Thus, the very case of the tuning-fork octave cited by A. A. Eichenwald refutes his theory, if one uses the result \(\Delta = 0\) that I obtained experimentally.

At the same time, one should in general call into question the possibility of the appearance of combination tones in air as a result of second-order terms in the velocity potential.

That, in the propagation of a harmonic wave over a large distance, this effect may occur has likewise not been proved by A. A. Eichenwald (see § 11), since the damping of the oscillations has not been taken into account; in this case it cannot be neglected and, moreover, the amplitude usually decreases because of the sphericity of the waves, whereas A. A. Eichenwald carries out his calculations only for plane waves.

Let us summarize.

In § 17 A. Eichenwald sums up the possible causes of the occurrence of combination tones.

The above allows one to conclude that methods 2, 3, and 4 for their occurrence have not been experimentally confirmed, and that there are weighty grounds for doubting their practical significance.

Those phenomena which, under heading 4, are described as apparent combination tones in fact constitute the only precisely established case of the appearance of combination tones in air, the cause of which lies, apparently, in the formation of vortices; A. A. Eichenwald’s theory, however, proves inapplicable here.

Moscow, VIEM, Department of Biophysics.

ACOUSTIC WAVES OF LARGE AMPLITUDE

(Reply to the comments of B. V. Deryagin).

A. A. Eichenwald, Milan

In my reply I shall try to be as brief as possible.

1. B. Deryagin’s article (1933) reached me only in August 1934, and I sent my article to the editorial office still earlier; thus I could no longer mention B. Deryagin’s work in my article. In any case, B. Deryagin’s work could not have had any influence on my article, since the result obtained by Deryagin coincides with the result of F. Lindig’s work (1903), and Lindig’s work was well known to me.

2. In contrast to B. Deryagin’s opinion, I must say that a “significant part” of my article is devoted to the phenomenon of wave deformation, and only on 3 or 4 pages (out of 32) is the occurrence of combination tones discussed. This is explained very simply by the circumstance that the very appearance of my report in Milan was prompted by Bothe’s completed experiments (1931) on the deformation of a traveling wave. As for the question of combination tones, I had to touch on it as well, so to speak in passing, in connection with the rest. I did not even find it necessary to cite the literature on this question, as I did for the other questions.

3. Now concerning the works of Lindig and Deryagin. Both observers confirmed the opinion of earlier investigators (beginning almost from 1830) that near a tuning-fork chamber an octave of the fundamental tone of the tuning fork appears in the air, as well as its octave. Both observers found that the amplitude of this octave is proportional to the square of the amplitude of the fundamental tone, and that the relative phases of both tones proved identical. It must be added that the methods of measurement used by Lindig and Deryagin are in principle identical, but, of course, Deryagin’s measurements themselves are incomparably more accurate than Lindig’s measurements. How, then, do matters stand with the theory? Lindig applied Helmholtz’s theory of combination oscillations to his experiments, and it turned out that

theory and experiment are in complete agreement with one another both with respect to the amplitude and with respect to the phase of the oscillations under study. More precisely, this agreement was found in B. Deryagin’s experiments. It would seem that everything is in good order. Unfortunately, however, the theory, in the form in which it was proposed by Lindig, cannot be accepted, since it assumes that particles of air have a certain natural period of oscillation, which in fact they do not have. If, however, this assumption is rejected and the theory of waves (and not of oscillations) is applied, as naturally suggests itself, then it turns out that theory and experiment agree only with respect to the amplitude and disagree with respect to the phase of the oscillations. All this was known as long as 30 years ago; moreover, it is easy to see it directly from the formulas, as B. Deryagin himself observes. How, then, can it be: an inapplicable theory proves to be correct in experiment, while an applicable theory proves to be incorrect. B. Deryagin cites in this connection the opinion of P. Lazarev, which consists in the assertion that vortices arise near the sharp edges of the tuning fork during oscillations and that these are the cause of the combination tones. It is undoubtedly true that vortices arise near an oscillating tuning fork, and it is likewise undoubtedly true that combination tones may be produced by vortices (since the relations are nonlinear, § 15; I also mentioned vortices in § 17); but other assumptions can also be made. However, all such assumptions, without an exact mathematical formulation and without application to one or another real experiment, cannot have great significance. B. Deryagin, however, as I am certain, assumes that the supposition of vortices is the only possible one, and therefore regards the occurrence of combination tones without vortices (according to Lindig) as refuted by experiment. Moreover, B. Deryagin supposes that if the tuning fork is replaced by a membrane with trimmed edges, then vortices will not be formed and the combination octave also will not appear. I must, however, disappoint B. Deryagin: vortices will still form (see the experiments of Gartmann-Kempf in § 14 of my paper). Nevertheless, further experiments are, of course, desirable.

