Recent Advances in Applied Acoustics

F. Trendelenburg, Berlin

III. Sound Receivers and Sound Emitters

1. Sound Receivers

Among the sound receivers used for technical purposes, we shall first of all consider the condenser microphone. Condenser microphones are connected both according to Wente’s low-frequency circuit41 and according to Rieger’s high-frequency circuit42, 43. The high-frequency circuit has found application in research work in many areas of acoustics and mechanics44. C. Kawazoe45 applied to the condenser microphone the circuit of a high-frequency voltage divider, the action of the microphone under changes of the high frequency being essentially unchanged if these changes do not exceed certain limits.

Systematic investigations were undertaken of the actual receiving system of the condenser microphone—its capsule; on the one hand, the constancy of its sensitivity was investigated, and on the other, the distortions of the sound field introduced by the microphone itself. C. Ballantine46 investigated the phenomena of “aging” in a Wente-type condenser microphone. The mechanical tension of the membrane of such a microphone changed over the course of \(3 \tfrac{1}{2}\) years by no more than 4.5%; the change in sensitivity over the same interval of time was likewise comparatively small. Some changes in sensitivity that depend on temperature are caused by the different expansion upon heating of the duralumin and the steel frame. C. Ballantine found,

* Jahrbuch d. drahtl. Tel., u. Tel. 37, 38, 1931, translated by N. D. Ershova. Continued; see Uspekhi Fizicheskikh Nauk, XI, issue 4, 650 (1931).

F. TRENDELENBURG

that temperature changes in sensitivity are expressed by the figure of 0.6% per degree Celsius.

Besides condenser microphones, ribbon microphones and Reiss microphones are used predominantly for high-quality transmission. The former, in their new design, providing for an extension of the transmitted band toward high frequencies, have found application, among other things, in sound-film technology.^47

Reiss microphones were tested by electrostatic excitation: the microphone cup was filled with carbon powder and closed with a carbon plate, and the natural oscillations of the plate did not affect the result because of the considerable damping caused by the carbon powder. A metal foil was placed on the plate, constituting one plate of a capacitor; the other plate was a metal grid situated at a small distance from the membrane.

Since the katodophone is no longer used for technical purposes, we shall briefly mention only the work of E. Meyer,^49 which elucidates the physical mechanism of its action. In an investigation in standing waves in a Kundt tube it turned out that the katodophone reacts most strongly when it is placed at distances of \(1/4\lambda\), \(3/4\lambda\), \(5/4\lambda\), etc., from the closed end of the tube. Thus the katodophone is not a pressure receiver, since the pressure oscillations have nodes at the indicated points. To clarify further the question of whether the katodophone is a receiver of velocity or of displacement, analogous measurements were carried out over a large frequency range, showing that the katodophone is a displacement receiver. E. Meyer supposes that the space charge of negative ions located between the anode and cathode oscillates together with the air particles in the sound wave, as a result of which the anode current is modulated.

In questions of quantitatively correct sound transmission, an important role is played by the distortions of the sound field introduced by the microphone itself. In receiving high-frequency waves it is practically difficult to satisfy the condition that the над-

the largest dimension of the microphone should be small in comparison with \(\lambda/2\pi\) (\(\lambda\) is the wavelength of sound); if this condition is not fulfilled, then reflection and diffraction phenomena arise. Let \(P/P_0\) denote the ratio of the sound pressure at the receiver to the pressure in the undistorted sound field; the ratio \(P/P_0\) has its greatest value when the microphone is located in a rigid wall whose dimensions are large in comparison with the wavelength. In this case \(P/P_0 = 2\), i.e., the intensity of the sound field near the sound receiver thereby increases fourfold.

C. Ballantine\({}^{50}\), on the basis of considerations given by Rayleigh, calculated the values of \(P/P_0\) for a rigid sphere (on the side facing the sound source) for various ratios of the radius of the sphere to the wavelength. Fig. 15 gives a graph of the function \(P/P_0\).

Fig. 15. Distortions of the sound field near a rigid sphere.

If a standard microphone is mounted in the surface of a rigid sphere, then, using the curve shown in Fig. 15, one can determine the correction coefficient for eliminating errors caused by the phenomenon of diffraction.

