, passes only oscillations lying between 0 and 20 hertz; the recording instrument will therefore show something only when the difference between the oscillator frequency and one of the frequencies present in the sound field does not exceed 20 hertz, because only then will the difference tone of the oscillator and the corresponding tone in the sound field be able to pass through the sound filter. If, in the case under consideration, the oscillator is smoothly tuned from 60 to 10,000 hertz, then the recording instrument will act at those moments when the oscillator frequency lies between 180 and 220 hertz, then between 380 and 420, and also between 780 and 820 hertz; on the recording drum with photographic paper, rises of the curve will appear in the corresponding places. It is possible to ensure that the amplitude of these rises is, with great accuracy, proportional to the amplitude of the corresponding component of the sound field; in this way an automatic analysis can easily be performed. It should be noted, however, that the change in pitch must be carried out so slowly that it is possible to avoid-

to avoid errors depending on the time of excitation of oscillations in oscillating systems; thus one can study only such acoustic processes as last several seconds.

The methods we have analyzed make it possible to measure those acoustic processes in air for which it is necessary to determine the amplitude. We have not yet touched upon the direct determination of the phase shift between pressure, velocity, and elongation. Generally speaking, phase relations play no role for acoustic processes in an undisturbed sound field, at distances from the sound source large in comparison with the wavelength; from equations (1), (2), and (3) we saw that in these cases the phase angle is equal to \(90^\circ\) or to zero. The relations obtained here are the same as for an alternating electric field at a large distance from the radiating antenna. But the question becomes very complicated when we wish to measure acoustic processes in closed media—for example, in acoustic filters.

Recently our knowledge of processes in closed acoustic systems has expanded considerably; here it will be appropriate to examine the theory of these systems, which may also find application as auxiliary means in acoustic measurements. After this it will be possible to proceed to methods developed specially for the practical study of such systems. In expounding the theory of these systems, we shall use the circumstance that they are in many respects analogous to electric oscillatory circuits; in particular, we shall make use of the theory of electric filters.

In a free sound field, for distances from the sound source large in comparison with the sound wavelength, the relation following from equations (2) and (3) is valid:

\[ \frac{p}{v} = \rho_0 u = S. \tag{5} \]

The quantity \(S\) is called the acoustic resistance. The simple dependence (5) is not valid for closed acou-

acoustic systems—for example, for resonators or acoustic filters. Here the principal role is played by the phase difference between pressure and velocity. In this case one may put

\[ \frac{p}{v}=S. \tag{6} \]

\(S\) is in general a complex quantity. Theoretical considerations acquire the greatest clarity for closed systems if, instead of the particle velocity \(v\), we introduce a somewhat different quantity, namely the volume flow \(V'\). This quantity is given by the equation:

\[ V'=\frac{dV}{dt}, \]

where \(V\) denotes the volume displacement (the cross-section of the conductor multiplied by the displacement); in this case the expression

\[ \frac{p}{V'}=Z \tag{7} \]

is called the acoustic impedance. We shall see that the introduction of this concept proves very useful in the analysis of processes in closed acoustic systems; it is analogous to the concept of electrical impedance, and this makes it possible to visualize the picture of sound processes on the basis of the corresponding electrical phenomena, which have been studied both theoretically and practically in all details.

As the starting point for our further considerations we shall take the differential equation, known from the theory of electricity, of an electrical circuit with self-inductance, ohmic resistance, and capacitance \(^{(19)}\):

\[ L\frac{d^{2}q}{dt^{2}}+R\frac{dq}{dt}+\frac{1}{c}q=E \tag{8} \]

or, in another form,

\[ L\frac{di}{dt}+Ri+\frac{1}{c}\int i\,dt=E, \tag{9} \]

where

\[ i=\frac{dq}{dt}. \]

The corresponding differential equation of mechanics for a material point bound by elastic forces to its equilibrium position is:

\[ m\frac{d^2x}{dt^2}+r\frac{dx}{dt}+cx=K. \tag{10} \]

Comparison of the two formulas shows that in formula (10) \(dx/dt\) has taken the place of \(i\), and \(m\) the place of \(L\).

Let us for the moment leave aside both the term expressing friction and the term expressing the elastic force; we shall therefore consider a process in which some mass is set in motion by a force \(K\), and, for simplicity, we shall assume that \(K\) is sinusoidal, i.e. \(K=K_0\sin\omega t\). The laws governing such a process, according to what has been said above, must be analogous to those which determine the character of the current in self-induction under the action of a periodic electromotive force. The question is: how can the acoustic analogue of electrical self-induction be realized?

Fig. 8. Acoustic self-induction.

Fig. 9. Acoustic capacitance.

Let us imagine that sound oscillations occur in a tube, in the side wall of which there is an opening, or, still better, in a short branch open at the end (Fig. 8). Under the action of the pressure oscillations taking place in the tube, the volume of air between \(A\) and \(B\) will move back and forth; if we denote by \(m\) the mass of this volume of air, and by \(x\) its displacement, then we obtain the relation:

\[ m\frac{d^2x}{dt^2}=F\cdot P_0 e^{i\omega t}, \tag{11} \]

where \(F\) denotes the cross section of the lateral branch, and \(P_0\) the pressure amplitude.

Let us introduce into this equation the quantity of volume flow defined above, and replace the mass \(m\) by the density of the air and its volume (the cross-section \(F\), multiplied by the length \(l\) of the side branch of the tube); then we obtain

\[ \rho_0 \frac{l}{F}\frac{d^2 V}{dt^2}=P_0 e^{i\omega t} \tag{12} \]

an equation entirely analogous to the equation

\[ L\frac{d^2 q}{dt^2}=E_0 e^{i\omega t}; \]

\[ \rho_0\frac{l}{F} \]

is therefore the expression for the acoustic self-induction of the side branch of the tube.

It should be noted that in determining the mass of air set in motion by pressure oscillations, we make one more simplification—we do not take into account the accompanying oscillations of the surrounding air; however, part of the external air located directly at the opening of the tube is, of course, set into oscillation. To introduce this correction, according to Rayleigh one must add to the length \(l\) the quantity \(\pi R/2\), where \(R\) denotes the radius of the side branch of the tube. As a result, for the acoustic self-induction one obtains the expression

\[ L_{ak}=\rho_0\frac{l+\frac{\pi R}{2}}{\pi R^2}. \tag{13} \]

In a similar manner we can also realize an acoustic capacitance. The lateral opening of the tube (Fig. 9) is connected with a closed chamber. The volume of air enclosed between the points \(A\) and \(B\), under the action of periodic pressure changes in the tube, undergoes elastic oscillations. It can be shown that the acoustic capacitance created in this way is determined by the equation:

\[ \frac{1}{C_{ak}}=\frac{\rho_0 u^2}{V_k}, \tag{14} \]

where \(V_k\) is the volume of the chamber, and \(u\) is the speed of sound.

Just as in electrical circuits we can choose the self-induction and capacitance so as to obtain

resonance or create a filter, so in acoustics we can make use of the acoustic self-inductance and capacitance elements considered above in order to obtain the desired effect.

As a first example we shall consider here a system analogous to a simple resonant electrical circuit, namely, the usual type of Helmholtz resonator.

This resonator is shown in Fig. 10. The mass of air \(\rho_{0}\pi l R_{1}^{2}\) enclosed in the neck of the resonator sets into elastic oscillation the volume of air located inside the resonator \(U_{\mathrm{res}}\); according to the formulas indicated above \((2^{0})\), the acoustic self-inductance will be equal to

\[ L=\frac{\rho_{0}l}{\pi R_{1}^{2}}, \]

and the acoustic capacitance

\[ C=\frac{V_{\mathrm{res}}}{\rho_{0}u^{2}}. \]

Accordingly, from the formula for the natural frequency of an electric circuit

\[ n_{0}=\frac{1}{2\pi\sqrt{LC}} \]

we obtain for the natural frequency of the Helmholtz resonator1

\[ n_{0}=\frac{u}{2\pi}\sqrt{\frac{\pi R_{1}^{2}}{lV_{\mathrm{res}}}}. \tag{15} \]

Fig. 10. Helmholtz resonator.

From this simple oscillatory system we shall now proceed to more complex acoustic systems and shall dwell first of all on acoustic filters. Stuart was the first to ascertain the possibility of combining acoustic self-inductances and capacitances to obtain systems which would delay a certain range of frequencies and pass other frequencies without attenuation. He also pointed out the broad analogies between electrical and acoustic filters.

The general scheme of such an acoustic system is shown in Fig. 11. A series of identical acoustic elements, with impedance \(\dot Z_{1}\), are arranged in succession—

additionally; elements of another kind with impedance \(Z_2^{\,1}\) are connected at the side.

Such a system is analogous to an electrical filter constructed according to the scheme of Fig. 12.

Fig. 11. General scheme of an acoustic filter.

The theory of electrical filters requires, as is known, first of all that the sum of the currents converging at the points \(A_n\), \(A_{n+1}\ldots\) be equal to zero. In exactly the same way the theory of acoustic filters proceeds from the assumption that the sum of the volume flows at these nodal points is equal to zero

Fig. 12. General scheme of an electrical filter.

(Kirchhoff’s law for acoustics). If one writes the differential equation for the voltage and the current (and, correspondingly, for the pressure and the volume flow in acoustics), then it can be shown that the resulting filter passes only those frequencies for which1

\[ 0 > \frac{Z_1}{Z_2} > -4. \]

As a first practical example let us consider an acoustic filter which, in the character of its operation, closely resembles the circuit of a capacitive filter. This circuit is shown in Fig. 13. Capacitors connected in series have a large resistance for low frequencies, whereas chokes connected in parallel have a small resistance for them. Thus low frequencies cannot pass through the filter, while high frequencies pass practically without hindrance. The “cutoff frequency” for such a system is equal to

\[ \omega_0=\frac{1}{2\sqrt{L_1 C_1}}; \]

higher frequencies are passed by the filter.

The scheme of an acoustic filter for screening out low frequencies is given in Fig. 14.

Fig. 13. Capacitive filter.

An acoustic filter consists of a tube having a series of open side branches. These branches are acoustic self-inductances—they correspond to chokes connected in parallel, as indicated in Fig. 13. The acoustic capacitance lies in the intermediate sections of the tube between each two self-inductances. It must be noted, however, that the analogy with the simple electrical circuit described above cannot be carried through to the end, because these intermediate parts of the tube themselves possess considerable acoustic self-inductance. It can be shown that in a tube of this kind the self-inductance and the capacitance have to be regarded as connected in parallel. If we wish to obtain predominantly a capacitive effect, for the circuit composed of capacitances and self-inductances, it is necessary to take a very low period of natural oscillations. For frequencies exceeding the natural frequency, the tube acts chiefly as a capacitance. By giving the proper dimensions to the portions of the tube intermediate between the branches, one can achieve the predominance of one or the other effect.

As a second example, let us consider a choke filter (Fig. 15), which does not pass frequencies higher than the limiting frequency:

\[ \omega_0=\frac{2}{\sqrt{L_2C_2}}. \]

Fig. 14. Acoustic filter passing high frequencies.

An acoustic system operating analogously to a choke filter is shown in Fig. 16. In this type of filter,

Fig. 15. Choke filter.

to the side wall of the tube, in which the effect of self-inductance predominates, there is attached a series of side cavities closed at the end. It is still more expedient, in order to reduce the unavoidable internal self-inductance of these side cavities, to give them a form similar to a resonator cavity, as shown in Fig. 16.

Fig. 16. Acoustic filter passing low frequencies.

Finally, one can also construct a filter (Fig. 17) which passes only one definite frequency band:

\[ \text{from }\ \omega_1=\frac{1}{\sqrt{L_4(C_4+4C_3)}}\ \text{ to }\ \omega_2=\frac{1}{\sqrt{L_4C_4}}. \]

An acoustic filter possessing similar properties is shown schematically in Fig. 18.

Fig. 17. Electrical filter passing only one frequency band.

What has been said is sufficient to show how fruitful the method of analogies between processes in electrical and acoustic systems is (21). But it must once again be emphasized that carrying out here a completely rigorous analogy often proves

Fig. 18. Acoustic filter passing only one frequency band.

to be impossible,—the example just considered of the practical inseparability of the acoustic compliance of the tube from its acoustic self-inductance is sufficiently

Fig. 19. Determination of the transmission capacity of a filter.

clearly indicates this. It must be said, however, that such compromises have to be allowed in electrical engineering as well,

Fig. 20. Frequency response of a filter passing high frequencies.

however there—at least for frequencies corresponding to

Fig. 21. Frequency response of a filter passing low frequencies.

the frequencies of conversational speech—the discrepancies are insignificant and appear only in one case, namely in the case of [[unclear: abbreviated word beginning “dr”]]

...networks with their ineradicable inherent capacitance; yet in electrical engineering there is nothing analogous to a capacitance with an intractable self-induction.

