Large-Amplitude Acoustic Waves

A. A. Eichenwald, Milan

1. Introduction

As is known, the differential equations of aerodynamics are nonlinear, and solving them in general form presents considerable mathematical difficulties. However, in applications to acoustics these equations are usually simplified by discarding terms in which the factors are small quantities of degree higher than the first.

This is called linearizing the equations, i.e., making them linear and thus more accessible to solution. One may say that almost all of theoretical acoustics is based on such linearized equations (acoustics of the first degree), and only in rare cases are corrections resorted to, applying the method of successive approximations to the basic differential equations.

Riemann (1859), however, showed that in some cases the equations of acoustics can be integrated in exact form.

Suppose that, applying Riemann’s method, we have found for some quantity \(S\) its dependence on another quantity \(s\) in the form:

\[ S = f(s). \]

If the quantity \(s\) itself is comparatively small, then we can expand the solution found in the form:

\[ S = a_0 + a_1 s + a_2 s^2 + a_3 s^3 + \ldots, \]

and, in calculations, take into account only those terms of this series which are of primary importance for us. Thus, for example, if the quantity \(s\) is so small that its higher powers may be neglected in comparison with the first degree, then in the calculation we may restrict ourselves to the formula:

\[ S_1 = a_0 + a_1 s. \]

But this formula will represent precisely the solution that we could have obtained directly from the linearized equation.

If, however, we compute by the formula:

\[ S_2=a_0+a_1s+a_2s^2, \]

then we obtain a result that we could have obtained from the linearized equation by adding to it second approximations, etc.

Of course, an exact solution is always preferable from the mathematical point of view, and we shall apply Riemann’s method to traveling waves. However, in application to standing waves Riemann’s method proves too complicated and insufficiently transparent from the physical point of view; and since in applications one must in any case be satisfied with approximations, I have applied the method of successive approximations, but not to the fundamental equation, as is usually done, but to an intermediate stage of the calculation—to the so-called characteristics of the fundamental differential equations. This method has physical transparency and can be successfully applied in other branches of theoretical physics as well.

2. Adiabat

Before passing to the equations themselves, we must agree on the properties of the medium in which the acoustic waves under consideration propagate. We shall assume air of uniform density \(\rho_0\), which, during the passage of waves, changes somewhat and assumes at different places and at different times different values \(\rho\); in exactly the same way the pressure \(p_0\) may change into \(p\); we shall take these changes to be adiabatic, applying Poisson’s law:

\[ \frac{p}{p_0}=\left(\frac{\rho}{\rho_0}\right)^k . \]

Here \(k\) is the ratio of the heat capacity of the gas at constant pressure to the heat capacity at constant volume.

The application of the adiabatic law to acoustic waves is confirmed by direct experiments on the velocity of propagation of sound and is explained by the fact that, in rapid acoustic oscillations and at comparatively small amplitudes, the heat of compression of the gas does not have time to be transferred by thermal conductivity.

For what follows it will be convenient for us to introduce the quantity of relative compression:

\[ s=\frac{\rho-\rho_0}{\rho_0};\qquad \rho=\rho_0(1+s). \]

In this case Poisson’s law will be written as:

\[ p=p_0(1+s)^k =p_0\left[1+ks+k(k-1)\frac{s^2}{2}+\ldots\right]. \]

Below we shall see that the speed of propagation of sound is determined by the formula:

\[ C=\sqrt{-\frac{dp}{d\rho}}, \]

and, consequently, under adiabatic compression we obtain:

\[ C=\sqrt{\frac{p_0 k}{\rho_0}\cdot\left(\frac{\rho}{\rho_0}\right)^{\frac{k-1}{2}}}= \]

\[ =C_0\left[1+\frac{k-1}{2}s+\frac{k-1}{2}\cdot\frac{k-3}{2}\cdot\frac{s^2}{2}+\ldots\right]. \]

Thus, for small amplitudes of the relative compression \(s\) (i.e. in the first approximation), we may use the formulas:

\[ p=p_0(1+ks), \]

\[ C=C_0=\sqrt{\frac{p_0 k}{\rho_0}}. \]

The last of the formulas written above was obtained by Laplace and is confirmed by experiment:

\[ C_0=\sqrt{\frac{p_0 k}{\rho_0}}=332\ \text{m/sec}. \]

Whereas Newton’s formula (isothermal compression)

\[ C_0=\sqrt{\frac{p_0}{\rho_0}}=280\ \text{m/sec} \]

does not agree with experiment.

It is necessary, however, to bear in mind that for large amplitudes neither the one nor the other formula describes the observed phenomena with sufficient accuracy.

3. Linearization of the Equations

We shall study phenomena occurring in a plane acoustic wave, whose plane is parallel to the coordinate plane \(OYZ\) and in which all motions occur parallel to the \(+OX\) axis. Let us denote by \(u\), \(p\), \(\rho\) the velocity of motion, the pressure, and the density of a certain element of the volume of air, located at time \(t\) at the point \(x\). For this case aerodynamics gives us two Euler equations:

Equation of momentum:

\[ \frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x} = -\frac{1}{\rho}\frac{\partial p}{\partial x} = -\frac{1}{\rho}\frac{dp}{d\rho}\frac{\partial \rho}{\partial x} = -\frac{C^2}{\rho}\frac{\partial \rho}{\partial x}. \]

The continuity equation:

\[ \frac{\partial \rho}{\partial t}+u\frac{\partial \rho}{\partial x} = -\rho\frac{\partial u}{\partial x}. \]

Substituting into these equations the formulas of the preceding paragraph, we obtain, in the first approximation, the linearized equations:

\[ \frac{\partial u}{\partial t}=-C_0^2\frac{\partial s}{\partial x}; \]

\[ \frac{\partial s}{\partial t}=-\frac{\partial u}{\partial x}. \]

Differentiating the first equation with respect to \(t\), and the second with respect to \(x\), we can easily eliminate the quantity \(s\) and obtain for \(u\) the differential equation of the second order:

\[ \frac{\partial^2 u}{\partial t^2}=C_0^2\frac{\partial^2 u}{\partial x^2}, \]

whose solution has been well known since the time of d’Alembert:

\[ u=F_1\left(t-\frac{x}{C_0}\right)+F_2\left(t+\frac{x}{C_0}\right), \]

where \(F_1\) and \(F_2\) are functions of arbitrary form; they may be chosen in accordance with the other conditions of the problem. D’Alembert’s solution represents two waves propagating along the axis \(\pm OX\) in opposite directions, but with equal velocities.

4. Harmonic Waves

Let us consider the case when, at the origin of coordinates, a harmonic oscillation of the form

\[ u_0=A\sin\frac{2\pi}{T}t, \]

is prescribed, where \(A\) is the amplitude and \(T\) is the period of oscillation. From this oscillation, waves propagate along the positive axis \(+OX\):

\[ u=A\sin\frac{2\pi}{T}\left(t-\frac{x}{C_0}\right). \]

This expression is nothing other than the first term of d’Alembert’s solution. Let us return again to the continuity equation, writing it for the present case in linearized form:

\[ \frac{\partial s}{\partial t}=-\frac{\partial u}{\partial x}. \]

Substituting here the value of the derivative \(\dfrac{\partial u}{\partial x}\) and integrating with respect to \(t\), we obtain:

\[ s=\frac{A}{C_0}\sin\frac{2\pi}{T}\left(t-\frac{x}{C_0}\right). \]

And, taking into account the expression for \(u\), we see that between the velocity of motion of an element of the volume of air and its density there is the relation:

\[ s=+\frac{u}{C_0}. \]

For a traveling wave going along the negative axis \(-OX\), we obtain in the same way the relation:

\[ s=-\frac{u}{C_0}. \]

Substituting these relations into the pressure formula, we obtain the relation between pressure and velocity:

\[ p=p_0\left(1\pm k\frac{u}{C_0}\right). \]

5. Small and large amplitudes

It is asked in what cases we can use the linearized equations and first-order formulas. To resolve this question we must turn to experiment; experiment shows that for the strongest sounds used in music (fortissimo), the air pressure oscillates within the limits from 10 to 50 bar*. Consequently, the ratio of the excess pressure \((p-p_0)\) of the air to the normal pressure \(p_0\) in strong sounds is approximately equal to:

\[ \frac{p-p_0}{p_0}=\text{from }10^{-5}\text{ to }5\cdot 10^{-5}\,\mathrm{atm}. \]

The formulas of the preceding paragraph show that the magnitude of the compression \(s\) and the magnitude of the ratio of the velocity of motion of the air to the velocity of propagation of sound will be of the same order:

\[ \frac{p-p_0}{p_0}=k\frac{u}{C_0};\qquad s=\frac{u}{C_0}. \]

Proceeding from this, we shall agree to regard amplitudes as small if they give ratios less than

\[ r_1=10^{-4}. \]

* One bar is equal to a pressure of one dyne per square centimeter, or approximately one millionth of a normal atmosphere: \(10^{-6}\,\mathrm{atm}\).