All the remarks made so far concern either questions of the possibility of reducing the distortions of the sound field introduced by the microphone itself, or, in some cases, estimating their magnitude. Conversely, in many practical questions it is precisely the fact that has great significance,

that, by arranging the microphone in an appropriate manner (for example, placing it in the throat of a funnel-shaped horn), we can obtain high sensitivity of the microphone in certain directions.

N. Obata and Y. Yoshida1 investigated in sufficient detail the dependence of the sensitivity of a condenser microphone on the direction of the sound; moreover, the microphone was either provided with a funnel-shaped horn or was placed

Fig. 16. Directional sensitivity of a parabolic sound mirror.

Fig. 17. Directional sensitivity of a funnel-shaped horn.

in the focus of the reflector. The sound source in these experiments was a loudspeaker located at a considerable distance from the microphone.

Some of the results of their studies are given in Figs. 16 and 17. Figs. 16 a and b give the graph of the directional sensitivity of a reflector with a cross section of 200 cm and a depth of 48 cm. Figs. 17 a and b give the same for a curved exponential horn, the section of which is indicated in the figure. Table I gives numerical values of the amplification (increase

amplitude of the sound pressure at the sound receiver when using a reflector or a horn).

TABLE I

The table shows that, with the aid of the indicated auxiliary means, one can obtain a significant increase in the sensitivity of the receiver for sound waves incident in the optimal direction.

Sound reflectors and horns of the type described have been successfully used^52 for localizing the sound of airplanes.

2. Sound Emitters

(General questions concerning electrical sound emitters)

A very large number of works is devoted to sound emitters, especially electrical sound emitters, which are of great technical importance. Alongside purely acoustic questions (for example, the properties of the sound field), questions of the electroacoustic efficiency coefficient were investigated in detail; whereas formerly, in order to avoid sound distortion, one was satisfied with an efficiency coefficient on the order of 0.1 percent, now the magnitude of this coefficient is being brought up to several percent without deterioration in the quality of transmission.

Let us first consider general questions of the configuration of the sound field, closely connected with the remarks made at the end of the preceding section.

K. Sato^55, on the basis of Rayleigh’s data, calculated the sound field of a conical horn, at the mouth of which there was a sound source. Figs. 18 a and b give some of the results of his calculations, the calculations referring to

to a horn with an aperture angle of \(20^\circ\) and to points situated at large distances from the horn. In Fig. 18a \(k \cdot a = 3.831\), in Fig. 18b \(k \cdot a = 7.662\) \((k = 2\pi/\lambda\) and \(\lambda\) is the length of the horn). In the sound field of a horn loudspeaker, interference maxima and minima are observed—phenomena similar to those which Backhaus and Trendelenburg\(^{54}\) observed in the sound field of a flat horn.

The calculation of the sound field of a circular piston membrane was formerly possible only for those points whose distance from the membrane is large in comparison with the radius of the membrane\(^{55}\), and moreover—

a                b

Fig. 18. Directional action of a horn radiator.

only for points lying on the perpendicular to the middle of the membrane\(^{56}\). Backhaus\(^{57}\) gave a general solution of the Rayleigh integral expressing the velocity potential of a piston membrane for all points lying outside a sphere with radius equal to the radius of the piston membrane. The results of the calculation were checked experimentally, and, generally speaking, good agreement between theory and experiment was found. The calculation of the sound field was based on the assumption that the membrane vibrates in a rigid wall, the dimensions of which are large in comparison with the wavelength\(^{58}\),

M. Strĕt2 investigated the properties of an acoustic screen in approximately the following way.

Two point radiators are placed on the surface of a rigid sphere in such a way that they are exactly at the opposite ends of one of the diameters. Both radiators oscillate with opposite phases. It is evident that if the diameter of the sphere is very small \((D \ll \lambda/2\pi)\), then at large distances from it there will be practically no sound, since at all points sufficiently far from the sphere the condensation produced by one radiator is neutralized by the rarefaction produced by the other. If the sphere is large in comparison with the wavelength \((D \gg \lambda/2\pi)\), then the radiation corresponds to a radiator of zero order set into a rigid wall of very large dimensions; one half of the radiation propagates to one side of the sphere, the other—to the other side.