In practice, a number of methods have been developed for determining the acoustic properties of filters. An especially simple method, possessing sufficient accuracy for trial measurements, is presented in Fig. 19 (²²). Two telephones, \(T_1\) and \(T_2\), are connected to the observer’s ears: one by a simple tube, the other through the filter; with the aid of the switch \(U\), either the sound passing through the tube or the sound passing through the filter can be supplied to the ear. By shunting the

Fig. 22. Frequency characteristic of a filter passing only one frequency region.

Fig. 23. Measurement of acoustic impedance.

telephone with a parallel ohmic resistance, one can weaken one sound to such an extent that both sounds will seem equal in strength. Provided that the operation of the telephones is identical, the magnitude of the parallel resistance will be a measure of the transmission capacity of the filter. In Figs. 20–22 are shown the results of measurements made by Stewart (²³) by an analogous method.

In conclusion, let us indicate one more method of measurement (²⁴), which makes it possible to obtain, in absolute measures, the magnitude of the impedance of a given acoustic system (Fig. 23). The sound source (telephone \(T_1\)) sends a continuous series of sound waves through a long tube that gives no reflections. The resulting sounds can be picked up through an ear tube attached at some point to the given tube. Another telephone \(T_2\) makes it possible to compen-

to zero the sound perceived by the ear. If now, by means of a side branch, the tube is connected with the device whose acoustic impedance we wish to measure, for example with a resonator or a horn, the compensation will be disturbed. By varying the magnitude of the excitation of the auxiliary telephone and the phase relation of the two telephones, the compensation can be restored; from the relations between the excitations and phases before and after the connection of the apparatus under investigation, it proves possible to calculate the impedance.

§ 2. Processes in a Sound Field.

Questions of Room Acoustics

In considering processes in a sound field, the questions of greatest interest at the present time are those connected with the study of directional sound radiation. The directional action of a sound source depends essentially on the dimensions of the sound source in comparison with the length of the sound wave; a separate and point-like sound source does not give directional action; it radiates uniformly in all directions diverging spherical waves. The relations become, however, very complicated for such sound sources whose dimensions are large in comparison with the length of the emitted sound wave.

To elucidate the question of the directional action of sound emitters, Bakhaus and Trendelenburg (25) investigated the directional radiation of piston membranes; the radiation of such a sound source is comparatively easy to investigate, and in this case it is especially easy to verify experimentally the results of theoretical calculations. In Riegger’s “Blatthaller” we have an emitter which in practice, with a very high degree of approximation, operates on the principle of a piston membrane and at the same time covers almost the entire region of auditory perception, so that experiments can be carried out with it for all possible frequencies.

For the theoretical investigation of questions of sound

It is also convenient, along with the principal quantities of the sound field defined above—elongation, particle velocity, and pressure oscillation—to introduce the concept of the velocity potential \(\Phi\). The velocity potential is such a function whose derivative with respect to a spatial coordinate gives the velocity; the derivative of the velocity potential with respect to time, on the other hand, is proportional to the pressure at the corresponding point of the sound field. We have:

\[ p=-\rho\frac{\partial\Phi}{dt}\qquad v=\frac{\partial\Phi}{\partial r}. \]

A general expression for the sound radiation of a piston membrane was given by Rayleigh. This expression makes it possible to calculate the velocity potential in the sound field of a piston membrane oscillating in an infinitely extended wall:

\[ \Phi=-\frac{1}{2\pi}\iint \frac{\partial\varphi}{\partial n}\frac{e^{-ikr}}{r}\,dS. \tag{16} \]

Here \(dS\) is an element of the membrane area, and the integration is to be carried out over this area; \(r\) is the distance of the point under consideration from the corresponding area element, \(k=2\pi/\lambda\), where \(\lambda\) is the wavelength, and \(n\) is the direction of the normal to the surface of the membrane.

The integration is easily carried out for all those points that lie on the perpendicular to the middle of the membrane, taken as a circle of radius \(R\). The pressure amplitude at these points takes the value

\[ p_0=k_1\frac{2v'}{k}\sin\left[\frac{k}{2}\left(\sqrt{z_0^2+R^2}-z_0\right)\right], \tag{17} \]

where \(z_0\) denotes the distance of the point under consideration from the center of the membrane, \(v'\) is the amplitude of the velocity of the piston membrane, and \(k_1\) is a constant.

Thus the amplitude, depending on the distance, varies from 0 to \(2v'/k\). At the points where

\[ \frac{k}{2}\left(\sqrt{z_0^2+R^2}-z_0\right)=n\pi \]

(\(n\) is an integer), complete disappearance of the sound is obtained. For these points we may write the relation

\[ z_0=\frac{(R^2/\lambda^2)-n^2}{2n/\lambda}. \tag{18} \]

There can be only a finite number of points of complete extinction. They are obtained only so long as \(n < R/\lambda\). Such interference points are therefore obtained in greater number the larger the ratio of the radius of the membrane to the wavelength.

A sound receiver sensing the pressure amplitude, when moved along the perpendicular to the center of the membrane, should, according to equation (17), show maxima and minima; this was also confirmed by means of a condenser microphone placed in the sound field. Fig. 24 gives the course of the pressure along the perpendicular to the center of the membrane; the pressure was measured by means of a condenser microphone.

Fig. 24. Variation of pressure along the perpendicular to the center of a piston membrane.

In the work cited, it was not possible to take the integral for points lying outside the perpendicular to the center of the membrane, but it is nevertheless possible, presumptively, to find the distribution of intensity outside the central perpendicular on the basis of an analogy with the corresponding problems of optics \((^{26})\). In optics, theoretical considerations show that the angle between the central perpendicular and the generatrix of the cone on whose surface the amplitude is zero obeys the condition

\[ \varphi = 0.61\,\frac{\lambda}{R}. \]

The acoustic maxima and minima in the sound field excited by a piston membrane of radius \(R\) are arranged in a quite analogous way. The product \(2 \times 0.61\,\lambda/R\) represents the solid angle at the vertex of the cone containing the entire principal part of the radiation of the sound source; the secondary maxima lying outside this cone turn out to be considerably smaller than the principal maximum and constitute only an insignificant part of the total radiation. Experimentally these relations were veri—

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measured by means of a condenser microphone mounted on a rod 3 m long; the rod can be rotated, and thus the pressure can be investigated, point by point, along the circumference described from the center of the membrane; we shall return later to the results of these investigations.

The diffraction phenomena obtained at a distance large in comparison with the size of the membrane were quantitatively calculated long ago by Stenzel (²⁷).

We shall briefly present here the results of his calculations for a square piston membrane; this form most closely corresponds to the Blatthaller membrane. In this way we shall be able to compare the experimental data directly with the theoretical calculation. It should also be mentioned that Stenzel’s work treats, in a general form, the questions of the sound field created by a row of point sources of sound; from sound point sources arranged in a row Stenzel then passes to a radiating surface. Considering the distribution of the sound field in a plane passing through the center of the membrane perpendicular to it and at the same time parallel to one of the side edges of the membrane, we obtain the following expression: the amplitude of the velocity potential (which, as was mentioned above, is proportional to the pressure amplitude) can be expressed in the form

\[ p_0 = k_2 \cdot \frac{\sin\left(\frac{a\pi}{\lambda}\sin\alpha\right)} {\frac{a\pi}{\lambda}\sin\alpha}, \tag{19} \]

where \(k_2\) is a certain constant, \(\alpha\) is the angle between the perpendicular to the middle of the membrane and the direction from the center of the membrane to the point under consideration; \(a\) is the side of the square membrane.

Substituting the quantity \(z\) for \(\frac{a\pi}{\lambda}\sin\alpha\), we obtain

\[ p_0 = k_2\left(\frac{\sin z}{z}\right). \tag{20} \]

The curve of the function \(\sin z / z\) is shown in Fig. 25.

For a sufficiently large membrane, when the angle \(\alpha\) is varied

from 0 to \(\pi/2\) a whole series of maxima and minima of excitation is obtained.

In Fig. 26a the pressure distribution in the plane is presented, obtained for \(a/\lambda = 1.5\) and \(a/\lambda = 5\).

Fig. 25. Graph of the function \(\dfrac{\sin z}{z}\).

In Fig. 26b the pressure distribution is shown as it was actually measured. The corresponding values of \(a/\lambda\) in these experiments ranged from 1.9 to 4.7.

Fig. 26a. Directional action of a piston membrane according to Stenzel’s calculations.

The agreement between Stenzel’s calculations and the measurements of Backhaus and Trendelenburg may be regarded as good. The position of the maxima and minima gives satisfactory agreement with the values calculated

theoretically, although the oscillations of excitation in the experimental curves are not as large as is assumed in the calculation; what accounts for these quantitative discrepancies has not yet been clarified.

The sound radiation of a piston membrane has been discussed here in comparatively great detail, because the situation is analogous for other types of sound radiators, for example, for the radiation of a horn. Namely, if the length of the horn is large in comparison with the diameter of the aperture, then the particles in the plane of the aperture oscillate approximately

Fig. 26b. Directional action of a piston membrane according to the measurements of Backhaus and Trendelenburg.

in the same phase, and if, in addition, the aperture itself is large in comparison with the wavelength, then the sound field in front of the aperture becomes similar to the sound field in front of a piston membrane. In Fig. 27 the pressure distribution in the sound field in front of a loudspeaker with a horn is presented, the drawing being made somewhat differently from Fig. 26. Namely, in Fig. 27 curves of equal pressure are plotted; from the drawing it is easy to see how rapidly the pressure falls off when moving away from the central axis; for the formation of secondary interference maxima the ratio of the horn aperture to the wavelength is still not large enough, and the condition of sufficient horn length

with respect to the diameter of the aperture has also not been fulfilled here to a sufficient degree.

We have so far touched upon questions connected with the excitation of a sound field by means of sound sources of finite dimensions. In doing so we considered only sound fields extending without limit; the only limitation here was the sound source itself. We also confined ourselves to considering only stationary processes. Nonstationary processes are of interest only for sound fields comparatively closely bounded in space; thus, for example, the processes of build-up and decay of sound inside rooms (the reverberation of rooms) depend essentially on the configuration and characteristics of the room.

Fig. 27. Sound field in front of a loudspeaker with a horn, according to Meyer’s measurements.

Recently, research in the acoustics of rooms has also become of interest for radiotelephony: in all cases where it is required to obtain the best conditions for radio transmission of some speech or opera, it is necessary to take careful account of the acoustic qualities of the room. In view of this, we shall briefly examine the newest works in this field, and it will not be possible to leave aside even works earlier than most of those considered up to now, if the aim is to give, as far as possible, a complete survey of this field.

Research in architectural acoustics concerns mainly the processes that occur in the reflection and absorption of sound. The study of sound reflection can be greatly facilitated by experience drawn from optics; there is a far-reaching analogy between the reflection of light and that of sound. It must be noted, however, that in ordinary cases of optical reflection the condition is almost always satisfied that the reflecting surface is large in comparison with the wavelength; whereas in acoustic reflections this condition is often not fulfilled. Studies of the reflection of sound inside rooms have nevertheless also been carried out by purely acoustic methods. W. C. Sabine \(^{(28)}\) made experiments on sound reflection in reduced-scale models of buildings: the traveling wave was produced by the crack of a spark; the various phases of propagation were recorded photographically by the method of Toepler. We shall not describe Toepler’s method itself, as an optical method.

What, then, is the disturbing effect of reflections? Between two syllables in normal speech there elapses a time of approximately \(1/5\) second; if, after this interval has expired, a reflected train of waves reaches the listener, then it will coincide with the next syllable coming directly from the sound source, and the intelligibility of speech will be considerably impaired. Sabine’s experiments with models are an excellent example of how the conditions of reflection should be checked in advance, before the corresponding room is built, and, if any interference is discovered, how it should be eliminated by changing the shape of the building’s contours.

Multiple reflections are especially dangerous. E. Michel \(^{(29)}\) recently made a report on such repeatedly recurring echoes. He also carried out experimental investigations of these questions with the aid of waves on water, the water being poured into tanks whose walls represented a reduced model of the sectional contours of the room under test. As a practical means against such repeatedly recurring echoes, sound-absorbing mate...

materials introduced into a room; the great influence of sound absorption on the acoustics of rooms compels us to touch briefly upon this question.