We shall call this quantity a quantity of the first order of smallness. In that case the quantity

\[ r_2 = r_1^2 = 10^{-8} = 10^{-4} r_1 \]

will be a quantity of the second order. If the experiment or observation that we carry out does not possess an accuracy of \(10^{-4}\), then quantities of the second order of smallness may be neglected in comparison with quantities of the first order; this is what is usually done in theoretical acoustics. Further, the quantity

\[ r_3 = r_1^3 = 10^{-12} = 10^{-8} r_1 \]

will, under our condition, be a quantity of the third order, and so on.

But, on the other hand, experience shows that the sensitivity of our ear is limited and that pressure oscillations smaller than \(10^{-9}\) atm are not perceived at all by our hearing.

On the basis of these data we may say that in the theory of most acoustic phenomena we may confine ourselves to the first approximation, i.e. use linearized equations. However, in some cases phenomena of the second order will also be audible, since nevertheless

\[ 10^{-8} > 10^{-9}. \]

But phenomena caused by amplitudes of the third order of smallness will not only be inaudible \((10^{-12} < 10^{-9})\), but will even be beyond the sensitivity limits of the most sensitive modern acoustic instruments.

However, in laboratory experiments we can produce oscillations whose amplitude is 100 and even 1,000 times greater. But such strong sounds become unbearable and may even be dangerous to our hearing. Finally, explosions, which produce so-called shock waves—waves which, in the stricter sense, no longer belong to acoustics—may have still larger amplitudes, but we shall not consider them here.

6. A Traveling Wave of Large Amplitude

Formulas for a traveling acoustic wave of large amplitude were first given by Poisson (1808), then by Earnshaw (1859); but the general solution of the equations for both traveling and standing waves was first given by Riemann (1859). Riemann’s method of integration has at present acquired great importance not only for one-dimensional acoustic waves, but in general for the theory of equations with partial derivatives.

We must omit the mathematical derivations here, but shall try to present the results in the most transparent form (for a physicist).

We need to know, first, the velocity of propagation of the passing waves of large amplitude and, second, the relation between the velocity of propagation and the velocity of motion \(u\). Both of these relations we can obtain from the following (schematic) considerations.

We have already said (§ 2) that the velocity of propagation of waves in quiescent air is determined by the formula:

\[ C=\sqrt{\frac{dp}{d\rho}}. \]

But during the passage of a wave the air is no longer quiescent, and its parts have velocities of motion; hence it follows that the velocity of propagation of sound relative to immobile space must be taken equal to:

\[ C_1=C+u. \]

For waves propagating in the opposite direction (along \(-OX\)), we shall have the velocity:

\[ C_2=C-u. \]

Next let us imagine that a sound wave is already propagating in the air, and that at some point of it we have the velocity of an elementary volume of air \(u\) and density \(\rho\). Upon this wave we mentally superpose an additional wave of very small amplitude \(du\) and, correspondingly, of additional density \(d\rho\). We already know that for waves of small amplitudes propagating along the \(+OX\) axis there is the relation (§ 4):

\[ u=C_0s=C_0\frac{\rho-\rho_0}{\rho_0}. \]

For our additional wave this relation becomes

\[ du=C\frac{d\rho}{\rho}. \]

But if the wave of large amplitude now under consideration, taken as a whole, with all its parts, propagates only in the \(+OX\) direction, then the written relation must also hold for all parts of the wave. Therefore, integrating this relation, we obtain for the entire wave of large amplitude the relation:

\[ u=\int_{\rho_0}^{\rho} C_0\left(\frac{\rho}{\rho_0}\right)^{\frac{k-1}{2}}\frac{d\rho}{\rho} =\frac{2C_0}{k-1}\left[\left(\frac{\rho}{\rho_0}\right)^{\frac{k-1}{2}}-1\right]. \]

Comparing the result obtained with the formula for the velocity \(C\), we obtain the exceedingly simple relation:

\[ u=\frac{2}{k-1}(C-C_0), \]

whence

\[ C=C_0+\frac{k-1}{2}u. \]

Now we can express the velocity of propagation of a large-amplitude wave by the following formulas.

For a wave traveling in the positive direction:

\[ C_1=C+u=C_0+\frac{k+1}{2}u. \]

For a wave traveling in the opposite direction:

\[ C_2=C-u=C_0+\frac{k-3}{2}u. \]

Thus, for a harmonic wave of large amplitude propagating along \(+OX\), we have the following formulas:

\[ u=A\sin \frac{2\pi}{T}\left(t-\frac{x}{C_0+\frac{k+1}{2}u}\right); \]

\[ \frac{\rho}{\rho_0}=1+s=\left(1+\frac{k-1}{2}\frac{u}{C_0}\right)^{\frac{2}{k-1}}, \]

and the adiabatic law:

\[ \frac{p}{p_0}=\left(\frac{\rho}{\rho_0}\right)^k =\left(\frac{\vartheta}{\vartheta_0}\right)^{\frac{k}{k-1}}. \]

All these formulas are completely exact, as may be verified by substituting them into the fundamental equations of aerodynamics; and in the following paragraphs we shall consider what phenomena may be expected in traveling waves of large amplitude, and in what respects these waves differ from the acoustic waves usually considered in acoustics of the first approximation.

7. Deformation of Waves

The formula for the velocity of propagation of a wave,

\[ C_1=C_0+\frac{k+1}{2}u, \]

shows us that the magnitude of this velocity is not the same for different parts of the wave. Only the zero points of the wave, where \(u=0\), propagate with the normal velocity according to Laplace’s formula:

\[ C_0=\sqrt{\frac{p_0 k}{\rho_0}}. \]

In those places where the velocity \(u\) is positive, i.e. directed in the direction of propagation of the wave, the velocity of propagation is greater than \(C_0\), and conversely, where the velocity \(u < 0\), the velocity of propagation \(C < C_0\). But if different parts of the wave have different velocities, then the wave, as it moves forward, must change its form; if at the beginning, at the very source of the waves, the oscillations and the wave had a sinusoidal form, then at some distance \(x\) from this source this will no longer be the case. The wave will be deformed.

Fig. 1. Wave deformation

In order to trace this deformation most clearly, we may apply the following graphical method. Let us draw, on the coordinate plane \(OXu\), the distribution of velocities in the wave under consideration in the form of a sinusoid (Fig. 1) and mentally impart to the whole wave a motion in the opposite direction with velocity \(C_0\); then all zero points of the wave (where \(C_1 = C_0\)) will remain in place, while the other points will have a relative motion with velocity:

\[ \frac{k - 1}{2}\,u. \]

This relative velocity turns out to be proportional to \(u\); we may therefore obtain the deformed wave by carrying out a simple displacement (cf. the theory of elasticity) of the upper parts of the wave to the right, and of the lower parts to the left, relative to the zero line, which remains at rest. The magnitude of this displacement (which, after all, occurs with uniform velocity) will be the greater, the greater the distance \(x\) the wave has traversed in the time under consideration. Thus we shall successively obtain wave forms (distributions of velocities) represented in Fig. 1 by the curves \([Oa_1O_1a_2O_2]\), \([Ob_1O_1b_2O_2]\), and so on.

8. Absorption of Waves

From the mathematical point of view, the deformation of a wave indicated in the preceding paragraph entails a certain difficulty, which was pointed out by Stokes (1848). Indeed, if the wave continues to progress farther and farther and at the same time is deformed in the indicated sense, then there will come a moment when the positive parts of the wave overtake the zero point, while the negative parts of the wave lag behind it, and we shall obtain on

in the drawing an S-shaped curve, in which three different ordinates will correspond to one and the same abscissa (for example, \(O_1\)) (see the curve \(|Oc_1O_1c_2O_2|\)). This would mean that at one and the same point we have three different velocities of motion \(u\): a result that has no physical meaning.