Denoting the total power by \(L\), one may put

\[ L = k Q^{2} e, \]

where \(k\) is a factor including the density of air, the speed of sound, etc., \(Q\) is the output of one sound source, and \(e\) is a factor characterizing the joint action of both radiators; according to what has been set forth above, \(e\) depends on the ratio \(D/\lambda\). In the limiting case \((D \ll \lambda/2\pi)\) \(e\) is equal to zero, while for \(D \gg \lambda/2\pi\) \(e\) reaches the value \(0.5\). Strĕt calculated the course of the variation of \(e\) as a function of the ratio of the sphere diameter to the wavelength.

Of particular technical interest is the question of how a piston membrane set into a rigid wall behaves. If one assumes that the shortest sound path between the two point radiators is decisive in the neutralization of the field caused by the opposition of their phases, then from the idealized system of two sound radiators located on the surface of a sphere one can pass to the radiation of a piston membrane. The decisive significance in this case will belong to the path of the sound from the reverse side of the membrane around the edge of the acoustic screen to its front side. The idealized system described above may be taken as the equivalent of a piston membrane

with a circular baffle whose diameter is equal to half the length of the circumference described with a radius equal to the radius of the sphere.

The course of the coefficient \(e\), which is extremely important for sound radiation, is given in Fig. 19, where \(2a\) denotes the diameter of the circular baffle. This relation may be illustrated still more clearly by a practical example. Suppose that the lowest of the tones to be transmitted is a tone with a frequency of 100 hertz (wavelength 3 m). If the power delivered at this frequency is to amount to not less than 80% of the maximum power output, then the baffle must have a diameter of not less than 1 m.

Fig. 19. Increase of radiation by a sound baffle.

Here it is necessary to note the following as well: the introduction of the baffle is often attributed to Rice and Kellogg\(^ {60}\); also, in the above-mentioned work by Stret, reference is made only to the extensive work of these investigators. However, O. Nesper\(^ {61}\) points out that an acoustic baffle was first used by V. Burstin\(^ {62}\) for a tuning fork. In addition, it should be emphasized that Rieger\(^ {63}\), in his work “On the Theory of the Loudspeaker,” investigated in detail precisely the action of a piston membrane oscillating in a rigid wall, and was the first to show that such a sound radiator satisfied the requirements of undistorted transmission.

Works concerning the design of electric sound radiators deal, on the one hand, with improving the frequency response, and, on the other, with increasing the electroacoustic efficiency. Particular attention was given to reducing nonlinear distortion. Electromagnetic systems are especially subject to this danger. By a special arrangement of the armatures in the “Gealion” loudspeakers\(^ {64}\), it was possible to achieve a reduction of the nonlinear distortions that arise as a result of the increase in force when the armature approaches the poles; this was achieved by the fact that, when the armature approaches the poles, the arm of the automatic lever...

…decreased practically, and therefore the torque remained constant.

Still earlier, the question had been discussed^65 that electrodynamic systems offer greater advantages in the sense of reducing nonlinear distortions; these advantages are especially obvious when it is necessary to obtain high acoustic power. Wherever high acoustic power and irreproachable fidelity of transmission are required, the electrodynamic principle is used. In such cases loudspeakers with a moving coil and a conical diaphragm are employed,^66 Blatthaller^67 and loudspeakers with a corrugated diaphragm^68 (a further development of the loudspeaker with the folded diaphragm of Gerlach).^69 Neumann^70 improved the Riegger Blatthaller by various structural measures and by strengthening the magnetic system. The expression given by Riegger for the electroacoustic efficiency of a piston diaphragm excited electrodynamically can, after certain transformations, be written in the following form:^*

\[ \eta = C \frac{B_L^2 V_L F^2}{\rho M^2}, \]

where \(C\) is a constant proportionality factor, \(B_L\) is the magnetic induction in the air gap, \(V_L\) is the volume of the conductor, \(F\) is the surface of the diaphragm, \(\rho\) is the specific resistance of the conductor material, \(M\) is the total mass (the mass of the diaphragm \(M_m\) + the mass of the conductor \(M_l\) + the co-vibrating mass of air \(M_v\)). To increase the efficiency, the following measures may be taken:

1. Increasing the radiating surface. The limit of this increase is the increasing directivity of action, which must not exceed certain limits.

2. Reducing the masses of the diaphragm and conductor.

^* The relation is valid only for a small efficiency and for tones whose wavelength is large in comparison with the cross-section of the diaphragm.