The duration of the decay of sound (reverberation) inside a room depends on the absorption of sound by the walls and by the people and objects present in the room, and also on the internal surface and volume1 of the given room. By the reverberation time, according to Sabine’s definition (³¹), is understood the time during which a sound whose intensity exceeds the intensity at the threshold of audibility by a factor of \(10^6\) is weakened to its value at the threshold of audibility. For the reverberation time \(t\) in a room of volume \(V\), Sabine established the formula

\[ t = 0.164 \frac{V}{A_R}, \tag{21} \]

where \(A_R\) denotes the total sound absorption in the given room. \(A_R\) is composed of the following terms:

\[ A_R = \sum a_n' f_n + \sum a_m'' s_m + \sum a_p''' v_p. \]

Here:

\(a_n'\)—absorption of sound on \(1\,m^2\) of the surface of the hall, in comparison with \(1\,m^2\) of an open window;

\(f_n\)—the corresponding area in \(m^2\),

\(a_m''\)—sound absorption by objects located in the hall (per object),

\(s_m\)—the number of corresponding objects,

\[ \left. \begin{aligned} a_p'''&\text{—absorption of sound per }1\,m^3,\\ v_p&\text{—volume in }m^3 \end{aligned} \right\} \quad \begin{gathered} \text{for all materials whose}\\ \text{absorption depends on volume.} \end{gathered} \]

The values of the various absorption coefficients are given in the following table:

Table 2

Material or object	Absorption coefficient
open window per 1 m²	1,00
audience	0,96
curtains with many folds	0,5—1,00
felt, 5 cm, unpainted	0,7
felt, 2,5 cm, unpainted	0,55
felt, 5 cm, painted on top	0,4—0,6
felt, 2,5 cm, painted on top	0,25—0,45
openings of heating and ventilation ducts	0,5
open stage, depending on the wings	0,25—0,40
heavy carpets	0,29
oil paintings in frames	0,28
curtains	0,15—0,25
dense wooden paneling	0,061
plaster	0,033
brick wall	0,017—0,25
marble	0,01
listeners (per person)	0,44—0,55
chairs, depending on upholstery (per chair)	0,14—0,28
indoor plants	0,11

The absorption coefficients given in the table refer to sound with a frequency of 512 hertz. The coefficients depend strongly on frequency; for lower frequencies the absorption, generally speaking, is smaller, while for higher ones it is at first considerably greater. This dependence can be seen in Table 3 (³²).

Concerning the course of the reverberation curve, it must be said that the total amount of sound energy in a room decreases according to an exponential law.

According to Eyring (³³), during reverberation the amount of sound energy in a room may be expressed by the following relation

\[ E = E_0 e^{-\frac{a u F t}{4v}}, \tag{22} \]

where \(E_0\) denotes the energy in the stationary state.

In order that reverberation not serve as an obstacle to auditory perception, it is best for rooms intended for speech to have a reverberation of 0,5—1 sec; for musical performances a longer reverberation is suitable; here even \(1^{1/2}\) seconds may be allowed.

Table 3

Material	128	256	512	1024	2048	4096
“Acoustic Zeniterm” (Zenitherm)—cork grains pressed into porous hard bricks, thickness 1.14″, weight 2.06 lb/sq. ft.	0.03	0.13	0.33	0.42	0.42	0.15
“Acoustolith”—1/2″ plaster on lime cement, thickness 1/4″	0.21	0.24	0.29	0.33	0.37	0.42
“Acoustic asbestos” (Asbestos-Akustikos)—felt, a mixture of felt with asbestos, thickness 1/2″	0.10	0.18	0.36	0.60	0.63	0.57
Same, thickness 3/4″	0.18	0.30	0.54	0.64	0.68	0.57
Wool “Balsam”—fluffily felted wool fibers, pressed loosely, thickness 1″, weight 0.26 lb/sq. ft.	—	0.18	0.44	0.62	0.66	—

From what has been said it follows that the determination of the reverberation time has great practical significance for judging the acoustic properties of a given room as a whole; in detail, however, the course of reverberation cannot be predicted. Likewise, the methods described above for determining sound reflections can give an idea only of the first moments of the decay of sound, since on a model it is practically impossible to reproduce exactly the absorption conditions actually existing in a room. In order to imagine what the actual course of reverberation in a room is, it is necessary to place a sound receiver in the hall and make a recording of the decay of the sound; the same method, of course, may also be used to record the growth of sound, and in general to investigate all those sound phenomena that fall under the concept of equalization processes. Meyer (³⁴) carried out similar investigations with the aid of an electrical sound receiver and an oscillograph. He recorded the equalization processes of sound in rooms with different damping and for different distances between the source and the sound receiver. His investigations

Fig. 28.

concerned mainly the question of how the influence of the acoustic conditions of a room affects the reception of speech by an electrical sound receiver, and how this influence changes in connection with the distance between the sound source and the receiver. His results are presented in Fig. 28, where the curves \(a, b, c, d\) refer to a room with well-reflecting surfaces, and the curves \(a', b', c', d'\)—to a strongly damped room. At the top is shown the curve of the current in the loudspeaker, and at the bottom the curve of the microphone current; the latter curve therefore represents the curve of the change in pressure at the corresponding point of the sound field. In both rooms the measurements were made for two frequencies (500 and 1300 hertz) and for two distances between the microphone and the loudspeaker (curves \(a, a'\)—approximately 4–5 m; \(b\) and \(b'\)—approximately \(1/2\) m; curves \(c\) and \(c'\), as well as curves \(d\) and \(d'\), were taken at the same distances, but at the higher frequency). The photographs seem to indicate a certain discrepancy between the experimental data and the customary notion that the decay of sound in a room with good acoustics follows an exponential curve. However, the exponential law applies only to the total amount of sound energy in the room. The amplitudes of pressure at any given point of the room are composed of a whole series of separate oscillations arriving at that point partly from the sound source itself, partly from the reflecting walls, and all of them arrive with the most varied phases; this explains how it sometimes happens, for example in Fig. 28, that the pressure amplitude still increases when the sound source has already been switched off. From these photographs one can also approximately estimate the influence of the acoustic conditions of the room on the quality of sound transmission. As the sound source is moved away from the microphone, the influence of sound oscillations reflected from the walls begins to predominate (Fig. 28), and the new syllable reaching the microphone may, under certain circumstances, coincide with the strong echo of the preceding syllable; whereas at a small distance between the sound source and the sound receiver, the predominant importance belongs to the sound

waves directly incident on the microphone, so that no noticeable echo is felt.

Similar experiments on an objective investigation of the quality of speech transmission by means of a loudspeaker in a room with good acoustic conditions were carried out by Trendelenburg \((^{35})\) in Cologne Cathedral.

Fig. 29 gives the course of the curves of the processes of equalization of sound energy at a certain point in the middle part of the cathedral, approximately \(14\ \text{m}\) from the microphone; after about two seconds the pressure amplitude fell only by \(10\%\). Let us note that, despite the comparatively large

Fig. 29. Rise of sound and reverberation in Cologne Cathedral. Frequency 170 hertz.

distance from the loudspeaker to the observation point, here the directly incident sound ray still predominates,—the intelligibility of speech was still sufficient; in the drawings it is seen that during reverberation the pressure amplitude very rapidly falls approximately to \(40\%\) of the value of the stationary amplitude.

In these experiments as well, for high frequencies the absorption proved to be considerably greater. Fig. 30 shows the course of the curves of the processes of equalization of sound at the aforementioned point of the cathedral for a frequency above 3000 hertz.

We have already said that reception was carried out at those points where speech intelligibility was good. With greater distance from the loudspeaker the transmission rapidly deteriorates and speech is lost in the general din. The intelligibility of speech depends to a considerable degree on the speed of speech (utterance); if one speaks too quickly, then with each new syllable the pre-

…still has not had time to die away. This process is an even greater obstacle to the intelligibility of speech than the considerable residual dying-away sound, in comparison with the increasing sound of the new syllable arriving at the given moment.

Fig. 30. Rise of sound and reverberation in Cologne Cathedral. Frequency above 3000 hertz.

Sufficient intelligibility of speech can therefore be obtained only at those points at which the predominant value is the direct action of the sound source. This circumstance makes it necessary to divide rooms with sharply pronounced reverberation into a whole series of sections with separate loudspeakers.^1

Let us give one more example of measurements of the processes of sound equalization in a hall with very great reverberation (Fig. 31), carried out in such a place in the room where the acoustic effects of the room are strongly manifested. This concerns a recording made in one machine hall. The room was about 30 m long, and the receiver was placed at a distance of 20 m from the loudspeaker; the intelligibility of speech at this place was very poor, so that one could follow the reading only by straining greatly

Fig. 31. Rise of sound and reverberation in a machine hall. Frequency 500 hertz.

^1 Such a network of loudspeakers distributed throughout the room was used in 1925 at the opening of the Deutsches Museum (36).

attention. To determine sound absorption coefficients, E. Meyer (37) recently applied the recording of reverberation curves by an electrical method. In order to eliminate the interfering influence of interference, the sound field was excited not by a constant tone, but by a tone whose pitch constantly fluctuated slightly. The reverberation curves obtained in this way show an almost uniform decay along an exponential curve; from the magnitude of the decay one can find the sound absorption coefficient.

By recording reverberation curves first in an empty room and then after introducing sound-absorbing materials into it, one can determine, from the change in the course of the curve, the sound absorption coefficient of the materials introduced. The survey we have given of investigations in room acoustics indicates that, on the basis of theoretical and practical work in this field, we are able to predict in advance, in sufficient detail, the acoustic conditions of buildings being designed and to improve the spatial acoustics of existing buildings. ^1

Architect and acoustician, working in close contact, can achieve still more significant successes in this field; it must, however, be noted that the requirements set by acousticians and architects often contradict one another, and one has to choose not very satisfactory compromise solutions. An interesting study on the design of rooms in acoustic terms is represented

^1 In broadcasting studios it has often proved necessary to increase the influence of the room acoustics. If the entire surface of the hall—walls, floor, and ceiling—is covered with sound-absorbing materials, as was done in the first years of radio broadcasting, then, when receiving with headphones, in which the acoustic qualities of the room do not make themselves felt, the transmission turns out dull. If, for example, the rear wall of the studio is made of a strongly reflecting material and the hall is sized so that the echo reflected from the rear wall reaches the microphone only after the appropriate intervals of time, then in transmission a far fuller sound can be achieved (38). By known technical means one can also create artificial reverberation; we shall return to this question below, in the section on sound radiators (the principle of the ultraphone) (39).

of F. Osswald’s work (40), “On the problem of the acoustics of the large hall for public assemblies in the League building in Geneva” (41). This work examines in detail what acoustic qualities may be expected from each of nearly 20 designs for the large hall of the League Assembly, and also offers a new design for the shape of the hall with optimal acoustic qualities.

§ 3. Sound emitters

Both the theoretical study and the construction of sound emitters are based on entirely different principles, depending on the purpose for which these sound emitters are intended: whether for operation over a broad frequency range or for operation in only one narrow region of tones. For the second group of devices, the output of the instrument is the primary concern; for the first group, alongside this, the requirement is advanced that the instrument act uniformly over the entire range of sounds of interest; in other words, the transmission must provide an accurate reproduction of sounds. We shall first turn to the consideration of sound generators suitable for transmitting a broad frequency range. We shall add here several remarks on the new “tuned” emitters (“Ton”-sender); at the end of the chapter the question of musical instruments and the sounds they produce will be touched upon; in this field our knowledge has expanded considerably in recent times.

The principal requirement that must be met for the faithful transmission of a broad frequency range is that, within the transmitted range, the reproducing apparatus should not possess its own weakly damped oscillations, whether electrical, mechanical, or acoustic; if such natural oscillations are present, then the corresponding region of tones is transmitted with an excessively large amplitude. In older models of sound emitters—almost without exception—there were such oscillating systems, chiefly mechanical and acoustic. In the field of mechanics, difficulties arise

whenever, for the conversion of mechanical oscillations into acoustical ones, the intervening link is a membrane fixed at its edges. The possibility of numerous natural oscillations of a membrane fixed at its edges is well known,—and increasing the damping of these oscillations can help matters only to a certain extent. In order to obtain sufficiently strong sound radiation, especially at low frequencies, a horn has been used; but the horn, in its turn, introduces the further possibility of the appearance of natural acoustical oscillations of the system,—although, by a successful choice of the shape of the horn, e.g. with a horn curved along an exponential curve, these oscillations can almost be eliminated (41). To show to what extent one may count on approximately uniform transmission in an ordinary electromagnetic loudspeaker with a horn, we give Fig. 32, which presents the results of measurements by Meyer (42), made by him with the aid of a Rayleigh disk. It is evident from the drawing that this type of loudspeaker can reproduce about two octaves with sufficient accuracy.

Fig. 32. Frequency characteristics of an electromagnetic loudspeaker.