A similar phenomenon is possible for (longitudinal-transverse) waves propagating on the surface of water, and is indeed observed in the surf of sea waves at the shore, when the crests of the waves, running ahead, curl over and form foamy vortices. Something similar to our Fig. 1 is obtained (the curve \(Oc_1O_1c_2O_2\)).

But what can happen in such cases in acoustic (purely longitudinal) waves?

The point is that our original equations are not complete; they do not take into account a whole series of circumstances that occur in reality, namely: the internal friction of air, the phenomenon of heat conduction, diffusion, the formation of vortices, etc. All these phenomena cause a scattering of the energy of waves and appear in the phenomenon of absorption of waves. The influence of friction and heat conduction can be introduced into the linearized equations (Stokes, Helmholtz, Kirchhoff), and as a result one obtains a decrease in the amplitude of waves according to the law:

\[ e^{-rx}, \]

where the coefficient \(r\) turns out to be proportional to the square of the frequency of the oscillations. Consequently, high sounds must be absorbed much more strongly than low sounds, which is confirmed by experience. However, the experiments of N. Neklepaev (1911), carried out in P. N. Lebedev’s laboratory in Moscow, showed that acoustic waves of length \(2.5\) mm and \(0.8\) mm, i.e. waves of very high frequencies, \(130\) thousand to \(400\) thousand oscillations per second, are absorbed in air much more strongly than theory requires. Pierce (1925) confirmed Neklepaev’s result; in Pierce’s opinion, this anomalous absorption must be attributed to the presence of carbon dioxide in the air, and he showed that carbon dioxide does indeed strongly absorb short acoustic waves. But it is very possible that Neklepaev’s waves should be classed as waves of large amplitude, for which the linearized equations, even with the addition of friction and heat conduction, proved insufficient. If, however, we introduce all these factors into the more exact equations of aerodynamics, their complexity will increase so much that Riemann’s method will be of little help to us.

Let us try, nevertheless, to clarify, even without detailed calculations, what happens in waves of large amplitude. We have already found that, as the wave moves forward, it must deform, and the front side of its positive part (near the point \(O_1\), Fig. 1) will become steeper and steeper; because of this, volume elements of air with large differences in pressure and temperature will be located closer and closer to one another, and the influence

the thermal conductivity must increase; diffusion will act in the same direction, not to mention the fact that the law of the adiabat loses its force. In addition, the phenomenon of thermal conductivity and diffusion will smooth out these differences, and the steepness of the wave must decrease. Thus, in waves of large amplitude we have two causes acting simultaneously in opposite directions: on the one hand, the difference in the velocities of propagation of the separate parts of the wave increases the steepness of its forward descent; on the other hand, this steepness is smoothed out by thermal conductivity. When these two causes come into equilibrium, the steepness of the curve will reach its maximum value, and with the further motion of the wave it will even begin to decrease. Thus the multivaluedness of the function \(u\), to which the incomplete theory leads, is in reality eliminated of itself.

We have already said that the differential equations of such an extended theory become extremely complicated; nevertheless, Becker (1921) succeeded in determining the form acquired by the wave when its steepness has reached the extreme limits. It turns out that this maximum steepness of the wave, under ordinary experimental conditions, becomes so considerable that it may already be taken as a discontinuity; in other words, the tangent at the point \(O_1\) (Fig. 1) may be taken as vertical. Such waves, in which the transition from positive values of the quantities \(u, p, \rho\) to negative ones occurs over a negligible length (discontinuously), are called shock waves. We thus arrive at the conclusion that every wave, including a wave with harmonic oscillations, such as we usually consider in acoustics, in its forward motion gradually turns into a shock wave. Riemann, having foreseen this, took up shock waves and gave their theory, which was later improved by Hugoniot and other scholars. At present the theory of shock waves has also acquired significance in explaining the propagation of explosions and detonation (see A. Eichenwald, “Theoretical Physics,” Part IV, Ch. XV).

9. Vieille’s Experiments

The dependence of the velocity of propagation of acoustic waves, and especially of shock waves, on amplitude had already been observed long ago (Regnault, Tyndall, and others). However, the most interesting in this respect must be acknowledged to be the experiments recently performed by Vieille. Vieille made use of a newly constructed water main in Lyon, not yet filled with water. The water main had a diameter of \(1\ \text{m}\) and a length of \(1\,660\ \text{m}\). At one end of this water main small explosions of gunpowder and other explosive substances were produced, and at some distance from this place an instrument was inserted into the pipe of the water main, based on the interference of light and allowing the pressure (density) of the air to be recorded during the passage of the wave at this

of the instrument. Although the water conduit was only \(1^{1/2}\) km long, multiple reflections of waves from the ends of the conduit made it possible to record the form of one and the same wave several times, even after it had traveled a distance of 15 km. Vieille’s experiments are so instructive that I consider it useful to present here one of his numerous diagrams (Fig. 2), prepared both by him and by his collaborators.

The first of the curves (Fig. 2) represents the form of the wave at the very beginning of its appearance; in the explosion of black powder the greatest pressure obtained was 10 atm. In the drawing the wave should be imagined as moving from right to left. The second curve represents the same

Fig. 2. Vieille’s experiments

wave, but after it has already traveled a distance of 3,224 m. Here it is clearly visible how the wave has become deformed: its front (left-hand) part has become much steeper. In the next image we then see that the steepness of the front part of the wave has increased so much that the wave may be regarded as a shock wave.

Simultaneously with the deformation of the wave, its absorption also took place. In the first drawings the absorption is revealed only in the general decrease of the amplitude; however, in the fourth drawing (after the wave has traveled 6,505 m) we see that the steepness of the front part has also decreased; and in general the absorption has increased.

Thus the whole phenomenon proves to be clarified both in theory and in experiment.

10. Transition of a Harmonic Wave into a Shock Wave

Let us return again to our initial equations and determine the distance \(x_m\) at which the steepness of the wave becomes maximal, i.e., for which we may set

\[ \frac{\partial u}{\partial x}=\infty . \]

For greater clarity we shall assume the initial form of the wave to be a harmonic oscillation and write:

\[ u=A\sin\varphi =A\sin\frac{2\pi}{T} \left[ t-\frac{x}{C_0+\frac{k+1}{2}u} \right]. \]

Computing the derivative with respect to \(x\) of this expression and taking into account, of course, that the quantity \(u\) also enters under the sign of \(\sin\), we obtain:

\[ \frac{\partial u}{\partial x} \left[ 1+A\cos\varphi\cdot\frac{2\pi}{T}\cdot \frac{k+1}{2C_0^2}\cdot x \right] = A\cos\varphi\cdot\frac{2\pi}{C_0}. \]

In order that the slope of the line be vertical

\[ \left(\frac{\partial u}{\partial x}=\infty\right), \]

it is necessary first of all that \(\cos\varphi\) be negative; this means that the maximum steepness should be expected in the front part of the wave (in Fig. 1 at the point \(O_1\)). We obtain the maximum steepness by putting

\[ \cos\varphi=-1, \]

\[ \left[ 1-A\frac{2\pi}{T}\cdot\frac{k+1}{2C_0^2}\cdot x_n \right]=0. \]

Hence it is determined that:

\[ x_m=\frac{C_0}{A}\cdot\frac{C_0T}{2\pi}\cdot\frac{2}{k+1}. \]

Thus, for example, for the sound of a whistle at 1,000 oscillations per second

\[ (T=10^{-3}; \text{ high up to soprano}) \]

and for a sound intensity corresponding to the ratio:

\[ \frac{A}{C_0}=10^{-3}, \]

we should obtain a shock wave at a distance

\[ x_m=\text{approximately }2\text{ kilometers}, \]

if absorption phenomena are not taken into account. This result also closely agrees with the experiments of Vauthier, although Vauthier did not perform experiments with harmonic waves.

From this (approximate) calculation we see that the higher the sound produced and the greater its amplitude, the sooner the wave of such a sound must transform into a shock wave.