The effect of the co-oscillating mass of air leads to the fact that reducing the masses of the diaphragm and the conductor can improve the efficiency only so long as these masses considerably exceed the mass of the co-oscillating air. In addition, an excessively large reduction in the mass of the diaphragm is also hindered by the requirement of sufficient rigidity, which is necessary in order to reduce the “softening” of the diaphragm with respect to the high-frequency components. It should further be pointed out that reducing the mass of the conductor (assuming that its material remains the same) causes a reduction of its volume; consequently, the efficiency does not increase.

Fig. 20. Blattthaller with “slotted” excitation.

Fig. 21. Blattthaller with pole-piece excitation.

A detailed investigation of the dependencies mentioned here only briefly shows that the optimal efficiency is obtained in the case when the masses of the conductor and the diaphragm are equal.

1. A considerable increase in efficiency can be obtained by strengthening the field, since the efficiency increases proportionally to \(B_l^2\). By a special arrangement of the coils it was possible to reduce the leakage flux. Fig. 20 shows a Blattthaller system with “slotted” excitation; Fig. 21 gives a diagram of the previous device with pole-piece excitation. A device analogous to that shown in Fig. 21 naturally entails a larger leakage flux; the expenditure for exciting a field of 20,000 gauss with pole-piece excitation is approximately 100% greater than with slotted excitation. In addition, the latter device is distinguished by better thermal—

duct. From a large Blatthaller with slit excitation it proved possible, without the use of a horn and with an expenditure of 800 watts of electrical power, to obtain \(1/4\) HP of acoustic power[^71].

The question of increasing the efficiency of an electrodynamic loudspeaker by means of pressure transformation and a horn was investigated by Ganna[^72]. For the investigation a moving-coil loudspeaker was chosen, operating in the frequency band from 100 to 5000 hertz, with an output of 30% when a copper conductor was used and an output of 40% when an aluminum conductor was used. Owing to the large amplitudes of vibration of the diaphragm at low frequencies, the power of a loudspeaker with a diaphragm of a given size is limited. Ganna, in cases where this was necessary, divided the frequency band among several loudspeakers: for the transmission of low tones a loudspeaker with a large diaphragm surface was used; for the transmission of high tones, a loudspeaker with a small diaphragm surface.

Fig. 22. Frequency curve of the total power of the Rice-Kellogg loudspeaker.

The efficiency of a sound radiator may conveniently be determined either by determining the total radiated acoustic power from the mean density of sound energy in a room whose absorption is known, or by electrical measurements with the field switched on and off, first in air and then in vacuum[^73].

Meyer and Just[^74] found that the efficiency of an electrodynamic loudspeaker is approximately 1%; the efficiency of a powerful electromagnetic loudspeaker reaches 3.5%; that of an electrostatic loudspeaker, up to 2%; and that of an electromagnetic horn loudspeaker, up to 7% (in the case of resonance).

In Fig. 22 the frequency characteristic of the total power of the Rice–Kellogg loudspeaker is given. The characteristic gives the mean intensity for the case when the loudspeaker operates in a room with well-reflecting surfaces. For a room with large absorption this characteristic gives an incomplete picture of sound radiation, since in taking it the inevitable directivity of the loudspeaker’s action was not taken into account; we spoke above about directional action. Fig. 23 gives, for the same loudspeaker, the curve of the dependence of the sound intensity on frequency directly in front of the membrane, along the normal to its middle; from a comparison of Figs. 23 and 22 it is evident that, along the normal to the middle, the influence of the directivity of action of the high-frequency components is strongly manifested.

Fig. 23. Dependence of sound intensity on frequency along the normal to the middle of the membrane.

3. Musical instruments^75 and their sound

The physical features of the violin—the principal instrument of the modern orchestra—received their further illumination in the works of Backhaus^76. Whereas formerly experimental investigations were carried out by optical methods (on the vibrating strings of the violin) and by methods of analyzing the sound of the violin in the sound field, it has now proved possible to obtain objective conclusions about the form of the vibrations of the violin body at various frequencies. The forms of vibration of the violin body were studied^77 in the following way: an extremely light metallic foil, approximately 0.8 cm in diameter, was glued to the regions of the violin body under investigation. The foil constituted one plate of a capacitor, the other plate of which was a stationary electrode situated at a distance of 0.5 mm

from foil. During vibrations of the violin body, changes arose in the capacitance of this capacitor. The changes in capacitance were converted, by Rieger’s method, into changes in current and were recorded oscillographically.