It fell to the late Rieger first to clarify, on the basis of broad theoretical considerations, by what means accurate transmission can be achieved for the entire practically important range of sounds. In his work on the theory of loudspeakers, completed as early as the beginning of 1924,${}^{43}$ he laid the foundations for this whole complex of questions. One of the types of loudspeakers constructed by him already at that time is still often used in practice. The necessity of combating distortions dependent on resonance leads to the requirement that, for transmission, only such systems be used for which the period of the natural oscillations lies below or above the transmitted frequency range. We shall see that, for acoustic reasons, the question can concern only systems tuned sufficiently low. To avoid the acoustic resonances of the horn, it was completely eliminated, and instead, in order to obtain sufficient strength of action, a piston diaphragm was used, excited practically uniformly over its entire surface. The piston diaphragm itself is placed in an opening of a fixed wall whose dimensions are sufficiently large in comparison with the wavelengths of the transmitted range of sounds. Rayleigh${}^{44}$ gave the theory for calculating the sound field of a piston diaphragm. According to Eigner,${}^{45}$ the radiated acoustic power is equal to

\[ L=\frac{\rho\cdot\pi\cdot R^{4}\omega^{4}x_{0}^{2}}{4c}, \]

where \(\rho\) denotes the density of air, \(R\) the radius of the diaphragm, \(x_{0}\) the amplitude of the diaphragm oscillations, and \(c\) the speed of sound. This formula is applicable insofar as the wavelength \(\lambda\) is large in comparison with \(R\).

It follows from the formula that the sound power increases in proportion to the fourth power of the frequency; consequently, to ensure accurate transmission of sounds, the amplitude of the diaphragm must be reduced in proportion to the square of the frequency. This condition is fulfilled in systems whose natural period of oscillation lies below the transmitted range of oscillations. Rieger${}^{46}$ carried out the calculation of the resis-

radiation pressure also for frequencies lying outside the limits of applicability of formula (23), i.e., also for those wavelengths that cannot be considered sufficiently large in comparison with the diameter of the membrane; it turns out that the radiation resistance increases too slowly to compensate for the decrease in the amplitude of the membrane’s oscillations. Then a decrease in output is obtained, which at first, however, is not too pronounced, because at shorter wavelengths the directional action of the membrane begins to manifest itself, so that in the part of the room close to the mean normal to the membrane the transmission continues to remain almost correct, and only farther away, to the sides, do the higher tones begin to weaken. We shall not touch here on the question of the directional action of the membrane, since we considered these problems in detail in Chapter 2.

Fig. 33. Cross-section of a Blatthaller.

To excite the oscillations of the membrane, Riegger applied the electrodynamic principle. To the membrane of the Blatthaller (Fig. 33) a copper strip is firmly attached along a Meander curve (in zigzag fashion). The telephone current being transmitted flows along this copper strip; the strip is situated in a correspondingly arranged magnetic field. The interaction of the current-carrying conductor and the magnets excites oscillations of the membrane. We cannot here analyze in detail the technical construction of this apparatus and refer the reader to other articles (47). In the present work we shall confine ourselves to showing to what extent the Blatthaller makes it possible to achieve faithful transmission. In Fig. 34 are presented the frequency characteristics of various models of the Blatthaller, obtained with the aid of Rayleigh’s disk.

The Rice–Kellogg loudspeaker \((^{48})\), reports of which arrived from America a year after the publication of Riegger’s principal work, has a radiating

Fig. 34. Frequency characteristics of various models of Blattthaller loudspeakers.

surface, operating approximately on the principle of a piston membrane. The sound-radiating surface of the Rice–Kellogg loudspeaker is made in the form of a cone with

Fig. 35. Frequency characteristics of loudspeakers: electrostatic (No. 1) and electrodynamic (No. 2).

a truncated end, on which sits a coil encompassed by the field of a pot-shaped magnet. The cone, made of paper, at low frequencies vibrates with its entire surface; at higher frequencies the cone ceases to be rigid and the amplitude of its vibrations toward

toward the edges diminishes. A partial characteristic \(^{(49)}\) of a loudspeaker operating according to this principle is presented in Fig. 35 (curve 2). In the same figure (curve 1) the characteristic is given of a loudspeaker with electrostatic excitation, in which the membrane is made of rubber.

Let us also mention further the loudspeaker with a folded membrane (Faltenlautsprecher), on which Gerlach \(^{(50)}\) reported in 1926 at the annual meeting of the Union of Electrical Engineers. The layout of this loudspeaker is shown in Figs. 36 and 37. The radiating organ of the loudspeaker is a certain surface: a sheet of pertinax folded into pleats. This surface has a very low transition of natural vibrations. Excitation is obtained either electrodynamically, by means of a copper conductor fastened to a fold, or, in another model (the “Protos” loudspeaker), also electromagnetically. An idea of the range of frequencies covered is given by Fig. 38, obtained by Gerlach with the aid of the automatic recording method developed by him \(^{(51)}\); a ribbon microphone was used as the receiver. Let us emphasize that, unlike the other characteristics, on this curve the amplitudes are plotted in

Fig. 36. Diagram of a loudspeaker with a folded membrane.

Fig. 37. Diagram of an electrodynamic loudspeaker with a folded membrane.

on a logarithmic scale, so that direct comparison of the curves is impossible.

We have discussed here only those works whose aim was the systematic determination of the acoustic properties of loudspeakers. We shall not go into an exposition of the numerous technical achievements in this field (52). It seems appropriate to us, however, to present a number of curves obtained by Grotzmacher and Meyer (53) in investigating the frequency range of head telephones. In this case the receiver was a condenser microphone, which was connected to the telephone under test by means of an artificial ear.

Fig. 38. Frequency characteristic of the “Protos” loudspeaker.

Fig. 39. Frequency characteristic of an electromagnetic telephone.

The artificial ear in this case consisted of a small cavity surrounded by brass. We note that this cavity, unlike the human ear, does not absorb sound, and therefore the resonance points are characterized by less damping than is the case in practice.

In Fig. 39 is given the frequency characteristic of a large low-resistance electromagnetic telephone, and in Fig. 40—the characteristic of an electrostatic telephone, in which a rubber membrane covered with carbon dust vibrates in front of a grid electrode.

We have so far examined the question of faithful sound transmission only from one point of view: how large is the frequency region transmitted by the loudspeaker? Now we must examine the question from another, also very essential, point of view, namely from the point of view of the linearity of transmission. What substantial errors in transmission are caused by deviations from linearity can be judged from the following considerations. Suppose that the force applied to the sound-radiating system has two components with angular frequencies \(\omega_1\) and \(\omega_2\). Thus let

\[ k = k_1 \sin \omega_1 t + k_2 \sin \omega_2 t . \tag{24} \]

If there are deviations from linearity, then in the forced oscillations, alongside the original frequencies \(\omega_1\) and \(\omega_2\), new frequencies \(\omega_k\) will appear (combination tones), which are formed according to the law \(\omega_k = m\omega_1 \pm n\omega_2\), where \(m\) and \(n\) take the values of a sequence of consecutive integers. It is clear that the formation of such new tones, which were entirely absent in the radiated sound, leads to distortions. Suppose, for example, that the sound radiator must transmit the vowel \(a\) (the principal formants are 800 and 3000 hertz); during transmission this \(a\) will pass into \(e\) (3000 − 800 = 2200, i.e. a tone lying in the region of the formant of \(e\)).

Fig. 40. Frequency characteristic of an electrostatic telephone.

With respect to the linearity of transmission, systems excited electrodynamically possess great advantages, manifested chiefly when it is required to obtain a large radiated power. Thus, for example, a Blatthaller membrane can execute oscillations with a very large amplitude, and nevertheless the conductors do not leave the field and the linearity is not disturbed; in other cases as well, the assumed linearity of transmission of precisely this type of loudspeaker has been fully confirmed.

We shall now briefly report on measurements of transmission nonlinearity recently carried out by E. Meyer (54). His measure of distortions depending on the nonlinearity of transmission is the ratio of the effective pressure amplitude for overtones to the effective pressure amplitude for the fundamental tone. Kopfmüller (55) proposed calling this ratio the chatter factor (Klirrfaktor). To measure the chatter factor, Meyer excites the radiator under test by means of a sinusoidal tone, while measuring the pressure distribution in the sound field by the compensation method with the aid of a condenser microphone.

Table 4 gives the chatter factors of two electromagnetic diaphragm loudspeakers with a permanent magnet, as a function of frequency. One of the loudspeakers (a) has an ordinary tin horn; in the other, the iron diaphragm is rigidly connected to a conical paper diaphragm.

Table 4

Hertz:	160	200	250	300	400	500	700	1000	1400
a	2.8	1.2	0.6	0.05	0.1	0.02	0.02	0.005	0.005
b	0.3	0.6		0.5	0.3		0.1	0.04	0.04

Both loudspeakers were excited by a current of 2 mA. The chatter factor is strikingly large for low frequencies. This circumstance is due to the fact that, in this type of loudspeaker, low frequencies are radiated poorly; therefore the low fundamental tones are drowned out by the overtones that are transmitted better.

Table 5 gives the dependence of the chatter factor on the magnitude of the exciting current; the measurements for loudspeaker a were made at a frequency of 250 hertz, and for loudspeaker b at approximately 300 hertz.

Table 5

mA:	0.5	1	2	4
a	0.1	0.2	0.3	0.7
b	0.1	0.3	0.5	0.3

From the table it is evident that the tremble factor increases strongly with increasing excitation. E. Meyer indicates the following possible causes of these nonlinear distortions:

1. The amplitude of the membrane ceases to be sufficiently small in comparison with the distance between the pole of the magnet and the membrane.

2. The alternating magnetic flux in the membrane and the iron core ceases to be small in comparison with the constant magnetic flux.

3. The magnetization curve is not rectilinear.

Which of the possibilities listed is capable of exerting the greatest influence, we shall not examine here.

E. Meyer also investigated another type of electromagnetic loudspeaker, in which excitation is produced by a system operating like a polarized relay; the armature, on which an alternating-current winding is also placed, is located midway between the poles of a permanent magnet. Such an arrangement is distinguished by better linearity. Table 6 gives the values of the tremble factor of a loudspeaker constructed according to this type.

Table 6

Hertz:	160	300	600	800	1400
	0.1	0.1	0.03	0.02	0.01

Loudspeakers with electrostatic excitation also possess a fairly considerable tremble factor, due in part to the fact that here the magnitude of the exciting alternating voltage cannot be neglected in comparison with the constant voltage. On the other hand, the tremble factor is also large because the amplitude of the membrane’s oscillations in this case is not sufficiently small in comparison with the distance between the membrane and the electrode situated opposite it. Precisely this latter circumstance is difficult to eliminate, since with an increase in this distance the output rapidly falls. Table 7 gives the tremble factors for an electrostatic

loudspeaker with a free \((a)\) and with a taut membrane \((b)\).

Table 7

Hertz:	200	400	800
\(a\)	0.4	0.2	0.02
\(b\)	0.3	0.1

With respect to the flutter factor, the electrodynamic excitation proved to be the best method. Thus, for example, measurements showed that the flutter factor of the Riegger flat-panel loudspeaker, even for low frequencies, remains below 0.01.

For a brief survey of the latest development of the question of loudspeakers, we may content ourselves with the material presented above. These data make it possible, in particular, to form an idea of the extent to which accurate sound transmission is now accessible to us, and where the sources of error lie.

Let us also point to two works \((^{56})\) that are of significance for the question of the mechanical and acoustic properties of telephones. E. Wetzmann and K. Schuster investigated coupled vibrations of continuous systems (kontinuierliche Teilsysteme). The experiments were carried out in rooms, the air vibrations being excited by telephone membranes. The theory of coupled vibrations of telephone membranes and air spaces was given by Schuster. This problem can be solved to any desired degree of accuracy by means of Ritz’s method. The paper also gives a method for an approximate solution of the problem.

We shall now turn to studies of the recording and reproduction of sounds by means of speaking machines. In the interest of a more coherent presentation of the works that belong here, we shall already have to touch upon a number of questions which properly belong to the following chapter—on sound receivers. Such, first of all, is the question of modern improved methods of applying sound recording to discs.

Until recently, in the specialized literature and in textbooks one could find only very scanty and, for the most part, only the most general information about speaking apparatuses—a fact explained by the circumstance that the firms manufacturing the apparatuses tried in every way to keep the particulars of their designs secret; a broader treatment of the problems and methods of operation of speaking machines has begun to spread only very recently.

The phonograph invented by Edison in 1877 had, as its sound receiver, a membrane (for the most part of mica, wood, or glass) provided with a point (usually of sapphire). The vibrations of the membrane are recorded by the point on a wax cylinder in the form of a groove, the depth of which changes in accordance with changes in the amplitude of the vibration. The motion of the point here is perpendicular to the surface on which the recording is made. For reproducing sounds there is a similarly constructed membrane; its needle runs along the groove, in turn setting in motion the transmitting membrane. The phonograph was soon displaced by the gramophone, the description of which was given in 1898 by E. Berliner. The advantages of the gramophone lie in the fact that here greater sound intensity and more accurate transmission of it are achieved. The gramophone is arranged in such a way that the recording point moves not perpendicular to the surface on which the recording is made, but parallel to it, and the recording is received not by a cylinder but by a horizontal disc. Between the membrane and the point, as well as between the needle and the reproducing membrane, there is a lever transmission, by means of which a more considerable effect is achieved than in the simple phonograph.