11. Change in the Timbre of Sound

A change in the shape of a wave, from the musical-acoustical point of view, is equivalent to a change in the timbre of the sound. Let us show, indeed, that in a purely harmonic wave

\[ u=A\sin\frac{2\pi}{T}\left[t-\frac{x}{C_0+\frac{k+1}{2}u}\right] \]

an octave is first of all clearly detected. To this end we expand this expression in a series in ascending powers of the ratio \(\dfrac{A}{C_0}\). Bearing in mind that in this series we shall stop at terms of the second order of smallness, we may, under the sine sign, confine ourselves to terms of the first order of smallness and write:

\[ u=A\sin\frac{2\pi}{T}\left[t-\frac{x}{C_0}+x\frac{k+1}{2}\frac{u}{C_0}\right]= \]

\[ =A\sin\left[\varphi+x\frac{2\pi}{T}\cdot\frac{k+1}{2}\cdot\frac{u}{C_0}\right]. \]

In expanding this formula, limiting ourselves to the second approximation, we may conveniently use the relation:

\[ \sin(\varphi+\alpha)=\sin\varphi\cdot\cos\alpha+\cos\varphi\cdot\sin\alpha, \]

putting in it:

\[ \cos\alpha=1;\qquad \sin\alpha=\alpha. \]

Then we obtain:

\[ u=A\sin\varphi+x\cdot\frac{2\pi}{T}\cdot\frac{k+1}{4}\left(\frac{A}{C_0}\right)^2\sin2\varphi. \]

With this approximation we have indeed obtained, besides the fundamental sound, an additional sound of double frequency, i.e. the octave of the fundamental sound. The amplitude of this octave is proportional to \(x\), and consequently will gradually increase as the sound wave moves forward. With further expansion of the series we would obtain sounds of triple, quadruple, etc., frequencies. In general we may say that during the propagation of sound its timbre changes. A musician would say that the farther one stands from the source of sound, the brighter the sound seems.

The sounds of higher frequencies that appear in this process must be counted among the so-called combination tones. In order that this should become clearer, let us suppose that the source of sound simultaneously gives two simple tones:

\[ u_0=A\sin(mt)+B\cdot\sin(nt). \]

In the plane wave emanating from this source, we obtain:

\[ u=A\sin m\left[t-\frac{x}{C_0+\frac{k+1}{2}u}\right]+B\sin n\left[t-\frac{x}{C_0+\frac{k+1}{2}u}\right]. \]

Here the quantity \(u\), standing under the sine signs, contains both frequencies of oscillation \(m\) and \(n\), and therefore, expanding this expression in a series, we obtain a formula of the form:

\[ u=A\sin\varphi+B\sin\psi+ \]

\[ +x\{A_1\sin2\varphi+B_1\sin2\psi+C_1\sin(\varphi+\psi)+C_2\sin(\varphi-\psi)\}\ldots, \]

which indicates to us the appearance of sum and difference combination waves.

The octave obtained above by us is also a combination sum tone according to the formula:

\[ 2\varphi=\varphi+\varphi. \]

It is interesting to determine what the amplitude of this octave that has appeared will be when the wave reaches its maximum steepness. For this it is sufficient to substitute into the formula

\[ A_1=x\frac{2\pi}{T}\frac{k+1}{4}\left(\frac{A}{C_0}\right)^2 \]

the quantity \(x\) obtained by us in the preceding paragraph:

\[ x_m=\frac{C_0}{A}\frac{C_0T}{2\pi}\frac{2}{k+1}. \]

We obtain:

\[ A_{1,m}=\frac{1}{2}A. \]

As we see, the octave can grow so much that its amplitude becomes equal to half the amplitude of the initial tone. It must not be forgotten, however, that all such calculations are far from exact and can serve us only for general orientation in the phenomenon we are studying.

12. Euler and Lagrange Coordinates

The equations of hydrodynamics, as is known, may be represented either in ordinary coordinates \(x, y, z\) (as has been done here also in § 3), determining the position of a point relative to fixed space (local Euler coordinates), or else one may take as independent variables those coordinates \(a, b, c\) which the element of volume of the substance under consideration occupied at some definite instant of time \(t_0\) (substantial, material Lagrange coordinates). The difference between these two coordinate ...

is reflected both in the formulas themselves and in their interpretation. But for first-order theoretical acoustics this difference is of no significance.

Suppose that we have some (material) plane \(OYZ\), for example, a telephone membrane, oscillating harmonically in the direction of the axis \(\pm OX\) according to the law:

\[ u_{a,o}=A\sin \frac{2\pi}{T}t . \]

The displacement of this membrane will be expressed by the formula:

\[ y_{a,o}=-A\frac{T}{2\pi}\cos \frac{2\pi}{T}t . \]

From this oscillating membrane there will travel along the axis \(\pm OX\) acoustic (longitudinal) waves, which we shall take to be plane (as though they were propagating inside a tube, and not diverging in all directions).

The formula for a plane wave in Lagrangian coordinates is obtained in exactly the same way as in Eulerian coordinates; only the magnitude of the propagation velocity \(C_a\) comes out somewhat different than in Euler’s coordinates, but at present this difference is of no significance for us, and we may even write the wave formula as in the first approximation, putting \(C_1=C_0\):

\[ u_a=A\sin \frac{2\pi}{T}\left(t-\frac{C_1}{C_0}\right). \]

For the deviations of the oscillating points from their equilibrium position we therefore have the formula:

\[ y_a=-A\frac{T}{2\pi}\cos \frac{2\pi}{T}\left(t-\frac{a}{C_0}\right). \]

If we now wish to pass from Lagrangian coordinates to Eulerian coordinates, then we must take into account that each of the oscillating planes in the wave, which at time \(t_0\) occupied the position \(a\), at some other time \(t\) will occupy the position:

\[ x=a+y_a . \]

Substituting the value of \(a\) thus obtained into the wave equation, we obtain its expression in Eulerian coordinates:

\[ u=A\sin \frac{2\pi}{T}\left(t-\frac{x-y}{C_0}\right). \]

Expanding this expression in a series and restricting ourselves to second powers, we obtain:

\[ u_x=A\sin \varphi-\frac{A^2}{C_0}\cos^2\varphi =A\sin\varphi-\frac{A^2}{2C_0}-\frac{A^2}{2C_0}\cos 2\varphi . \]

This formula shows us that at each fixed point \(x\) of space the oscillations of the air will be accompanied by an octave (and, in general, by combination tones), whereas the wave motion of the air itself:

\[ u_a = A\sin \frac{2\pi}{T}\left(t-\frac{a}{C_0}\right) \]

is quite symmetric and contains only one fundamental tone of period \(T\).

This phenomenon was apparently observed by many experimentalists, such as, for example, Lindig (1903), who carefully investigated the appearance of an octave near the vibrating prong of a tuning fork, whereas the tuning fork itself did not produce an octave. Lindig called the sounds he discovered “asymmetric tones,” explaining their origin by the difference in the motion of the air during the half-period when the prong of the tuning fork displaces the air and when it returns again to its place. But if one takes into account that Lindig listened to the sounds with the aid of an ear tube, one end of which was inserted into the ear, while the other end was placed near the vibrating prong of the tuning fork at various fixed points of space, then this circumstance alone, according to what has been set forth above, would have been sufficient for hearing not only an octave, but also a duodecima (the fifth from the octave, a combination tone of the third order), whose oscillations are perfectly symmetric. Even earlier than Lindig, Stumpf (1895) investigated the sounds of a large tuning fork (100 oscillations per second) at very large amplitudes of oscillation and could hear not only the octave, but also all multiple oscillations up to the fourth octave inclusive. From the point of view of the theory presented here these were already sounds of the sixteenth order!—they evidently had an entirely different origin.

13. Mean quantities.

In comparing theory with experiment, it is often necessary to determine the time averages of various quantities. For calculating these averages it will be convenient for us to have expansions in series, since we have decided to restrict ourselves to the second approximation (cf. § 6):

\[ \rho-\rho_0 = \rho_0 \left[ \frac{u}{C_0} + \frac{3-k}{4} \left(\frac{u}{C_0}\right)^2 +\ldots \right], \]

\[ p-p_0 = kp_0 \left[ \frac{u}{C_0} + \frac{k+1}{4} \left(\frac{u}{C_0}\right)^2 +\ldots \right], \]

\[ \vartheta-\vartheta_0 = (k-1)\vartheta_0 \left[ \frac{u}{C_0} + \frac{k-1}{4} \left(\frac{u}{C_0}\right)^2 +\ldots \right]. \]

Determining mean quantities in time, we shall therefore have to calculate the values of the integrals:

\[ \overline{u}^{\,t}=\frac{1}{T}\int_{0}^{T}u\cdot dt;\qquad \overline{u^{2}}^{\,t}=\frac{1}{T}\int_{0}^{T}u^{2}\cdot dt. \]

Since we are considering a wave of harmonic form, the second of these integrals, representing the mean of the squares of harmonic oscillations, will be equal to one half of the square of the amplitude \(A\);

\[ \overline{u^{2}}^{\,t}=\frac{1}{2}A^{2}. \]

As for the first integral, it will have different values depending on whether we determine substantial or local time means. In the first case, i.e. with respect to the oscillating air, the integral

\[ \overline{u}_{a}^{\,t}+\frac{1}{T}\int_{0}^{T}A\sin\frac{2\pi}{T}\left(t-\frac{a}{C_{0}}\right)\cdot dt=0 \]

is equal to zero, since the oscillations are completely symmetric. Meanwhile, calculating the integral in Eulerian coordinates, we obtain (see § 12):

\[ \overline{u}_{x}^{\,t}=\frac{1}{T}\int_{0}^{T}\left[A\sin\varphi-\frac{A^{2}}{C_{0}}\cos^{2}\varphi\right]dt=-\frac{A^{2}}{C_{0}}, \]

since the velocity function \(u_x\) with respect to local coordinates is asymmetric.