Of particular interest was the problem of determining, by investigations of this kind, what order of radiator the violin represents. For this purpose oscillographic records were made simultaneously with two such capacitors located at two points of the violin, and then the phase difference of the obtained vibration curves was determined. Fig. 24 gives the form of the vibrations of the violin body for a tone of frequency 192 hertz. The nodal line runs along the side; the measured points on the back and the belly of the violin vibrate in opposite phases; at low frequencies, the side itself constitutes a rigid connection between the belly and the back of the violin. In addition, the nodal line runs from above downward through the back and the belly. Vibrations of such a form may be approximately taken as vibrations of a sectorial spherical radiator of the second order (whose axes correspond to the axes of the instrument).

Fig. 24. Form of vibration of the violin body at a frequency of 192 hertz (after Backhaus).

Backhaus had earlier come to the conclusion ⁷⁸, on the basis of the negligibly small strength of the fundamental tone in the acoustic spectrum of the violin at this frequency, that in the low-frequency region the violin vibrates as a radiator of comparatively high order and therefore in this frequency region cannot produce any appreciable radiation. This supposition was fully confirmed by the facts set forth above. At higher frequencies

F. Trendelenburg

the violin vibrates, as Figs. 25 and 26 show, almost like a dipole (i.e., like a first-order radiator). The base and the soundboard vibrate almost in the same phases; the ribs still remain rigid.

1. $\bar N = 10.0$
2. $\bar N = 1.25$

Fig. 25. Form of vibration of the violin body at a frequency of 287 hertz (after Backhaus).

Of particular interest is the fact that at high frequencies (683 hertz) a good violin vibrates almost like a zero-order radiator. In this case the ribs are no longer a rigid connection, but become dynamically “soft.” Backhaus attaches great importance to this circumstance: zero-type radiators are the most advantageous, since in this case the radiation of the violin is optimal.

Lüder[^79] investigated the distribution of the sound of individual instruments and of entire orchestral works; the investigations were carried out by the method described above of statistical analysis by octaves. These investigations made it clear how the components of individual sounds are distributed over various octaves and what maxima are observed over the course of a certain

1. $N_{95\%} = 59.0$
2. $N_{95\%} = 6.4$

Fig. 26. Form of vibration of the violin body at a frequency of 483 hertz (after Backhaus).

time interval (4 minutes). In Table II the mean pressure amplitudes in the overall sound picture are compared, i.e., without taking into account the distribution by octaves and the maximum values occurring in the overall spectrum. The latter data refer not to all maximum values encountered in general, but to a certain calculated value \(S_{95\%}\); by this is meant such a value below which 95% of all maxima lie. This value was chosen on the grounds that, technically, it is less essential to know what maxima can occur at all, but it is extremely important to know what “average” maxima one has to deal with, while completely random small excesses are almost imperceptible. The data of Table II refer to points located at a distance of 5 m from the sound source, in the direction of optimum radiation.

TABLE II

\(N = 2.0\)

Fig. 27. Spectrum of mean values and spectrum of maxima for a siren.

\(N_{95\%} = 16.0\)                 \(N = 1.8\)

Fig. 28. Spectrum of mean values and spectrum of maxima for a flute.

\(N_{95\%} = 28.0\)                 \(N = 9.1\)

Fig. 29. Spectrum of mean values and spectrum of maxima for a pipe.

Fig. 30. Spectrum of mean values and spectrum of maxima of the contrabassoon.

\(N_{95\%} = 6.96\)

\(\overline{N} = 1.1\)

Fig. 31. Spectrum of mean values and spectrum of maxima of the double bass.

\(N_{95\%} = 9.3\)

\(\overline{N} = 0.96\)

Fig. 32. Spectrum of mean values and spectrum of maxima of kettledrums (1) and cymbals (2).

Spectra of maxima and mean values of individual instruments and various orchestral pieces in the form in which they were obtained by successive measurements with the aid of an octave filter are given in Figs. 27–34. Dia-

\(N_{95\%}=34.5\)

\(N=1.9\)

Fig. 33. “The Magic Flute.”

grams designated by \(a\) refer to the spectra of maximum values. The parameter \(p\) for the individual curves is the total frequency. Total frequency

\(N_{95\%}=60.0\)

\(N=4.2\)

Fig. 34. Lohengrin.