In describing the individual parts of the gramophone, we shall first of all briefly acquaint ourselves with that part of it which is essentially the same in all its systems and the type of which has in practice been fully worked out: we are speaking of the gramophone record.

Data on the properties of gramophone records, and in particular on the recording curves obtained on them, may be found in the work of Hermershausen (⁵⁷). He indicates,

that the width of the groove is approximately 0.06 mm, and the average distance between two grooves about 0.24 mm. If the radius of the disk is 300 mm, then it is possible (leaving free from recording the central part of the disk, 120 mm in diameter) to cut on the disk a groove approximately 240 m long. It goes without saying that for the needle of the reproducing apparatus it is necessary to choose a material of especially high quality, distinguished by the greatest durability, since it must traverse this long path while wearing down as little as possible.

For the amplitude of the curve there remains, as is clear from the above, only about 0.1 mm. We shall now briefly examine how the recording should be made in order to make the best use of the available space[^58]. Suppose that the initial sound field is produced by an acoustic process of the form \(p_1\sin\omega t\). This acoustic process, when recorded on the disk, gives the curve \(x=x_1\sin\omega t\), and we are now concerned above all with the question: what dependence on the frequency of the oscillations must we establish for \(x\), in order to make the best use of the disk; and when this dependence has been established, then in constructing the reproducing apparatus we shall have to adhere strictly to it, since otherwise we shall not be able to obtain an accurate reproduction of the frequencies of the original acoustic process.

At first glance it seems most advantageous to make the recording in such a way that, when the frequency \(\omega\) changes, the amplitude of the curve cut on the disk remains constant; or, in other words, so that the ratio of the pressure in the sound field to the amplitude of the curve does not depend on the frequency. Such a method of recording would, to be sure, make it possible to use well the space between two neighboring curves, but it would lead to a considerable inconvenience. As the frequency increases, the length of the individual waves on the disk decreases, and thus the radius of curvature of the sinusoidal waves at the turning points becomes smaller and smaller, and finally becomes so small that the recording, and especially the subsequent reproduction of such a curve for high-

often encounters insurmountable difficulties. Therefore the recording has to be made so that equal pressures in the sound field correspond not to equal amplitudes, but to equal velocities. Consequently, it is required that

\[ x=\frac{P}{\omega}. \tag{25} \]

For very low frequencies (approximately below 200 hertz) deviations from the requirement (25) have to be allowed; if, for the acoustically most important regions, we wish to attain the most faithful possible transmission, then the amplitude of the curve cannot be taken too small (in absolute magnitude), otherwise the noise of the moving needle is very disturbing. Under these conditions, however, for the lowest—acoustically less important—frequencies the curve would already take hold of the neighboring groove. Therefore, for such lowest frequencies one has to accept a compromise and work already with constant amplitude. On the other hand, for the highest frequencies (above 5000 hertz) it becomes necessary to take not constant velocity, but constant acceleration, in order to avoid too small a value of the radius of curvature. These compromises, forced by the conditions, lead to distortions of sounds in the very lowest and very highest registers.

Let us now turn to the methods of applying the curve to the disc. Here we may leave aside purely mechanical methods of recording, although until quite recently they were generally accepted in the technique of talking machines: now they have been completely displaced by the electro-mechanical methods that have developed in recent times. The disadvantages of the purely mechanical method of recording were the unavoidable distortions of sounds, connected chiefly with the resonance of the diaphragm and horn, and also the low sensitivity of the apparatus, which could not be increased to a sufficient degree. The low sensitivity of the instruments led, for example, to the fact that, when recording orchestral pieces, it was necessary to arrange the instruments in the orchestra in an entirely artificial way, and this of course could not fail to be felt as a great inconvenience both by the persons conducting the recording and by the arti-

impressions, and at the same time the very impression from the music was disrupted.

A major step forward was the use of electrical sound receivers. In the best electrical receivers the threshold of sensitivity is not perceptible. They possess almost uniform sensitivity over the entire acoustically important range of frequencies and reproduce amplitude relations to a high degree faithfully. The transmission of music acquires greater naturalness, since here there is no need to depart from the usual placement of instruments and performers (59). The transmission of orchestral and choral numbers benefits most of all in this respect. An electrical sound receiver can be placed at a comparatively great distance from the source of sound, whereby the picture obtained is considerably more uniform.

We shall return later, in the chapter on sound receivers, to the design of electrical sound receivers and to the quality of the transmission obtained with their aid; for the present we shall take up questions connected with obtaining a sound recording on a disc by means of a sound receiver operating into an amplifier. In doing so we shall assume that the transmission remains faithful up to the output tube of the amplifier, and consequently that the voltage on the grid of the output tube of the amplifier always remains proportional to the amplitude of the sound-field pressure, and, in particular, that this ratio does not depend on frequency.

Maxfield and Harrison (60) gave a detailed analysis of the electromagnetic apparatus for recording gramophone curves, which we shall briefly describe.

The apparatus is shown in Fig. 41. The writing stylus is set in motion by means of the armature of a polarized relay. To judge the operation of this instrument, Maxfield and Harrison proceeded from considerations of analogy between such a mechanical system and a correspondingly constructed electrical filter. Already in the first part of our survey we pointed out the analogy between acoustic and electrical oscillatory circuits; in particular, we showed that to acoustic filters one may apply—

to apply the same considerations, which have been developed in detail for electrical filters. Similar considerations can also be applied to mechanical systems capable of oscillation. Such a complex device as the electromagnetic apparatus for gramophone

Fig. 41. Electromagnetic apparatus for gramophone recording.

recording shown above may be regarded as a system of coupled oscillatory circuits and, accordingly, replaced by a suitable circuit.

On the basis of considerations similar to those to which we resorted in constructing the theory of acoustic filters in Chapter I, the following comparative

... diagram of the corresponding mechanical and electrical quantities.

Mechanical quantities	Electrical quantities
Force $f$ (dyne)	Voltage $e$ (volt)
Velocity $v$ (cm/sec)	Current $i$ (ampere)
Displacement $s$ (cm)	Charge $q$ (coulomb)
Impedance $z$ (dyne·sec/cm) (For dyne·sec/cm the expression “mechanical ohm” is also used)	Impedance $z$ (ohm)
Mechanical capacitance1 $c$ (cm/dyne)	Capacitance $C$ (farad)
Mass $m$ (gram)	Self-inductance $L$ (henry)

Fig. 42. Electrical analogy to Fig. 41.

The electrical circuit given by Maxfield and Harrison for interpreting the apparatus shown in Fig. 41 is presented in Fig. 42. In accordance with the foregoing, the apparatus must satisfy the requirement that, in acoustically important ranges of sounds, the amplitude of the velocity of the recording stylus be proportional to the amplitude of the voltage on the grid of the last tube—and, consequently, to the amplitude of the pressure—and, in particular, that the ratio remain independent of frequency. We must therefore consider whether an electrical filter constructed in this way has a uniform

passband over the entire acoustically important range of sounds. It can be shown that, by tuning individual parts of the system and by selecting a suitable damping, filters with a passband for a sufficiently wide range of frequencies can be obtained. The question of damping is especially important here. In the apparatus for obtaining a sound recording there is no damping of the vibrations due to radiation (Strahlungsdämpfung), whereas in the sound-reproducing apparatus, which will be considered below, it is present. It therefore proved necessary to create artificial damping in the first apparatus. Such damping is produced by a special device made of rubber.

Fig. 43. Frequency response of the apparatus for gramophone recording.

We wish further to show, by giving the corresponding calibration curve, to what extent the apparatus shown in Fig. 41 satisfies the requirements of accuracy in the transmission of sounds. In Fig. 43 a curve is given, obtained by Maxfield and Harrison; let us note that the ordinates here are taken on a logarithmic scale and that in reality the fall of the curve on passing to low and high frequencies should be much stronger than it appears here. Nevertheless, the transmission in the most important frequency range, approximately from 300 to 5000 hertz, may be regarded as very uniform. Unfortunately, from the work we are considering it is impossible to discern by exactly what method this frequency characteristic was obtained.

Fig. 44. Reproducing mechanism.

In a completely similar manner, Maxfield and Harrison studied reproducing apparatus. The scheme of the sound receiver—the needle with the transmission lever and horn—

can be seen in Fig. 44; as a horn, in order to save space, a folded funnel bent along an exponential curve was used. The electrical “equivalent circuit” is given in Fig. 45, and the calibration curve—again on a logarithmic scale—in Fig. 46.

Fig. 45. Electrical analogy to Fig. 44.

It should also be mentioned that for sound reproduction, especially in cases where high acoustic power is required, it is advantageous to use electrical devices for sound reproduction (Tonabnehmer) connected to a loudspeaker. This makes it possible, with a proper choice of loudspeaker, to bring out the very lowest frequencies better and, even at high sound intensity, to preserve sufficient linearity of transmission.

Fig. 46. Frequency characteristic of the reproducing mechanism.

Kellogg\(^{(61)}\) describes the construction of an electrical sound receiver. He points out that very much depends on the transmission lever, while of small weight, being sufficiently rigid. Kellogg also gives a survey of the designs of the magnetic systems of such apparatus.

In connection with the question of gramophones, we shall also briefly indicate here the latest improvements in photographic recording of sounds—an area of work having special importance for talking films. On the so-called “Tri-

“Ergon” is discussed in his book by Engle (62). In this same field Rankin (63) worked. It would, however, be inappropriate to enter here into a detailed consideration of these questions, since the problems pertaining to them concern photographic optics more than acoustics or electroacoustics. The acoustic problems that arise in connection with talking films pertain to the question of the perception of sounds and their transmission, which has already been discussed above.

In passing, let us touch here also on the so-called “Ultraphone” principle (64). If two sound pickups are placed on one and the same groove of a gramophone record so that one needle precedes the other by approximately \(1/10\) sec., then subjectively the loudness is considerably increased, provided that each pickup has its own horn. It must be noted that this gain in sonority is purely apparent; the auditory illusion is due to a strongly expressed stereo-acoustic effect. If, for example, speech or singing is being transmitted, then immediately after each syllable the ear at once perceives a rapidly following echo; this creates the illusion of greater fullness of sound.

Up to now we have dealt with apparatus for which the requirement of faithful sound transmission over a wide frequency range is brought to the fore. Let us now turn to works devoted to sound radiators tuned to a tone of definite frequency. Purely acoustic work on improving tuned sound radiators (“Tonsender”) had for the most part already been carried out earlier; by this I mean first of all tuning forks, which play such an outstanding role in acoustic measurements as frequency standards, sirens (65), whistles, etc. Let us mention here also the electrodynamic radiator of Feessenden and the electromagnetic radiator of the Submarine Signaling Company. In one special area, lying, strictly speaking, outside the field of acoustics, very rapid development has been achieved precisely in recent times. This is the area of ultrasonic radiators. The impetus for these works was given chiefly by practical considerations. It turned out that with ultrasonic oscillations it is comparatively easy to obtain

directional action, whereby a higher concentration of energy is attained in the sound receiver. For problems of transmitting sound under water (to mark the entrance to a harbor, for example, etc.) a directed ultrasonic beam proves very suitable.

For the excitation of ultrasonic waves, piezoelectric crystals are usually used—the method indicated by Langevin (66) for underwater telegraphy employing ultrasonic oscillations. As is known, a piezoelectric crystal can easily be brought into mechanical oscillations by the action of an alternating electric voltage. Piezo-crystals are successfully used, for example, as frequency standards (67) in the field of radiotelegraphy. A piezoelectric underwater sound emitter was constructed by Langevin. Since sufficiently large quartz crystals are not easy to find, Langevin made the emitter out of small pieces assembled in the form of a mosaic, the thickness of an individual piece barely reaching 2 mm.

Fig. 47. Underwater sound emitter.

In Fig. 47 a cross section of the emitter is given: \(a\)—the mosaic gasket, \(b\) and \(c\)—steel plates, about 3 cm thick, to which the quartz is glued. An alternating voltage is applied to the steel plates, setting the apparatus in action. The natural frequency of such an oscillatory circuit lies in the region of 40,000 hertz. The technical applications of this ultrasonic emitter, such as, for example, measuring sea depths by the echo method, lie outside the scope of our survey. We refer the reader on these questions to other works (68).

Piezo-quartz was also successfully used for producing high-frequency oscillations in air. Thus Pierce (69) applied similar emitters for measuring the speed of sound in air at the very highest frequencies, up to

up to \(1.5 \cdot 10^6\) cycles. We shall also mention here the investigations of Wood and Loomis \((^{70})\) on piezoquartz immersed in oil; in these investigations the quartz was subjected to the action of very powerful sources of electrical energy, and the experiments performed revealed new and peculiar phenomena. Wood and Loomis excited the piezoquartz with an electron tube of 2 kW power; the voltage on the crystal was about 50,000 V. Quartz plates from 7 to 14 mm thick were used in the apparatus, making it possible to work in the region from 100,000 to 700,000 cycles. The sound emitter was at the bottom of a glass vessel filled with oil. Under the action of the sound pressure, on the surface of the liquid above the emitter there appeared a mound of oil up to 7 cm high. Drops of oil were thrown upward to distances of up to 40 cm. The pressure on a glass plate 8 cm in diameter applied to the surface was so great that it could be loaded with a weight of 150 g. The energy of the oscillations produced quite unusual effects in the vessel with oil; thus, for example, interesting biological effects were observed: small fish and frogs exposed to the action of the sound waves were killed after several minutes.