Taking this remark into account and, in the calculations, restricting ourselves to quantities of the second order of smallness, we obtain the time mean values for all three quantities of interest to us, as compared in the following table:

	\(a=\mathrm{const.}\)	\(x=\mathrm{const.}\)
\(\overline{\rho-\rho_{0}}^{\,t}\)	\(+\dfrac{3-k}{8}\rho_{0}\left(\dfrac{A}{C_{0}}\right)^{2}\)	\(-\dfrac{k+1}{8}\rho_{0}\left(\dfrac{A}{C_{0}}\right)^{2}\)
\(\overline{p-p_{0}}^{\,t}\)	\(+\dfrac{k+1}{8}kp_{0}\left(\dfrac{A}{C_{0}}\right)^{2}\)	\(-\dfrac{3-k}{8}kp_{0}\left(\dfrac{A}{C_{0}}\right)^{2}\)
\(\overline{\vartheta-\vartheta_{0}}^{\,t}\)	\(+\dfrac{k+1}{8}(k-1)\vartheta_{0}\left(\dfrac{A}{C_{0}}\right)^{2}\)	\(-\dfrac{5-k}{8}(k-1)\vartheta_{0}\left(\dfrac{A}{C_{0}}\right)^{2}\)

It is interesting to note that the values of the mean quantities with respect to substantial and with respect to local coordinates differ from one another not only in magnitude, but also in sign: the substantial means are in general positive, whereas the local means of the same quantities are negative.

If we focus our attention on the substantial quantities, we obtain the following result. In sound waves of large amplitude, the mean pressure and mean temperature are slightly higher than the normal pressure and the normal temperature of the air in the absence of sound. It is true that this difference is insignificant and for strong sounds has a relative magnitude of only \(10^{-6}\) (one millionth). Nevertheless, with modern experimental means such differences in temperature are accessible to observation and measurement. Moreover, all the means obtained by us, both for substantial and for local (Eulerian) coordinates, are of the same order of magnitude as the sound pressure that was observed and measured, not in progressive but in standing waves, by Altberg, Zernov, and others (see below).

14. Acoustic wind

Let us compute the flow of air formed in an acoustic wave. The mass of air passing through a square centimeter placed perpendicular to the direction of propagation of the wave, in each second, is determined from the formula:

\[ \rho \cdot u_x = \rho_0 \left[ 1+\frac{k-1}{2}\frac{u}{C_0} \right]^{\frac{2}{k-1}} = \rho_0 u_x+\rho_0\frac{u_x^2}{C_0}. \]

If the quantity \(u_x\) oscillated symmetrically with respect to the fixed point of space \(x\), then we would obtain:

\[ \overline{u_x}^{\,t}=0; \qquad \overline{u_x^2}^{\,t}=\frac{1}{2}A^2; \]

the time-mean flow of air would be equal to:

\[ \overline{\rho u_x}^{\,t} = +\rho_0\frac{A^2}{2C_0}; \]

i.e., we would obtain a one-sided motion of the air, which could be called an “acoustic wind.” But we have already seen in § 13 that waves issuing from a symmetrically oscillating source are expressed, with respect to Eulerian coordinates, by an asymmetric function, and that in the formula written we must put:

\[ \overline{u_x}^{\,t} = \operatorname{av.} \left[ A\sin\varphi-\frac{A^2}{C_0}\cos^2\varphi \right] = -\frac{A^2}{2C_0}. \]

And in that case the mean value of the flux is determined by the formula:

\[ \rho \overline{u_x}=-\rho_0 \frac{A^2}{2C_0}+\rho_0 \frac{A^2}{2C_0}=0 . \]

This means that no acoustic wind is obtained! Nevertheless, Hartmann-Kempf (1909) observed an acoustic wind which was produced by the vibrating membrane of a telephone. The membrane was set into vibration by an alternating current with a frequency of 600 oscillations per second. In this case a strong one-sided flow of air was obtained along the axis of the telephone in the direction of wave propagation; at the same time there was also a reverse flow of air, which went around the axial flow from all sides and supplied new quantities of air to the membrane. But from this description we already see that near the vibrating membrane an annular vortex was formed, whose axis coincided with the axis normal to the plane of the membrane. Such vortices can be formed only under the action of friction, which we did not introduce into our initial equations. Similar vortices are also formed in standing waves; they were observed by Dvořák and explained theoretically by Rayleigh.

15. General theory of combination tones

Combination tones have been known since the time of Sorge and Tartini (1754), and their theory was given by Helmholtz (1856). Nevertheless, to this day certain misunderstandings on this subject are encountered in the literature, which should be eliminated; and therefore I allow myself to dwell on this theory in somewhat greater detail.

We have already seen in several examples that the relation between various acoustic quantities is often not linear. Therefore, when one of these quantities, for example the density \(\rho\), oscillates harmonically, producing, for example, two oscillations with frequencies

\[ \frac{2\pi}{T_1}=m,\qquad \frac{2\pi}{T_2}=n, \]

and oscillates according to the law

\[ A\sin(mt)+B\sin(nt), \]

then at the same time some other quantity (for example the pressure \(p\)) will be expressed by a formula with powers:

\[ [A\sin(mt)+B\sin(nt)]^q. \]

But such formulas contain products of the form:

\[ C[\sin(mt)]^\alpha[\sin(nt)]^\beta, \]

where \(\alpha\) and \(\beta\) are integers. And these products, in turn, may be represented in the form of harmonic oscillations with frequencies

\[ \nu=\alpha\cdot m \pm \beta\cdot n . \]

These considerations underlie Helmholtz’s theory: a theory as ingenious as it is simple.

In the particular case when \(m=n\), or when we are dealing with a system that gives one fundamental tone, we obtain combination tones with frequencies:

\[ \nu=(\alpha \pm \beta)n . \]

All these higher tones have frequencies that are multiples of the frequency of the fundamental sound. However, they need not be confused with the ordinary harmonic (also multiple) overtones of vibrating systems (string, pipe, etc.). The harmonic, as well as the inharmonic (non-multiple), overtones of various acoustic apparatus have an entirely different origin and are not due to the nonlinearity of the relations. Unfortunately, however, the term overtones is often applied (especially in electrical engineering) also to such oscillations as are in fact combination oscillations.

Another very widespread misunderstanding consists in seeking the cause of the appearance of combination tones in the asymmetry of the system (cf. § 12). Helmholtz himself gave grounds for this by applying his theory to the formation of combination tones in the tympanic membrane of our ear; the tympanic membrane is indeed asymmetric. However, Helmholtz did not mean thereby to say that combination tones arise only in asymmetric systems. According to the considerations set forth here, we may say that all combination tones of even degree are asymmetric, as, for example:

\[ [A\sin(nt)]^{2}=\frac{1}{2}A^{2}-\frac{1}{2}A^{2}\cdot\cos 2nt, \]

but alongside these there are combination tones of odd degree (or order), which are perfectly symmetric.

What is important is not asymmetry, but the nonlinearity of the relations.

It is otherwise when we compute and observe mean quantities: here asymmetry plays the principal role. In the second approximation, at which we stopped, the mean values were determined by the second powers of small quantities; but, strictly speaking, these mean quantities, such as, for example, the experimentally observed sound pressure, are determined not by the terms of second degree alone, but by the sum of all terms of even degree of the corresponding expression.