1. The formation of vortices during oscillations was studied, among others, by Dvořák (1876, 1901), and the theory of Dvořák’s experiments was given by Rayleigh (1883). However, as applied to the experiments of Lindig and Deryagin, where the acoustic probe is placed in the immediate vicinity of the oscillating foot of the tuning fork, such a theory would present considerable mathematical difficulties.

2. In the end B. V. Deryagin believes that “one should in general call into question the possibility of the appearance of combination tones in air as the result of terms of the second order in the velocity potential,” and that “Eichenwald has not proved ... since the influence of attenuation has not been taken into account and the calculations were performed for plane waves” (and not spherical ones, as B. Deryagin supposes), and so on and so forth. I hasten to agree with my opponent: Eichenwald has proved nothing at all... because all this had already long since been proved by others and without Eichenwald. Nevertheless, perhaps further explanations will not be entirely superfluous.

3. Riemann’s theory as applied to plane waves, without attenuation and even without vortices (and I shall add: without turbulent motions as well), shows us that an acoustic wave, as it propagates in air, must become deformed (like a sea wave as it approaches the shore). Vothe’s experiments (Fig. 2 in § 9 of my paper) demonstrated this deformation. There is no need that Vothe’s wave was probably not quite plane and was attenuated, and was probably accompanied by vortices (the motion of air without vortices would indeed be a great rarity).

4. Further, Riemann’s theory leads to the conclusion that a deforming wave must ultimately turn into a shock wave. And this is confirmed at the present time in experiments with the detonation of explosives. Riemann’s theory is also confirmed in measurements of the velocity of propagation of waves. In general, at the present time there is no need to doubt the phenomenon of wave deformation. And if the wave, in its propagat-

deformed in space, then combination tones undoubtedly also arise in it; the two assertions are entirely equivalent.

1. From this theory of Riemann it follows directly that in acoustic waves (progressive and standing) the oscillations are asymmetric. In other words, in air there must arise, among other things, combination waves of even degrees and, of course, an octave must appear first of all. Asymmetric oscillations differ in that the mean values of their amplitudes are different from zero, and the magnitudes of these mean values are equal to the amplitude of the corresponding combination tone. But the magnitudes of the mean pressure values of a sound wave were carefully measured by Altberg (1903, 1907) and by Zernov (1906), and, what is especially important, they were measured not only relatively, but also absolutely. The results of these measurements fully confirmed Rayleigh’s theory, which also follows from Riemann’s theory (§ 21).

From this we may draw the following conclusion: the sound pressure measured by Altberg and Zernov may be regarded as an indirect measurement of the amplitudes of even combination tones (of course, chiefly of the octave). These measurements may be called quite objective, and they may be trusted even more than if someone with his own ear had heard the combination octave in air.

1. Riemann’s mathematical theory for acoustic waves is so far confined to cases of plane undamped waves, and generalizations of this theory including other factors (air friction, its thermal conductivity, vortex motion, turbulence, etc.) are, of course, highly desirable. But it cannot be denied that Riemann’s theory, even in its simplest form, already gives very much, and moreover not only in questions of a scientific nature, but also in questions of a purely practical character. As for Helmholtz’s basic idea of combination tones (nonlinearity of the relations), it is irrefutable and will remain in force in all further improvements of acoustic theories.

Responsible editor: E. V. Shpolsky. Technical editor: A. V. Smirnova.

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