We consider it impossible to conclude the survey of works on sound emitters without touching, at least briefly, on yet another group of them which, by its diversity, would deserve special interest; however, precisely because of this diversity it has hitherto remained little accessible to systematic study from the physical point of view; this is the field of musical instruments. Since the time when, through the works of Helmholtz and Lord Rayleigh, classical acoustics was created, during the long period that has elapsed from then to our days only a small number of systematic investigations in the physical acoustics of musical instruments have been carried out, and only quite recently has the detailed experimental and theoretical study of this type of sound emitter begun. Here, first of all, one must name the Indian scholar Raman \((^{71})\), who, together with his collaborators, has gathered an exceptionally—

valuable material on musical instruments. Raman dealt especially thoroughly with the theory of bowed and struck string instruments. He carried out extensive theoretical investigations of the motion of strings. The dependence of a string’s vibrations on the pressure and speed of the bow or of the striking hammer was studied. Any detailed exposition of these questions would require too great a digression into particular problems of the theory for our survey, and we may all the more readily refrain from it here since Raman himself gave a brief survey of these questions. Other investigations were devoted to the study of the character of the sounds of musical instruments; the task was to find a possibility, from the quality of the sound, of drawing conclusions about the peculiarities of the operation of the instruments themselves. Familiarity with the properties of the sounds of musical instruments is at the same time of direct practical significance—let us recall, if only, the problems of accurate transmission of sounds, for which knowledge of their qualities is highly important. In view of this, we shall allow ourselves here to dwell on works precisely in this field.

Stumpf, whose extremely important works on the investigation of conversational speech we have already had occasion to discuss earlier (72), subsequently gave, as an appendix to his book on the sounds of speech (73), an exposition of the question of the sounds of musical instruments.

We noted earlier that the fundamental question has still not yet been definitively resolved—whether the sounds of musical instruments are determined only by the relative distribution of overtones, or whether they are characterized by the absolute position of certain and constant frequency regions in the sound spectrum (similar to the formants of conversational speech).

According to Stumpf’s investigations, alongside mobile formants there are indeed also constant formants here; and it must be said that, from the point of view of physics, it is quite natural to expect, for oscillatory systems having a definite spatial configuration, certain regions of predominantly prevailingныя фреквенцис.

tones or resonance bands. Stumpf gives the following table of characteristic pitch regions for various instruments:

Instrument	Fundamental tone	Approximate constant maxima	Movable principal formant
Trombone	\(c\)	\(c^{1}—e^{3}\)	\(c^{2}—c^{3}\)
Trombone	\(c^{1}\)	\(c^{1}—e^{3}\)	\(c^{2}—c^{3}\)
Trumpet B	\(c^{1}\)	\(c^{2}—c^{3}\)	\(g^{2}—g^{3}\)
Trumpet B	\(c^{2}\)	\(c^{2}—c^{3}\)	\(g^{3}—c^{4}\)
Trumpet B	\(cis\)	\(c^{2}—c^{3}\)	\(cis^{2}—cis^{3}\)
Clarinet A	\(cis^{1}\)	\(cis^{4}—gis^{4}\)	\(gis^{2}(cis^{3})—gis^{3}(cis^{4})\)
Clarinet A	\(fis^{1}\)	\(cis^{4}—gis^{4}\)	\(fis^{3}—cis^{4}\)
Clarinet A	\(fis^{2}\)	\(cis^{4}—gis^{4}\)	\(fis^{4}—cis^{5}\)
Clarinet A	\(c^{1}\)		\(e^{3}—b^{3}\)
Clarinet B	\(c^{2}\)	\(d^{4}—b^{4}\)	\(c^{4}—d^{5}\)
Clarinet B	\(c^{3}\)	\(d^{4}—b^{4}\)	\(c^{5}\)
Contrabassoon	\(c\)		\(g^{1}—c^{2}\)
Contrabassoon	\(g\)		\(d^{1}—g^{2}\)
Bassoon	\(c\)	\(c^{2}—g^{2},\ c^{4}—e^{4}\)	\(c^{2}—g^{2}\)
Bassoon	\(c_{1}\)	\(c^{2}—g^{2},\ c^{4}—e^{4}\)	\(c^{3}—g^{4}\)

From the study of the properties of the sounds of musical instruments one may expect—especially at the present time—valuable results in the sense of deepening our acquaintance with the properties and the physical aspect of the operation of the instruments themselves; thanks to broad systematic work on improving the recording of sounds, it is now possible to record and analyze sounds with great accuracy, sufficient for the most detailed investigation. Such investigations, carried out thus far, it is true, only with respect to one instrument—the principal instrument of the modern orchestra, the violin—were reported by Backhaus \((^{74})\) in 1927.

As the sound receiver in these experiments, Richter’s high-frequency microphone was used; the microphone acted through a resistance amplifier upon the loop of an oscillograph tuned to a high pitch.

Let us note first of all that one very peculiar observation, made earlier by researchers who work-

together with simpler apparatus, was again confirmed: it again turned out that for very low notes of the violin the fundamental tone is only slightly prominent (15). Thus, some analyses show that the amplitude of the fundamental tone ($g$, 192 hertz) amounts to only 3% in one case, and 32% in another, if we arbitrarily take the amplitude of the strongest overtone (in our case the second) as 100%. Bactaus seeks the explanation of this fact in the peculiarities

Fig. 48. Powers of spherical and sectorial sound radiators.

of the sound radiation of the body of the violin: namely, if one calculates, by Rayleigh’s formula, the power of spherical waves, then the relations shown in Fig. 48 are obtained. The abscissae in this diagram are the quantities $k \cdot r$, where $r$ is the radius of the sphere, and $k=\frac{\pi}{\lambda}$. The ordinates are proportional to the power referred to a constant amplitude of velocity of the radiating surface. One of the curves represents the function

\[ \frac{k^2 r^2}{1+k^2 r^2}, \tag{26} \]

which gives the expression for the power of spherical radia-

of order zero. The other curve corresponds to the function

\[ \frac{k^{6} r^{6}}{k^{6} r^{6} - 2 k^{4} r^{4} + 9 k^{2} r^{2} + 81}. \tag{27} \]

This expression gives the power of a sectorial radiator of the second order. On the basis of Seiffert’s investigations (76), the violin can apparently be equated to a sectorial radiator of the second order. If now, in Fig. 49, we follow the course of the curve for the power of vibrations of a radiator of the second order with increasing value of \(k \cdot r\), i.e., in other words, with increasing frequency, we shall see that the power, at a certain value of \(k \cdot r\) (approximately equal to 1.5), begins to grow rapidly; the higher frequencies are radiated much more strongly, while the low ones are very weak. If the radius of the sphere, in connection with the dimensions of the violin, is taken as equal to 18 cm and, under these conditions, \(kr\) is calculated for the frequency \((g)\) 192 cycles, which in the records obtained corresponded to the fundamental tone, then we obtain \(kr : 0.67\)—a value which, judging from our curve, indicates very weak radiation. Let us also note here that the physical, objectively very considerable weakening of the fundamental tone is subjectively almost not perceived, since the physically weak fundamental tone reappears for the ear owing to the formation of combination tones (77).

Also of interest are the records obtained for strings of different materials at the same pitch. The differences in the structure of the photographs obtained must, of course, be attributed chiefly to the difference in the magnitude of friction between the bow and one or another string. The friction and pressure of the bow, according to Raman’s investigations (78), affect the form of vibration of the string itself. Alongside such a form of the string’s vibration curve, in which in each period there is only one bend (a sawtooth-shaped curve, the Helmholtz type of vibration or first-order type of vibration), there are also forms of vibration with several breaks; their order is designated by the number of bends. On the curve of each po-

partial vibrations of the corresponding order prevail. One of the analyses gives a segment of a recording of the complete motion of the bridge of a free metal string of a Stradivarius violin (tone \(e_2\)). Here one can see that the form of the vibrations corresponds to the first type only for a short time, and later passes over to a type of higher order. One may further expect interesting material from the investigations now being undertaken, concerning simultaneously both the vibrations of the string and the sound it emits.

Fig. 49. Resonance regions of various violins.

From numerous photographs of various sounds of violins of different types, conclusions were drawn concerning the position of the resonance regions for violins. Fig. 49 shows the qualitative results of these investigations. We see that the existence of a strong resonance between 3000–4000 hertz was established; the resonance region lay the higher and stood out the more sharply, the “better” the instrument tested was. Backhaus also took up the question of what accounts for the difference between good and bad violins. This question had earlier been studied by Hewlett (\(^{73}\)), who used for his investigations Rayleigh’s disk and resonators. Hewlett came to the conclusion that a violin seems to be the better, the more energy is concentrated

in the fundamental tone as compared with the overtones. Backhaus was able to verify this on excellent material. Among the violins selected for study were ten old Italian instruments, each distinguished by high qualities and universally acknowledged exceptional beauty of sound. At the same time, Hewlett’s assertion was not confirmed. On the contrary, photographs obtained from a Carlo Bergonzi violin made in 1737 and from a Stradivarius violin of 1707 showed that overtones even of considerable pitch are present, with a fairly large amplitude; and, apparently, for the high quality of a violin it proves very important that the very high range of tones from 3000–4000 hertz should be transmitted with great strength. Such a distribution of sound intensity is seen, for example, in Fig. 50 (upper spectrum) for the Stradivarius violin. Backhaus’s work shows how far an investigator, armed with modern technique built on exact physical foundations, can now penetrate into these most interesting fields. One may hope that further experimental material will gradually be collected, and that these investigations will come into still closer contact with the theoretical work already done in this field.

§ 4. Sound Receivers

In comparison with the numerous works in the field of emitters, which in part have important fundamental significance, little has recently been done in the field of sound receivers. Work in this field concerns not any fundamentally new designs, but chiefly the further development of methods already described by us in the review of works on the analysis of sounds (⁸⁰). In these investigations the greatest successes were achieved in the area of precisely determining the operation of receivers, in particular obtaining precise amplitude and frequency characteristics of receivers.

Let us first of all mention a series of investigations by Grützmacher and Meyer (⁸¹). They used the electro-

of acoustic compensation, and used a microphone constructed on the model of Wente’s microphone. The diaphragm of this microphone was made of brass (thickness 0.005 cm, diam. 25 cm) and was strongly tensioned. The frequency characteristic of the microphone is shown in Fig. 50; its form is determined chiefly by the air layer behind the diaphragm, which, as the frequency increases, acts more and more against the diaphragm. The cause of the fall of the curve at low frequencies lies in the properties of the resistance amplifier used.

Fig. 50. Frequency characteristic of a condenser microphone.

In order to study still more closely the operation of the condenser microphone, characteristics were taken with a reduced influence of the air layer. In Fig. 51a is shown the frequency characteristic taken at a pressure of 40 mm Hg. At this pressure the lowest natural period of the diaphragm lies approximately at 6000 cycles. For diaphragms fixed at the edges, the appearance of high, non-harmonic natural periods in relation to the fundamental oscillation is typical.

Fig. 51a. Frequency characteristic of a condenser microphone at a pressure of 40 mm Hg.

Fig. 51b. Frequency characteristic of a condenser microphone at a pressure of 160 mm Hg.

Fig. 51b gives the characteristic of the diaphragm at 160 mm Hg, on which the influence of the air layer is already clearly visible.

For Ritter’s condenser microphone, F. Trendelenburg (82) took frequency and amplitude charac-

teristics by the method described in Chapter 1 (by comparison with Rayleigh’s disk). Fig. 52 gives the amplitude characteristic of this sound receiver. Fig. 53 gives its frequency characteristic. By the same method Hartmann investigated microphones (83). Fig. 54 shows the frequency characteristic of a normal telephone microphone; Fig. 55 gives the amplitude characteristic for the same microphone. Measurements with Gerlach’s ribbon microphone gave the results presented in Figs. 56 and 57. Let us once more draw attention to the fact that in the latter two frequency characteristics, unlike the first ones, the ordinates are plotted on a logarithmic scale—an circumstance that must not be overlooked when comparing the curves in Figs. 50, 53, 54, and 57.

Fig. 52. Amplitude characteristic of Riegger’s microphone.

Fig. 53. Frequency characteristic of Riegger’s microphone.