16. Relation to Music

In Helmholtz’s theory of musical dissonances, a large role is played by the so-called “beats” of two neighboring ...

sounds. If two such sounds fall upon a nonlinear system, then, in addition to them, we shall obtain a difference (very low) sound and a summation sound (close to the octave). From this two phenomena may arise: first, the timbre of the consonance will change and, second, unpleasant beats may arise in the overtones.

If some system (a musical instrument) produces only integral (so-called harmonic) overtones, then all the combination tones arising from them will also be integral multiples of the fundamental frequency. In this case the combination tones cannot produce any new sounds. Nevertheless they have musical significance, since in the sum they change the amplitudes of the audible overtones, and the perceived timbre of the resultant sound depends on the distribution of the amplitudes of the overtones.

If, however, the system itself gives a series of nonintegral overtones (a tuning fork, membranes), then the combination tones that appear will already differ in their frequencies from the overtones, and the timbre will change much more strongly.

The question has repeatedly been raised whether the timbre of a sound depends on the relative phase of the overtones. We can answer this question in the affirmative in those cases where combination tones are formed. Indeed, let us take, for example, a sound composed of a fundamental tone accompanied by an octave, as often occurs in various musical instruments:

\[ A\cdot \sin (nt)+B\cdot \sin (2nt+\delta). \]

If such a sound falls upon a nonlinear system, then the fundamental tone \(A\) first of all forms an octave, which, when added to the overtone octave \(B\), gives the sum:

\[ \frac{1}{2}A^2\cdot \cos (2nt)+B\cdot \sin (2nt+\delta), \]

depending on the relative phase \(\delta\). Further, the product

\[ 2AB\cdot \sin (nt)\cdot \sin (2nt+\delta) \]

will give a difference sound \((2n-n=n)\) of the same frequency as the fundamental sound, and upon addition we shall obtain the amplitude of the fundamental sound:

\[ A\cdot \sin (nt)+AB\cdot \cos (nt-\delta), \]

also depending on the relative phase of the initial tones \(\delta\).

This simple example already shows us that not only the amplitude, but also the relative phases of the overtones can influence the timbre of the sound we perceive, since the combination sounds depend on this phase. In practice, however, this observation is not of great importance, since there are many other reasons for the appearance of combination tones.

17. Origin of Combination Tones

The places where combination tones can arise are extremely numerous and varied. I shall indicate here only the chief ones.

1. In the majority of cases, the vibrating systems used in practice (in music) possess linear elasticity (the displacement is proportional to the acting force; Hooke’s law). In such systems no combination tones are formed. However, at large amplitudes of vibration, when the elasticity becomes nonlinear (deviations from Hooke’s law), we obtain combination tones. Such combination tones were in fact observed by Helmholtz and many others. Incidentally, in a tuning fork one can hear the octave, although the rod of the tuning fork itself, at small oscillations, gives no harmonic overtones; this octave is the combination tone \(1+1=2\). In sirens, and especially in König-type sirens, where the sound is produced by blowing air against the edge of a rotating disk whose rim is cut according to a sinusoid or some other periodic curve, the motions of the air producing the sound are so complex that combination tones must be formed to a considerable degree; this is also confirmed by experiment. In pipes, when air is blown in, a sound of complex composition is also formed, but it is then filtered by the construction of the pipe itself.

2. In all sources in which oscillations of air are used (organ pipes, clarinets, oboes, physharmonicas, etc.), the air itself serves as the cause of the appearance of combination tones, since the relation between the pressure and the compression of air is nonlinear, namely:

\[ \frac{p}{p_0}=\left(\frac{\rho}{\rho_0}\right)^k . \]

1. Further, when a wave formed by some source of sound propagates in air, we obtain combination sounds as a result of deformation of the wave (§ 11). These combination tones increase as the wave moves forward.

2. In some cases the combination tones may be apparent ones and due to the fact that the source is in oscillatory motion, whereas the receiving apparatus is usually at rest (cf. § 12).

3. Finally, the receiving apparatus itself may have nonlinear elasticity, as, for example, the tympanic membrane of our ear, and also the majority of modern radio receiving apparatus with numerous amplifiers. In such a case the apparatus may reveal oscillations that are entirely absent in the incoming sound wave. To all this must also be added the influence of friction, vortex formation, etc. (§ 14).

We have listed here only the principal causes of the origin of combination sounds, but, as we see, there are so many of these causes that producing a pure harmonic tone, devoid of combination tones, is an experimental task of extraordinary difficulty.

of extraordinary difficulty. In exactly the same way it is sometimes very difficult to indicate whence (§ 12) precisely the observed combination tone has arisen, and this may give occasion to the most varied erroneous judgments (cf. the end of § 12).

In the literature there has arisen several times a dispute as to whether the beats of two sounds can be perceived by us in the form of a difference tone (so thought Young, König, and many others); in exactly the same way there was dispute as to whether the relative phase of two simultaneously heard sounds affects the timbre of the sound perceived by us.

To both these questions there is an exhaustive answer in Helmholtz himself, namely: beats by themselves do not yet give a difference sound; in exactly the same way, phase does not affect the timbre of a sound; but all this is true only in the first approximation. As soon as combination tones appear (nonlinear relations), then at once the beats will give a difference tone, and the phase will have an influence on the timbre of the sound (see § 16).

18. Standing Waves of Small Amplitudes

Integration of the equations of § 3 for the case of standing waves (and in general for two waves traveling simultaneously toward one another), by Riemann’s method, is much more complicated than for a single traveling wave. But since we have already decided to restrict ourselves in our calculations to the second approximation, we can apply the method of successive approximations; moreover, from the physical point of view it is preferable to apply it to the differential equations of the characteristics. Here I must omit the details of the calculation and confine myself to a brief exposition of the general idea.

Suppose that the wave motion given to us is bounded along the axis \(\pm OX\) by two impermeable walls placed at a distance

\[ l=\frac{\lambda}{2} \]

from one another. As is known from acoustics of the first order, between these walls oscillations are possible with frequency

\[ \nu=\frac{1}{T}=\frac{C_0}{\lambda}, \]

where \(\lambda\) denotes the wavelength of the oscillation under consideration. Oscillations with frequencies that are multiples of the one indicated are also possible; but for the time being we shall not take them into account. Standing waves between the two impermeable walls given to us may be regarded as composed of two traveling waves of equal amplitude and period, but traveling toward one another:

\[ u_1=A\sin\frac{2\pi}{\lambda}(x-C_0t), \]

\[ u_2=A\sin\frac{2\pi}{\lambda}(x+C_0t). \]

If we add these two formulas and divide by two, we obtain the velocity distribution in a standing wave:

\[ u=\frac{1}{2}(u_1+u_2)=A\cdot \sin \frac{2\pi}{\lambda}x\cdot \cos \frac{2\pi}{\lambda}t . \]

If, however, we take the half-difference of the same expressions and divide by \(C_0\), then (cf. the relation between \(u\) and \(s\) in § 4) we obtain the distribution of relative compression in a standing wave:

\[ s=\frac{1}{2C_0}(u_2-u_1)=\frac{1}{C_0}A\cdot \cos \frac{2\pi}{\lambda}x\cdot \sin \frac{2\pi}{\lambda}t . \]

These formulas satisfy the conditions given to us. Indeed, at the walls \(x=0\) and \(x=l=\dfrac{\lambda}{2}\) the velocity \(u\) is always equal to zero: at the walls we have nodes of oscillation, while in the middle between the walls we have an antinode. Conversely, the relative compression \(s\) has an antinode at the walls and a node in the middle.

Fig. 3. Standing waves. Fig. 4.

I shall remind the reader of the method of graphical construction of standing waves. On two strips of paper one draws two identical sinusoids and, placing one sinusoid over the other, constructs with the aid of dividers the sums and differences of their ordinates, plotting them each time at the corresponding abscissae. In this way one obtains two resultant curves, which, of course, will again be sinusoids, but now of a different amplitude. Then one shifts one strip of paper to the right, and the other strip is shifted by the same amount to the left, and again repeats the graphical operation of addition and subtraction. Then the strips are shifted to the right and to the left once more, by the same distance, and again the sum and the difference are drawn, and so on. If the shifts are made each time by a length equal to \(1/8\) of the wavelength, then after eight constructions we shall obtain two drawings: Fig. 3, representing the distribution of velocities in a standing wave in various

moments of time separated from one another by \(1/8\) of a period, and Fig. 4, which depicts the distribution of the relative compressions \(s\) at different moments of time (the same moments as in the preceding drawing) in the very same standing wave.