\(p\%\) means that in the corresponding octave \(p\%\) of all maxima lie below the corresponding curve and \((100-p\%)\) of the maxima lie above this curve. Diagrams designated by \(b\) give the arithmetic mean com-

pressure moments in the corresponding octaves. It should be noted that the curves have been brought to practically equal scales by multiplication by factors depending on the arithmetic mean value over all frequencies. These factors are indicated under the diagrams; thus, for example, the ordinate values of the mean value for the flute are the products of the mean value by 1.8; the ordinate values of the curve for the trumpet—by 2.1, for the timpani—by 14.0. These factors may be used as measures of the sound load produced by the given instrument.

From consideration of the curves in Figs. 27–34 it is seen that the maxima of individual musical instruments lie in quite different octaves. Thus, for example, the principal components of the flute sound (Fig. 28) lie between 200 and 800 hertz; the maxima of the trumpet (Fig. 29) lie considerably above 1600 hertz; the maxima of the timpani and double bass lie between 50 and 100 hertz (Figs. 31 and 32).

The facts set forth above are very important for the calculation of apparatus for sound transmission. All elements of the device must be designed so that they can transmit without distortion also the maximum values of the sound components. It is evident from the curves that the maxima of individual instruments lie in quite different frequency regions and, consequently, different types of loudspeakers must be used accordingly; thus, for example, for the transmission of wind instruments a horn loudspeaker of the old type, which is wholly unsuitable for transmitting the double bass and timpani because of its extremely small output at low frequencies, may be used.

LITERATURE

1. E. C. Wente, Phys. Rev. (2), 10, 39, 1917.

2. H. Riegger, Wiss. Veröffentlich. a. d. Siemens-Konz., III/2, 67, 1924; F. Trendelenburg, Wiss. Veröffentlich. a. d. Siemens-Konz., III/2, 43, 1924.

3. A. H. Reeves, E. Commun., 7, 268, 1929; there is no reference to Riegger’s work.

1. Cf., for example, H. Backhaus, ZS. f. techn. Phys., 9, 491, 1928 (maintenance of the form of oscillations of a quartz oscillator); H. Gardien, Wiss. Veröffentlich. a. d. Siemens-Konz., VIII/2, 126, 1929; W. Mauksh, l. c., 130; C. Salomon, Loewe, 14, 117, 1929; H. Thoma, ZS. d. V. D. I., 639, 1929; O. v. Auwers, Wiss. Veröffentlich. a. d. Siemens-Konz., VIII/2, 137, 1929; H. Gardien, H. Pauli u. F. Trendelenburg, ZS. f. techn. Phys., 10, 374, 1929 (measurement of the mechanical force and measurement of the torque of the transmitter, electrodynamic transmitters, measurements with a seismic pendulum, etc.); K. Schnauffer, Jahrb. d. D. Vers.-Anst. f. Luftfahrt, S. 304 1930.

2. S. Kawazoe, Res. Elektrot. Lab. Tōkyō, Nr. 282, 1930.

3. St. Ballantine, Proc. Inst. Radio Eng., 18, 1206, 1930. In the paper calculations are given of the influence of the cavity immediately in front of the membrane; the cavity acts as an acoustic capacitance, and the mass of the oscillating air as an acoustic self-induction.

4. Hartmann’s communication on this recently appeared in E. N. T.

5. E. Reiss, AEG Mitt., 1929, S. 601.

6. E. Meyer, E. N. T., 6, 17, 1929.

7. St. Ballantine, Phys. Rev. (2), 32, 988, 1928; see also W. West, I. I. E. E., 67, 1137, 1929.

8. J. Obata u. Y. Yosida, Rep. of the Aeronaut. Res. Inst. Tōkyō Imperial Univers., V/9, 231, 1930.

9. In addition to paper (d), see also E. Waetzmann in Müller-Pouillets, Lehrb. d. Phys., 11. Aufl. I/3, 299, 1929. In addition, sound attenuators were described by R. Berger, Schalltechnik, 3, 90, 1930.

10. K. Satō, Jap. Journ. Phys., V, Nr. 3, 103, 1928/29. Cf. also the new paper by K. Satō, Proc. Imp. Acad. Tōkyō, VI, 27, 256, 1930; these works were carried out in the reverse direction (horn as receiver), and quite good agreement with the theory given above was obtained.