Electrical sound receivers have recently also been used for a whole series of special purposes. Thus, for example, a task very important for theoretical and practical medicine—the obtaining of an undistorted accurate recording of heart tones and pulmonary sounds—

Fig. 54. Frequency characteristic of a normal telephone microphone.

has until now presented great difficulties, because these sound processes, on the one hand, have very low intensity, and on the other hand embrace broad regions of frequencies. At present a high-frequency condenser microphone of Riegger is used for this purpose, which in this case proved to be a very suitable sound receiver.

Fig. 55. Amplitude characteristics of a normal telephone microphone.

Fig. 56. Amplitude characteristic of a ribbon microphone.

With the aid of such a sound receiver, Trendelenburg (84) carried out a detailed investigation of the physical properties of heart tones in healthy and sick persons; this investigation yielded a whole series of new data, essential in both physical and physiological respects.

Fig. 57. Frequency characteristic of a ribbon microphone.

In a similar way, E. Bass (85) applied this method very successfully to the study of pulmonary sounds.

E. Wetzmann (86) recently made a report on an electric underground sound receiver. Fig. 58 shows a sound receiver that was used during the war for listening to noises connected with mine-laying work. The base of the instrument is a heavy plate about 15 cm in diameter. On the underside of the plate there is a small cavity closed by a membrane of tinplate. The lower cavity, by means of a tube passing through the plate, is connected with a second small cavity, closed by the membrane of the microphone. The receiver is placed with its tinplate membrane against the ground, and noises in the ground can thus be perceived by the microphone.

Fig. 58. Diagram of an underground sound receiver.

5. Speech and Hearing

The fact that to each speech sound there correspond definite regions of tones, constant also in their absolute pitch—the regions of the formants—and that the overtones lying in these regions appear in the given sound with special intensity, was examined in detail by me in a report (87) cited more than once here.¹ In particular, data are given there on the position of the formant regions, insofar as it is determined by experimental study. I pointed out the outstanding significance of Stumpf’s investigations; later Stumpf himself published an extensive work in which he gave a survey of his investigations (88). In this work are given

¹ Curves of vowel sounds, from Trendelenburg’s recordings, were cut on the circumference of a circular disk and, with the aid of a photocell, were again converted into electrical oscillations that could be perceived by the ear. With a properly chosen fundamental frequency, the evaluation of the corresponding speech sound proved quite correct, whereas with changes in frequency the accuracy of the evaluation was quickly lost.

numerous data, drawn in part from as yet unpublished experiments aimed at elucidating the character of speech sounds. The data were obtained by the three methods which Stumpf used with such great success: analysis of sounds by means of resonating tuning forks, analysis and synthesis of sounds with the aid of interference tubes, and artificial synthesis of speech sounds.

Krendall, whose valuable work in the field of the study of speech from the physical point of view (carried out in part in collaboration with Sacia and McKenzie) we have already discussed earlier, insistently points out in his latest work \((^{89})\) the characteristic features in the position

Fig. 59a            Fig. 59b

Schematic representation of the larynx–oral-cavity system.

of the formant regions of vowels, leading to curious conclusions about the conditions of origin of these vowels. Namely, if one compares with one another the formant regions characterizing individual speech sounds, it is found that in the frequency spectrum of most vowels, for each sound two distinct formant regions are indicated; at the same time the conclusion naturally suggests itself that, in all probability, in the initial generation of these sounds a system of two resonators takes part. Such a double resonator may naturally be assumed in the combination of the larynx and the oral cavity. Fig. 59b gives a sketch diagram of the speech organs; Fig. 59a—the diagram of the corresponding double resonator; here \(V_1\) and \(V_2\) are the volumes of air in the first and second parts of the resonator, and \(K_1\) and \(K_2\)—

acoustic conductance¹ of the opening of the first resonator and of the connecting tube. In Fig. 60 the forms of the oral cavity characteristic of each of the vowels considered are shown schematically. Crandall set himself the task of deter—

Fig. 60

¹ Acoustic conductance is determined from the expression:

\[ V = K \cdot \Phi, \]

where \(V\) is the volume flux, and \(\Phi\) is the velocity potential inside the cavity (³). It can be shown that the natural frequency of the resonator may then be expressed by the formula:

\[ \omega_1 = c \sqrt{\frac{K}{V}}. \]

The acoustic conductance of an opening of radius \(r\) is \(2r\); for a tube whose length \(l\) is large in comparison with the radius,

\[ K = \frac{\pi r^2}{l}. \]

The relations given above can be verified by calculation on the basis of the expressions for acoustic capacitance and self-inductance (Ch. 1).

by dividing the formant position, and then calculate the values \(V_1\) and \(V_2\), or \(K_1\) and \(K_2\), corresponding to each form of the oral cavity. In doing so he was able each time to establish, for the sum of the volumes \(V_1\) and \(V_2\), suitable values fairly close to reality. The value \(K_1\) as well (the conductance of the lip opening) can be approximately calculated from the dimensions of the opening. The damping, which could have been approximately estimated only insofar as it is damping due to radiation (Strahlungsdämpfung), remained unconsidered.

With the aid of the indicated calculations one can also draw certain conclusions about the peculiarities of the coupling and tuning of the resulting double resonators and work out a scheme that satisfactorily explains the production of vowel sounds. Crandall’s work represents a further development of the fundamental resonance theory of vowels created by Helmholtz. Although in its time this theory proceeded from the notion of a system of simple single resonators, it also lies at the foundation of this more general theory.

Concerning the resonance theory of vowels as such, we shall make one further observation: according to Helmholtz’s resonance theory, the source of sound during the singing of vowels is the vibrations of the vocal cords. These vibrations contain numerous overtones. The overtones that coincide with the natural frequencies of the cavity of the mouth and pharynx are strengthened by resonance and are radiated into the surrounding medium with especially great intensity. In this way the sound is given its characteristic coloring.

In opposition to this resonance theory of vowels, another theory was advanced by Hermann. Hermann’s theory considers not the entire consecutively occurring process of vibration of the vocal cords, but singles out from it one period. It takes as the basis of its further considerations the fact (correct for the chest register) that the glottis remains closed during a considerable part of an individual period and opens each time only for comparatively short intervals of time. At the moment when the glottis

opens, by one rapid impulse a certain volume of air is pushed through it into the pharynx and excites the natural vibrations of the oral cavity; these vibrations then gradually die away. When the slit opens again, the impulse is repeated; the opening of the slit is renewed periodically, and along with it periodic series of damped waves are radiated into the surrounding medium. As to the correctness of one theory or the other, at one time a heated dispute arose, lasting for years, which, however, may now be regarded as resolved: already Rayleigh (⁹¹) pointed out that the contradiction between the two theories of the origin of vowels is only apparent. Both theories make use of the same initial data for calculating the forced oscillations of the oral cavity and the timbre of the radiated sound: the natural period and the damping of the exciting and resonating systems. They differ only in their view of the details of the excitation of the intermittent process in the larynx, caused by the vibrations of the vocal cords.

To sum up, one may say: “The theory of vowel sounds, created by Helmholtz, provided the general foundations. Hermann’s theory considers the special case in which the sound produced by the vocal cords forms a periodic series of brief impulses, and can easily explain certain features of the phenomenon, namely, damped series of waves” (⁹²).

These views, theoretically and practically sufficiently well founded, still continue to provoke attacks. Thus, recently there appeared works by Scripture (⁹³), in which the following is advanced as the first proposition in the theory of vowel sounds: “In vowel sounds there is no fundamental tone in the physical sense.” This assertion is derived from the fact that all analyses of vowel sounds give a picture in which the fundamental tone is either entirely absent or is quite insignificant. And yet it should be noted that precisely those same records of Trendelenburg (which Scripture also cites as evidence in favor of his view) show that the fundamental vibration is present in the sound of vowels, and moreover for the most part with considerable amplitude. This can be инпо-

directly in the corresponding recordings. It is precisely the strict periodicity of the fundamental tone in Trendelenburg’s recordings that is the most convincing proof of the correctness of Helmholtz’s theory of vowels.

This is not the place to enter into a detailed analysis of the views expressed by Scripture. We shall confine ourselves to the remarks made above concerning attempts to shake the physical laws established earlier.

But we consider it necessary to say a few more words about another work by V. S. Kazanskii and S. N. Rzhevkin (⁹⁴), who published their sound recordings. Let us note beforehand that, unfortunately, their valuable material was obtained with apparatus that does not fully meet modern requirements for accuracy in sound transmission. A membrane receiver was used, with a cork membrane; for amplifying the sound a horn was employed, and for recording the vibrations—a device similar to a phonautograph with a rotating Raman needle (⁹⁵).

We note here a curious fact. The authors found that the vibration curves for vowels sung by experienced singers not infrequently revealed pulsation; precisely in those singers whose voices seemed “the most beautiful,” such “vibrato” is observed especially often, which naturally leads to the result that individual periods are no longer strictly identical. These observations—we emphasize this especially—do not contradict the fact that the sounds of vowels sung on one definite note recur strictly periodically; they show only that an experienced artist is often inclined to enhance the brilliance of his voice by a slight vibration. The impression of greater beauty of sound thereby produced must evidently rest on physiological or psychological effects that still remain to be clarified. The authors also give data on the difference between the singing and the speaking voice. In this connection, for the low and middle registers of the male singing voice, a very sharp predominance of a few (one or two) harmonic overtones is observed, whereas for non-singing voices no such predominance of individual harmonic ...

…no such overtones are observed. On the contrary, many overtones prove to be intensified.

After the exhaustive investigations carried out, in particular, by American scholars on questions concerning the physical aspect of hearing, there have been no new substantial works that would consider the question primarily from the physical point of view. Works concerning questions of the threshold of excitation of hearing as a function of the magnitude of the pressure amplitude of a sound heard by the ear, as well as the so-called phenomena of sound masking, and also works on more general questions of the theory of hearing, have been analyzed in detail by E. Meyer (⁹⁶).

Let us also note here an interesting article by B. Knudsen (⁹⁷), although its content does not fall within the range of questions examined in the present chapter. Knudsen systematically investigated questions concerning “hearing” by means of touch. He touches on the following points:

1. Determination of the lowest and the highest frequency for a vibrating body whose motions can still be perceived by touch.

2. Dependence of the sensitivity of touch on frequency.

3. Sensitivity to differences in amplitude.

4. Sensitivity to differences in frequency.

It is especially curious that the threshold of sensitivity to differences in amplitudes is comparatively low (depending on the absolute magnitude of the amplitude it lies between 5 and 10%). The author believes that a valuable auxiliary means in the perception of speech by the deaf would be the possibility, by means of touch, of following the average amplitude of speech with the aid of suitable apparatus—alongside the commonly accepted observation of the speaker’s lips (which helps especially in catching consonants).

6. Conclusion

In our survey we have attempted to acquaint the reader, in general outline, with the newest works on acoustics and especially on electroacoustics. Experience shows that

the modern approach to questions of acoustics stands in the closest connection with those methods of work which have proved fruitful in other areas of physics. Numerous examples point, for instance, to the close connection between acoustical, electrical, and optical problems.

We shall allow ourselves, in conclusion, to quote a few words recently uttered by a prominent advocate of exact methodology and of the mathematical grounding of acoustical problems (98).

“Physicists’ indifference to the problems of acoustics led to the result that its connection with the other branches of physics was increasingly lost; things had already gone so far that acoustics, as a scientific discipline, began to be assigned not to pure physics but to physiology, although this idea had never arisen with regard to optics. Only recently have physicists perceived that the problems of hydrodynamics and of the theory of elasticity, and especially the general problems of acoustical oscillatory processes, are no less attractive than any other questions of physics. It is curious that the impetus for this turn was provided by the needs of technology—an exemplary illustration of the proposition that technology, by borrowing from pure science, repays it a hundredfold. How interest in questions of acoustics has risen in recent years can be judged already in a purely external way from the fact, for example, that America is producing a type of specialist—sound-engineers; that the demand for well-trained acousticians is growing more and more, and in Germany it cannot be satisfied; and that the Union of German Engineers several years ago founded a special Committee for the Study of Oscillations, which convenes annual congresses and carries on valuable work in the field of science and practice.”

Literature

1. In the journal Jahrb. d. drahtl. Tel. u. Tel., 28, 54, 84, 1926, I gave a survey of the methods and results of sound analysis. In the present review, compiled at the request of the editor of the same journal, I shall be able to consider the further development of this complex of problems, as well as a number of further works in the field of acoustics and electroacoustics.

2. A. v. Hippel. Ann. d. Phys. (4), 75, 521; 76, 590, 1925. Cf. also the note by J. Friese und E. Waetzmann. Ann. d. Phys., 76, 39, 1925.