19. Standing waves of large amplitude

Analogously to the way in which we proceeded in calculating standing waves of small amplitude, for large amplitudes as well we may regard standing waves as the result of the simultaneous motion of two traveling waves going toward one another. We shall prescribe the traveling waves in the same form as we did in the preceding paragraph, namely:

\[ u_1=A\sin\frac{2\pi}{\lambda}(x-C_1t)=A\sin\varphi, \]

\[ u_2=A\sin\frac{2\pi}{\lambda}(x+C_2t)=A\sin\psi. \]

It can be verified by direct substitution that each of these formulas fully satisfies the fundamental differential equation (§ 3). Nevertheless, the sum, or difference, of these expressions will no longer satisfy this equation. This is explained by the fact that our equations are nonlinear. From the physical point of view this means that, in the case of two waves, each of them propagates in air already disturbed by the opposing wave; between the two waves there arises a certain interaction. I have calculated this interaction, restricting myself to the second approximation, which, as we have found, is quite sufficient for our purposes (in the same way further approximations can also be obtained). It turns out that the interaction between two opposing waves is manifested in a periodic change of their phase, and in place of the formulas written above, valid for each wave separately, we now obtain the following formulas:

\[ u_1=A\sin\left(\varphi+\frac{k-3}{4}\frac{A}{C_0}\sin bx\cdot \sin mt\right), \]

\[ u_2=A\sin\left(\psi-\frac{k-3}{4}\frac{A}{C_0}\sin bx\cdot \sin mt\right); \]

where

\[ b=\frac{2\pi}{\lambda};\qquad m=\frac{2\pi}{T};\qquad \lambda=C_0T. \]

It is not difficult to verify that these formulas satisfy (in the second approximation) the fundamental differential equations. The expressions standing in the large parentheses, when set equal to some constant,

and are the equations of those characteristics which were discussed in § 1.

The half-sum of these expressions gives us, as in the preceding paragraph, the distribution of velocities in the resulting standing wave:

\[ u=\frac{1}{2}(u_1+u_2). \]

Meanwhile the half-difference

\[ v=\frac{1}{2}(u_2-u_1) \]

does not yet give us directly the distribution of densities \(s\), since for large amplitudes the relations are more complicated; in place of the linear relations of §§ 4 and 18 we now obtain (cf. § 6):

\[ 1+s=\left(1+\frac{k-1}{2}\frac{v}{C_0}\right)^{\frac{2}{k-1}}. \]

Restricting ourselves to the second approximation, we may write (cf. § 13, replacing \(u\) by \(v\)):

\[ s=\frac{v}{C_0}+\frac{3-k}{4}\left(\frac{v}{C_0}\right)^2. \]

But in any case, once we have found the expressions \(u_1\) and \(u_2\) with sufficient accuracy for us, then from the formulas written above we can determine all the remaining quantities in the standing wave as well: the distribution of velocities, densities, pressures, temperatures, and so on.

Expanding the expressions obtained (as we did in § 11) and restricting ourselves to the second approximation, we obtain:

\[ u=\frac{1}{2}(u_1+u_2)=A\sin bx\cdot\cos mt- \]

\[ {}-t\cdot\frac{2\pi}{\lambda}\cdot\frac{k+1}{8}A^2\cdot\sin 2bx\cdot\cos 2mt+ \]

\[ {}+\frac{k-3}{16C_0}A^2\sin 2bx\cdot\sin 2mt; \]

\[ v=\frac{1}{2}(u_2-u_1)=-A\cos bx\cdot\sin mt+ \]

\[ {}+t\cdot\frac{2\pi}{\lambda}\cdot\frac{k+1}{8}A^2\cdot\cos 2bx\cdot\sin 2mt+ \]

\[ {}+\frac{k-3}{16C_0}A^2\cdot\sin^2 bx\cdot\sin^2 mt. \]

Now we can verify that the formulas obtained satisfy not only the basic differential equations, but also the boundary conditions, namely: at the walls \(x=0\) and

\[ x=\frac{\lambda}{2} \]

we again obtain nodes of oscillation and density for the quantities \(v, s, p\), etc.

Putting \(t=0\) in these expressions, we obtain those initial conditions to which our solutions correspond, namely:

\[ u=A\sin\frac{2\pi}{\lambda}x;\qquad v=0. \]

However, in an analogous way one can also find solutions for other boundary and initial conditions.

But here we note one more feature of standing waves of large amplitude: in the second term of each of the formulas there appears a factor \(t\). This means that the form of the wave changes with time; in the progressive wave we had something similar (the factor \(x\)); it, too, changed its form as it advanced forward. As in the progressive wave, so here in the standing wave, what appears first of all is the octave of the fundamental tone, but afterward, of course, other higher combination tones will also appear.

In order to picture more clearly how the standing wave will be deformed with time, we may here too apply the graphical method (§ 18). To simplify matters we may discard the variable phase, since it does not grow with time and constitutes the half-sum and half-difference of two already deformed waves obtained by us, also graphically, by the “shift” method in § 7. Carrying out such a graphical construction with two deformed waves (as in Fig. 1, \(Ob_1O_1b_2O_2\)), and in the wave going to the right taking the inclination to the right, while in the wave going to the left the inclination to the left, we obtain Figs. 5 and 6 for \(u\) and \(v\). Comparing these drawings with the corresponding drawings of the sinusoidal undeformed wave (Figs. 3 and 4), we can trace the phenomenon of deformation of the standing wave. First of all, we are struck by the change in the curve \(u\) (Fig. 5), indicating the appearance of combination tones and, in the first place, of the octave. This conclusion is also confirmed by Fig. 6, which differs from the corresponding Fig. 4 in that in the middle between the walls we have obtained oscillations of the quantity \(v\), and consequently also of the quantities of condensation \(s\) and pressure \(p\). All these quantities have already ceased to form a node in the middle between the walls, as was the case at the beginning of the motion (Fig. 4).

Finally, what we said about the influence of friction and thermal conductivity in progressive waves is applicable here as well, namely: the standing wave will be deformed with time, becoming steeper and steeper (for example, in Figs. 5 and 6); in the end, in place of a sinusoid we shall obtain an almost sharp triangle. Then, besides diminish-

tion of the amplitude, friction and thermal conductivity will smooth out their steepness; the wave will again become more symmetrical with respect to the middle of Fig. 5, until, finally, it dies out altogether.

Of course, as is also evident from our exposition, the drawings we have obtained are only approximate, but they are quite sufficient to give us a general idea of what occurs in standing waves of large amplitude.

Deformed standing waves.
Fig. 5.              Fig. 6.

20. Forced standing waves

In the preceding section we saw that standing acoustic waves, left to themselves, gradually change their form, and in them there appears above all the octave of the fundamental tone (as in traveling waves). The appearance of this octave may be explained as follows. When, at the very beginning of the motion, oscillations of the air occur and the value of the relative compression \(s\) changes, then, owing to the nonlinearity of the adiabatic law:

\[ p=p_0(1+s)^k=p_0\left[1+ks+k(k-1)\frac{s^2}{2}+\ldots\right] \]

in different parts of the standing wave there begins to act a pressure proportional to \(s^2\), with twice the frequency; we obtain, as it were, a new force acting on the waves and pumping them with double frequency. But the double frequency of oscillations has a wavelength half as large, which also fits along the length \(l\) an integral number of times (two times); we must, consequently, obtain the phenomenon of resonance and a gradual increase of the amplitude of the octave proportional to the time \(t\) (if the influence of friction is not taken into account). This is indeed what we found in the formulas. For the same reasons, together with the octave there will begin to grow oscillations of the duodecime and of other combination tones. We might call this phenomenon self-excitation or autoresonance.

It follows further from this reasoning that if we ourselves, by an external action, begin to pump the air between the walls \(x=0\)

and \(x=\dfrac{\lambda}{2}\), but with a period that does not coincide with any of the natural oscillations of the system under consideration, then (owing to the nonlinearity of the adiabat) waves of double, triple, etc. frequency will again appear; but all these oscillations also will not coincide with any of the periods of the natural oscillations of the system, and we shall obtain no resonance. From this we conclude that if the period of the external force does not coincide with any of the periods of the natural oscillations of the system, then the formulas for the forced oscillations of a standing wave will not contain terms with the factor \(t\), and the wave will not be deformed; meanwhile terms with double frequency will nevertheless remain.