11. H. Backhaus u. F. Trendelenburg, ZS. f. techn. Phys., 7, 630, 1926.

12. H. Stenzel, E. N. T., 4, 239, 1927; 6, 165, 1929; ZS. f. techn. Phys., 10, 567, 1929. Cf. also the work of N. W. McLachlan, Proc. Roy. Soc., 122, 604, 1929, in which the author arrives at the same results as H. Stenzel in the above-mentioned article. Further, R. B. Lindsay, Phys. Rev., 32, 515, 1928; J. Wolff u. L. Malter, Phys. Rev., 33, 1061, 1929.

13. See above, 54.

14. H. Backhaus, Ann. d. Phys. (V), 5, 1, 1930.

15. Here it is necessary to note that Stenzel also succeeded in calculating the sound field for zones located at a great distance from the radiator. Technically the most important question is that of sound emitters with a conical membrane; Stenzel replaced the conical membrane by a spatial angle of equal surface, whose radiation field ...

could be calculated. The results of the calculation agree well with the experimental data. See on this H. Stenzel, Forsch. u. Techn., p. 349, 1930.

1. M. J. O. Strutt, Phil. Mag. (VII), 7, 537, 1929.

2. C. W. Rice u. E. W. Kellogg, Journ. Amer. Inst. Electr. Eng., 44, 982, 1925.

3. O. Nesper, Funkbastler, 3, 40, 1930.

4. W. Burstyn, ZS. f. techn. Phys., 3, 180, 1922.

5. H. Riegger, Wiss. Veröffentlich. a. d. Siemens-Konz., III/2, 67, 1924.

6. H. Benecke, AEGMitt., p. 588, 1929.

7. ZS. Hochfrequenztechn., 33, 134, 1928.

8. Cf., for example, F. A. Fischer u. Lichte, AEGMitt., p. 25, 1929 (the paper gives detailed data on the “Geaphon” and “Geakord” loudspeakers, frequency curves, etc.); H. Benecke, AEGMitt., p. 509, 1930; E. D. Cook, Gen. Electric. Rev., 33, 509, 1930; H. M. Clarke, Exp. Wireless, 6, 602, 1929. On electrodynamic loudspeakers with a flat circular diaphragm see R. W. Paul u. B. S. Cohen, Wireless World, 26, 374, 1930; Exp. Wireless, 7, 421, 1930; N. W. Mc. Lachlan, Phil. Mag. (VII), 7, 1011, 1929.

9. H. Riegger, Wiss. Veröffentlich. a. d. Siemens-Konz., III/2, 43, 1924.

10. Cf., for example, G. Kemna, u. M. Kluge, Siemens-Jahrbuch, 4, 361, 1930.

11. Few papers have been published on electrostatic loudspeakers; among them we shall cite here the works of V. F. Greaves, F. W. Kranz u. W. D. Crozier, Proc. Inst. Rad. Eng., 17, 1142, 1929; G. Green, Phil. Mag. (VII), 115, 1929.

12. H. Neumann, ZS. f. techn. Phys., 10, 548, 1929; Wiss, Veröffentlich. a. d. Siemens-Konz., IX/2, 226, 1930.

13. H. Neumann, Siemens-Ztschr., 10, 562, 1930 (detailed data on the construction).

14. C. R. Hanna, Journ. Amer. Inst. Eng., 47, 253, 1928.

15. For the literature see Uspekhi Fiz. Nauk, issue 4, 1931.

16. E. Meyer u. P. Just, ZS. f. techn. Phys., 10, 309, 1929.

17. On new ideas about musical instruments see A. Kalähne in Müller-Pouillets, Lehrbuch der Physik, hrsg. von E. Waetzmann, 1, 3, 226, 11. Aufl., Braunschweig, 1929. See further E. G. Richardson, The acoustics of orchestral instruments, London, 1929.

18. H. Backhaus, Naturwissensch. 17, 811 u. 835, 1929.

19. H. Backhaus, ZS. f. techn. Phys., 9, 491, 1928; ZS. f. Physik, 62, 143, 1930.

20. H. Backaus, ZS. f. techn. Phys., 8, 509, 1927.

21. H. Lueder, Wiss. Veröffentlich. a. d. Siemens-Konz., IX/2, 167, 1930.

1. Reference number as printed on the page. ↩

2. Strĕt. ↩