3. J. Friese und E. Waetzmann. ZS. f. Phys., 29, 110, 1924; 31, 50, 1925; 34, 131, 1925.

4. F. Trendelenburg. Jahrb. d. drahtl. Tel. u. Tel., 28, 54, 1926.

5. E. C. Wente. Phys. Rev. 10, 39, 1917; 19, 498, 1922.

6. H. Riegger. Wiss. Veröff. a. d. Siemenskonzern 3, H. 2, 67, 1924; F. Trendelenburg. Ebenda, p. 43, cf. also F. Trendelenburg. Jahrb. d. drahtl. Tel. u. Tel., 28, 54, 1926.

7. E. Gerlach. Wiss. Veröff. a. d. Siemenskonzern, 3, H. 1, 139, 1923.

8. E. Meyer. El. Nachr.-Techn., 4, 86, 1927.

9. W. König. Wied. Ann. 43, 43, 1891.

10. E. Meyer. El. Nachr.-Techn., 4, 509, 1927.

11. F. Trendelenburg. Wiss. Veröff. a. d. Siemenskonzern, 5, H. 2, 120, 1926.

12. C. A. Hartmann. El. Nachr.-Techn., 4, 375, 1927.

13. B. S. Cohen, A. J. Altridge und W. West. Journ. Am. Inst. El. Eng., 44, 1023; 1926.

14. M. Grützmacher und E. Meyer. El. Nachr.-Techn., 4, 203, 1927.

15. E. Gerlach. ZS. f. techn. Phys., 8, 515, 1927.

16. C. R. Moore and A. S. Curtis. Bell Syst. Techn. Journ., 6, 216, 1927.

17. E. Gerlach. ZS. f. techn. Phys., 8, 515, 1927.

18. M. Grützmacher. El. Nachr.-Techn., 4, 533, 1927.

19. Cf. also Fr. Canac. Filtres acoustiques. Journ. de Phys. (6), 7, 161, 1926. An exhaustive theory of acoustic filters and practical standards for such filters were given by Stewart (G. W. Stewart). His most important works are the following: Phys. Rev. 20, 528, 1922; 25, 90, 1925; 28, 1038, 1926; 29, 220, 1927; cf. further: H. B. Peacock. Phys. Rev. (2), 23, 525, 1924; W. P. Mason. Bell Syst. Techn. Journ., 6, 258, 1927; Phys. Rev. (2), 31, 283, 1928.

20. Cf. also E. G. Richardson. Sound. London, 1927, p. 224.

21. Electrical analogies proved especially applicable in the development of the theory of horns. See, for example, J. B. Crandall. Theory of vibrating systems and sound. New York, 1926, p. 166.

22. Cf. Fr. Canac. Journ. de Phys., 7, 166, 1926.

1. G. W. Stewart. Phys. Rev. (2), 20, 528, 1922. Data on the construction of filters are also given there.

2. G. W. Stewart. Phys. Rev. (2), 28, 1040, 1926.

3. H. Backhaus and F. Trendelenburg. ZS. f. techn. Phys., 7, 630, 1926.

4. Cf. F. Trendelenburg. ETZ, 48, 1685, 1927.

5. H. Stenzel. El. Nachr.-Techn., 4, 240, 1927.

6. W. C. Sabine. Collected papers on acoustics. Cambridge, 1923, p. 180.

7. E. Michel. Deutsche Bauhütte. 1927, p. 124.

8. Cf. G. Jäger. Wien. Ber., 120, H. 5, Abt. IIa, 613, 1911.

9. W. C. Sabine, ibid., p. 43.

10. Extracted from the table in the article: F. R. Watson. The absorption of sound by materials. Engineering experiment station, University of Illinois, Urbana, 25, No. 13, 1927 (Bulletin 172).

11. G. Jäger, ibid.

12. E. Meyer. ZS. f. techn. Phys., 7, 609, 1926; El. Nachr.-Techn., 4, 135, 1927.

13. F. Trendelenburg. Wiss. Veröff. a. d. Siemenskonzern, 6, H. 1, 276, 1927.

14. Cf. H. Gerdien. Telefunk. Ztg., VIII, Nos. 43 and 44, 1926; W. O. Schumann. ETZ, 47, 294, 1926; J. Zenneck. Jahrb. d. drahtl. Tel. u. Tel., 26, 177, 1925.

15. E. Meyer. El. Nachr.-Techn., 5, 293, 1928.

16. W. Schäffer. El. Nachr.-Techn., 4, 387, 1927.

17. Cf. H. Hollmann. El. Nachr.-Techn., 4, 180, 1927.

18. F. M. Osswald. Schweiz. Bauztg. 90, No. 5, 1927.

19. On horns see, for example, H. B. Crandall. Theory of vibrating systems and sound. New York, 1926, p. 152; for further considerations on this question see also G. R. Hanna. Journ. Am. Inst. El. Eng., 47, 253, 1928.

20. E. Meyer. El. Nachr.-Techn. 3, 293, 1926.

21. H. Riegger. Wiss. Veröff. a. d. Siemenskonzern, 3, H. 2, 67, 1924.

22. Lord Rayleigh. Theory of sound. II, § 302.

23. F. Aigner. Unterwasserschalltechnik, p. 114 ff.

24. H. Riegger, ibid.

25. For example, F. Trendelenburg. ETZ, 48, 1685, 1927.

26. C. W. Rice and E. W. Kellog. Journ. Am. Inst. El. Eng., 44, 985, 1925.

27. Borrowed from the article: E. Meyer. El. Nachr.-Techn., 3, 295, 1926.

28. E. Gerlach. V. D. E.—Fachber. d. 31. Jahresvers. d. Verb. d. Elektr., Wiesbaden, 1926.

29. E. Gerlach. ZS. f. techn. Phys., 8, 515, 1927.

30. See, for example, J. Engl. Der tönende Film, Braunschweig, 1927, p. 76 (there is a Russian translation: J. Engl, The Talking Film, GIZ, Moscow–Leningrad.

1928, see p. 80, Ed.). There an electrostatic loudspeaker by Vogt, Engel, and Massolle is described. For further data on various loudspeaker systems, see E. Gerlach, Lautsprecher, in the handbook Banneitz (Taschenbuch d. drahtl. Telegr., Berlin, 1927, p. 548 ff.); F. Trendelenburg. ETZ, 48, 1685, 1927. Here one should also point to the detailed exposition of the question of electrical sound emitters in the article by H. Lichte, Hdb. d. Phys., Bd. VIII, Kap. 6.

1. M. Grützmacher und E. Meyer. El. Nachr.-Techn., 4, 203, 1927.

2. E. Meyer. El. Nachr.-Techn., 4, 509, 1927.

3. K. Küpfmüller. Fachber. d. 31. Jahresvers. d. V. d. E., 1926, p. 87.

4. F. Waetzmann und K. Schuster. Ann. d. Phys. (IV), 84, 507, 1927; K. Schuster. Ann. d. Phys. (IV), 84, 525, 1927.

5. W. Gerwershausen. Helios, Fachzeitschr. f. Elektrot., 28, 229, 241, 1922. In particular one should also point to the work: L. Hajek. Mtschr. f. Ohrenheilkunde u. Laryngo-Rhinologie, 62. Jahrg., 808, 1928.

6. See in particular: J. P. Maxfield und H. C. Harrison. Bell. Syst. Techn. Journ., 5, 493, 1926.

7. See in particular H. Backhaus. Siemens-ZS. H. 5, 1928.

8. J. P. Maxfield and H. C. Harrison, ibid.

9. E. W. Kellog. Journ. Am. Inst. El. Eng., 46, 1041, 1927. On this question see also: K. Norden. ETZ, 48, 261, 1927.

10. J. Engl. Der tönende Film. The book contains data on the problems of the electrical engineering of the gramophone considered above.

11. A. O. Rankine. Proc. Phys. Soc. 31, 242, 1919; 32, 78, 1920; Nature, 108, 276, 1921; Proc. Opt. Convention, 2, 909, 1926.

12. See E. Lübcke. VDI, 70, 496, 1926.

13. On signal sirens see W. Kunze. Jahrb. d. Hafenbautechn. Ges., 9, 185, 1926.

14. Cf. Bureau hydrographique international, Monaco, Publication spéciale, No. 3, Oct. 1924 and No. 14, Août, 1926; La techn. moderne, 19, 425, 1927.

15. See A. Scheibe. Jahrb. d. drahtl. Tel. u. Tel. 29, 120, 1927.

16. For literature see note 66; in addition: E. Lübcke. VDI, 71, 1245; 1927.

17. G. W. Pierce. Proc. Am. Acad., 60, 271, 1925. A thorough work on piezoquartz as a radiator and receiver of high-frequency sound oscillations was published by F. W. Hehlgans. Ann. d. Phys. (IV), 86, 587, 1928.

18. R. W. Wood and A. L. Loomis. Phil. Mag. (VII), 4, 417, 1927. (See also P. N. Tsenkov. UFN, 8, 222, 1928. Ed.)

19. C. V. Raman. Hdb. d. Phys. Bd. VIII, Kap. 8.

20. F. Trendelenburg. Jahrb. d. drahtl. Tel. u. Tel. 28, 84, 1926.

21. C. Stumpf. Die Sprachlaute. Experimentell-phonetische Untersuchungen nebst einem Anhang über Instrumentenklänge. Berlin, 1926. Cf. also C. Stumpf. ZS. f. Phys., 38, 745, 1926.

1. H. Backhaus. ZS. f. techn. Phys., 8, 509, 1927; for further communications, in particular on the directional radiation of the violin and on the form of the vibrations of the violin body, see H. Backhaus. ZS. f. techn. Phys.

2. This fact has recently been confirmed again. See M. Grützmacher. ZS. f. techn. Phys., 8, 506, 1927, especially Fig. 4 and Table IX cited there; W. S. Kasansky and S. N. Rschevkin. ZS. f. Phys., 47, 233, 1928.

3. A. Seifert. Arch. f. Musikwiss., 4, 456, 1922.

4. On experiments on changing the timbre of a sound while excluding the fundamental tone and several overtones, see H. Fletcher. Phys. Rev., 23, 427, 1924.

5. C. V. Raman. Ind. Assoc. Bull., 15, 1—158, 1918.

6. C. W. Hewlett. Phys. Rev., 35, 359, 1912.

7. F. Trendelenburg, ibid. On the technical features of various electrical sound receivers, see F. Weichart. Jahrb. d. drahtl. Tel. u. Tel., 28, 120, 1926.

8. M. Grützmacher und E. Meyer. El. Nachr.-Techn., 4, 203, 1927. On the method of connecting a receiver operating on the Wente microphone principle, see A. J. Jakowleff. Jahrb. d. drahtl. Tel. u. Tel. 31, 85, 1928.

9. F. Trendelenburg. Wiss. Veröff. a. d. Siemenskonzern, 2, H. 2, 120, 1926. On the theory of the condenser microphone, see also A. J. Jakowleff. Jahrb. d. drahtl. Tel. u. Tel., 30, 151, 1927.

10. C. A. Hartmann. El. Nachr.-Techn., 4, 375, 1927.

11. F. Trendelenburg. Wiss. Veröff. a. d. Siemenskonzern, 5, H. 3, 175, 1927; 6, H. 2, 184, 1927. See also the literature there.

12. E. Bass. ZS. f. ges. experim. Med., 59, 133, 1928. Further works by Bass are to be published in the same journal.

13. E. Waetzmann. Naturwiss., 15, 401, 1927.

14. F. Trendelenburg. Jahrb. d. drahtl. Tel. u. Tel. 28, 84, 1926. Among further works devoted to the problem of formants, the following should be mentioned: V. Engelhardt und R. Gehrke. Wiss. Abt. d. Phys.-techn. Reichsanstalt, 11, H. 2, 390, 1928.

15. See note 73.

16. J. B. Crandall. Bell Syst. Techn. Journ., 6, 100, 1927.

17. Cf. H. Backhaus. Hdb. d. Phys, Bd. VIII, Kap. 4, Ziff. 16.

18. Lord Rayleigh. Theory of sound, II, p. 473. London, 1926.

19. F. Trendelenburg. Hdb. d. Phys. Bd. VIII, Kap. 10.

20. E. W. Scripture. ZS. f. Sinnesphysiol., 58, 195, 1927; 59, 83, 1928.

21. W. S. Kasansky und S. N. Rschevkin. ZS. f. Phys., 47, 233, 1928.

22. C. V. Raman and A. Dey. Phil. Mag., 39, 145, 1920.

23. E. Meyer. Hdb. d. Phys., Bd. VIII, Kap. 11.

24. V. O. Knudsen. Journ. of gener. psychol., 1, 320, 1928.

25. E. Waetzmann. Moderne Probleme der Akustik. Unterrichtsblätter f. Mathem. u. Naturwiss., 33, No. 12, 377, 1927.

1. The expression “mechanical capacitance” is given here in place of the word “Compliance” adopted in the American literature. The reciprocal of the quantity designated as mechanical capacitance is the directing force (Direktionskraft). ↩↩↩↩↩

2. The numbers in parentheses refer to the bibliography at the end of the article. ↩