On the basis of these considerations we can write the following formulas for forced standing waves:

\[ u=A\sin bx\cdot\cos mt-A_1\sin 2bx\cdot\cos 2mt+ \frac{k-3}{16C_0}A^2\cdot\sin 2bx\cdot\sin 2mt; \]

\[ v=-A\cos bx\cdot\sin mt+A_1\cos 2bx\cdot\sin 2mt+ \frac{k-3}{16C_0}A^2\cdot\sin^2 bx\cdot\sin^2 mt. \]

The coefficients \(A\) and \(A_1\) will depend on the amplitude of the external force and on the ratio of the period of the external force to the period of the natural oscillations of the system. It should not be overlooked, however, that these formulas refer only to forced oscillations, and that we assume that the natural oscillations of the system, if they arose for any reason, have already had time to die out sufficiently.

Starting from these formulas, we obtain the following formulas for the distribution of densities and pressure in forced standing waves:

\[ (\rho-\rho_0)=-\rho_0\frac{A}{C_0}\cos bx\cdot\sin mt+ \rho_0\frac{A_1}{C_0}\cos 2bx\cdot\sin 2mt+ \frac{3-k}{4}\rho_0\left(\frac{A}{C_0}\right)^2\cos 2bx\cdot\sin^2 mt; \]

\[ (p-p_0)=-\rho_0 C_0A\cos bx\cdot\sin mt+ \rho_0 C_0A_1\cos 2bx\cdot\sin 2mt+ \rho_0\left[\frac{k+1}{4}\cos^2 bx-\frac{3-k}{4}\sin^2 bx\right]A^2\cdot\sin^2 mt. \]

21. Mean Quantities

We produce forced standing waves, for example, in the well-known Kundt apparatus. If the Kundt tube is well closed, then the quantity of air that was enclosed in the tube before excitation

waves, must remain unchanged also during the oscillations. From the mathematical point of view this means that the mean density of the air along the entire length of the tube

\[ \bar{\rho}^{\,x}=\frac{1}{l}\int_0^l \rho\,dx=\rho_0 \]

must remain unchanged. The formulas obtained for \((\rho-\rho_0)\) in the preceding paragraph do indeed satisfy this condition, since the mean values of \(\cos bx\) and \(\cos 2bx\) are equal to zero.

Nevertheless, the mean values in time, determined over the time of one or several periods of oscillation, are not equal to zero:

\[ \overline{(\rho-\rho_0)}^{\,t} = \frac{1}{T}\int_0^T(\rho-\rho_0)\,dt = +\frac{3-k}{8}\rho_0\left(\frac{A}{C_0}\right)^2\cos 2bx. \]

Consequently, the oscillation of density in all parts of the wave occurs asymmetrically. The same must be said of the oscillations of pressure and temperature. We shall write here also the formula for the pressure, since the mean pressure can be observed experimentally:

\[ \overline{(p-p_0)}^{\,t} = \left[ \frac{k+1}{8}\cos^2 bx - \frac{3-k}{8}\sin^2 bx \right]\rho_0 A^2 = \]

\[ = \left[ \frac{k+1}{8} - \frac{1}{2}\sin^2 bx \right]\rho_0 A^2. \]

This distribution of the mean pressure along the length \(x\) is shown in our Fig. 7, in which, for the case of air with coefficient \(k=1.4\), we must take as the axis \(p=p_0\) the thick line 1.4, from which the pressures at the various points of the wave must be reckoned. But the same drawing can also serve for reckoning pressures for other values of the coefficient \(k\); it is only necessary correspondingly to raise or lower the axis of zero pressures, as is indicated in our figure.

Fig. 7. Mean values of quantities at various places of a standing wave.

From this figure, and also from our formulas, it follows that at the nodes of oscillation, and therefore also at the reflecting walls, the mean pressure will be equal to:

\[ \text{nodes:}\quad \overline{(p-p_0)}^{\,t} = \frac{k+1}{8}\rho_0 A^2. \]

This quantity is positive, and consequently the pressure at these points will be greater than in the absence of sound.

Meanwhile, from the same formulas it follows that at the antinodes

\[ \left(x=\frac{\lambda}{4}\right) \]

\[ \text{antinodes:}\quad \overline{(p-p_0)}^{\,t}=-\frac{3-k}{8}\rho_0 A^2 \]

the mean pressure will be negative.

The excess pressure at a reflecting wall was first observed by Dvorzhak (1876) and explained theoretically by Lord Rayleigh (1878). Applying the method of approximation to the fundamental equations, Rayleigh expressed the mean pressure at a reflecting wall by the formula:

\[ \overline{(p-p_0)}^{\,t}=\frac{k+1}{2}E, \]

where \(E\) denotes the mean energy density of the entire standing wave in time:

\[ E=\frac{1}{\lambda T}\int_0^\lambda \int_0^T E_0\,dx\,dt. \]

If we confine ourselves to the first approximation, then for a standing wave we obtain:

\[ E=\frac{1}{4}\rho_0 A^2. \]

From this we see that Rayleigh’s formula coincides with our formula in a particular case, namely in its application to the nodes of oscillation.

Rayleigh’s formula was confirmed quantitatively in the careful experiments of Altberg (1903) and Zernov (1906), carried out in the laboratory of P. N. Lebedev at Moscow University.

The formula we have obtained for the mean pressures for different parts of a standing wave,

\[ \overline{(p-p_0)}^{\,t}_x= \left[ \frac{k+1}{8}-\frac{1}{2}\sin^2 bx \right]\rho_0 A^2 \]

refers to the coordinates \(x\) (Euler’s). Meanwhile, an analogous calculation performed in Lagrangian coordinates gives us the formula:

\[ \overline{(p-p_0)}^{\,t}_a=\frac{k+1}{8}\rho_0 A^2, \]

which shows that with respect to the substantial coordinates the mean pressure is the same for all points of the wave, and moreover it is equal to the mean pressure of a traveling wave of the same amplitude.

All the results obtained here by us for calculations of mean quantities may be presented in the form of the following table:

	Traveling waves	Traveling waves	Traveling waves
Standing waves	$a=\mathrm{const}$	$a=\mathrm{const}$	$x=\mathrm{const}$
Standing waves	$a=\mathrm{const}$ everywhere	$x=\mathrm{const}$ at the nodes	$x=\mathrm{const}$ at the antinodes
$\overline{\left(\dfrac{\rho-\rho_0}{\rho_0}\right)^t}=s$		$+\dfrac{3-k}{8}$	$-\dfrac{k+1}{8}$
$\overline{\left(\dfrac{p-p_0}{p_0 k}\right)^t}$		$+\dfrac{k+1}{8}$	$-\dfrac{3-k}{8}$
$\overline{\left(\dfrac{\vartheta-\vartheta_0}{\vartheta_0(k-1)}\right)^t}$		$+\dfrac{k-1}{8}$	$-\dfrac{5-k}{8}$

In this table, in its first column are denoted those quantities for which the mean value in time is determined. The inscriptions $a=\mathrm{const}$ and $x=\mathrm{const}$ mean that in the first case the means refer to the substantial coordinates $(a)$, whereas in the second case the time means are referred to the local coordinates $(x)$. In the second and third columns are shown the coefficients by which the quantity

\[ \left(\frac{A}{C_0}\right)^2, \]

must be multiplied in order to obtain the corresponding mean value in the case under consideration.

Let us also note that all the means considered by us are of the same order of smallness.

Additional remark

Almost all our formulas may also be applied to waves propagating, or being formed, on the surface of water in a channel, with, of course, the limitation that waves in a channel can also be treated as a phenomenon determined only by one coordinate $x$ and by the time $t$, as is often done in calculations. Moreover, we can obtain the formulas for waves in a channel directly from our formulas by putting in them $k=2$. With this substitution $k=2$, Fig. 7 remains valid.

Literature

The article presented here is an abridged and revised exposition of certain chapters of the author’s Theoretical Physics (Part IV, Chapters 13 to 15, “Theory of Waves”). As regards its content, it was delivered in parts at the Moscow Physical Society named after P. N. Lebedev and at the Moscow Society of Theoretical Music in 1918. The theoretical part, namely the application of the method of successive approximations to the characteristics of differential equations, was reported to the Milan Mathematical Seminar in 1931 and printed in Rendiconti del Seminario Matematico e Fisico di Milano, Vol. VI, 1932.

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