LETTERS TO THE EDITOR

REMARKS ON A. A. EICHENWALD’S ARTICLE “ACOUSTIC WAVES OF LARGE AMPLITUDE” *

B. V. Deryagin, Moscow

Usually in acoustics, as a first approximation, “linearized” equations of hydrodynamics are used. A. A. Eichenwald’s article aims to find out what consequences are obtained if one does not stop at this approximation, but goes further, for example retaining terms of the second order of smallness relative to the amplitude.

In doing so, confining himself to the case of plane waves, A. A. Eichenwald thereby, which is extremely essential, assumes the existence of a velocity potential.

A considerable part of A. A. Eichenwald’s article contains applications of the theory to the explanation of the origin of combination tones, and moreover mainly in air.

It is quite beyond doubt that the formulas derived in this article show the possibility of the occurrence of combination tones in air as a consequence of the presence of second-order terms in the velocity potential. But the physical essence of the question does not lie in this plane; it is important to show that the magnitude of the effect obtained in the theory is sufficient to explain the practically noticeable combination tones and corresponds to their intensity measured experimentally, so that other possible causes of the appearance of combination tones need not be taken into account.

As far as I know, up to the present it has not in general been proved experimentally that objective combination tones appear in air, except in cases of the appearance of harmonic overtones near a body that itself oscillates strictly sinusoidally, for example near the prongs of a tuning fork; A. A. Eichenwald quite consistently assigns this phenomenon to the category of combination tones.

The only work containing quantitative measurements of the intensity, and also of the phase, of these (and, as far as I know, of any whatsoever) combination tones is a work published by me under the title “Messungen der Amplitude und Phase der Oktave bei der Stimmgabel” (Phys. Zs. d. Sowjetunion 3, 574, 1933).

The measurements of the intensity of the octave showed that it is proportional to the square of the intensity of the fundamental tone, which could serve as quantitative confirmation of A. A. Eichenwald’s theory (and in general is the first quantitative confirmation of the basic premise of Helmholtz’s theory of combination tones).

Otherwise, however, matters stand with measurements of the phase of the octave relative to the fundamental tone, as I shall now show.

This article, apparently, remained unknown to A. A. Eichenwald; otherwise he would have been convinced that it gives a negative answer to the question raised above concerning the physical significance of his hydrodynamic theory of the occurrence of combination tones. Since a considerable

* See Uspekhi fizicheskikh nauk, XIV, 552, 1934.

the circle of readers of Uspekhi Fizicheskikh Nauk, including specialists, may thus be incorrectly oriented; I consider it necessary to dwell on this fundamental point.

I quote in extenso the following passage from my above-cited work, which already contains the indicated negative answer: “It is easy to develop a more exact theory if, taking the hydrodynamic equations that the velocity potential must satisfy, one retains in them terms of the second order with respect to the amplitude.

If, however, this path is followed, then for the phase difference \(\Delta\) between the pressures of the octave and the fundamental tone of the tuning fork one obtains not zero*, but \(\frac{1}{4}\)**:

To explain such a discrepancy it is necessary to assume (as P. P. Lazarev pointed out to me) that the reason for the occurrence of the octave lies in the formation of vortices at the sharp edges of the prongs of the tuning fork. Since, according to our measurements, the phase difference \(\Delta\) is equal to zero with great accuracy, it is obvious that the formation of vortices must in practice be regarded as the sole cause of the occurrence of the octave...”

That, in the presence of a velocity potential, one inevitably obtains

\[ \Delta = \pm \frac{1}{4}, \]

was stated by me without proof, from a desire not to lengthen the article by expounding theories not confirmed by experiment. It is easy, however, to show the correctness of this assertion, just as for example of those calculations which, according to A. A. Eikhenvald, allow one to explain*** the appearance of the octave of a tuning fork (Fig. 568).

Indeed, for the case of a plane wave propagating in the direction of positive \(x\)’s from a sinusoidally oscillating surface, A. A. Eikhenvald obtains the following expression for the velocity \(u_x\) of an air particle, expressed through Eikhenvald’s local coordinate \(x\) (§ 12, p. 567):

\[ u_x = A \sin \frac{2\pi}{T}\left(t - \frac{x}{C_0}\right) - \frac{A^2}{2C_0} - \frac{A^2}{2C_0}\cos \frac{4\pi}{T}\left(t - \frac{x}{C_0}\right), \tag{1} \]

where \(t\) is the current time and \(C_0\) is the propagation velocity. The first term of the right-hand side corresponds to the fundamental tone, the third to the octave; since the corresponding pressures are proportional to the velocities, one must find the phase difference of the harmonic function:

\[ \sin 2\pi \frac{t'}{T} \]

and its octave:

\[ -\cos 4\pi \frac{t'}{T} = \sin 2\pi\left(\frac{2t'}{T} - \frac{1}{4}\right), \]

where

\[ t' = t - \frac{x}{C_0}. \]

The desired phase difference, in the definition adopted for it in my work (pp. 578–580), is equal to:

\[ \Delta = -\frac{1}{4}, \]

and so on.

* Obtained in my measurements with an accuracy of up to \(0.01\) period.
** Disregarding the sign \(\pm\).
*** Incidentally, A. Eikhenvald’s explanation itself causes bewilderment. After all, any person can hear the octave of a tuning fork even without any hearing tube!

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

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 great distance, this effect can occur has likewise not been proved by A. A. Eichenwald (see § 11), since the damping of oscillations has not been taken into account, whereas in this case it cannot be neglected; moreover, the amplitude is usually diminished because of the sphericity of the wave, whereas A. A. Eichenwald performs the calculations only for plane waves.

Let us summarize.

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

The foregoing makes it possible to conclude that methods 2, 3, and 4 of their occurrence are not confirmed experimentally, and there are weighty grounds for doubting their practical significance.

Those phenomena which in 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.