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VISUALIZATION OF SPATIALLY MODULATED SOUND WAVES
S. N. Rzhevkin
Spatially modulated waves, i.e., waves in which the amplitude does not remain constant along fronts of equal phase, arise in the most diverse cases. The usual method of obtaining such waves amounts to making the wave pass through a surface or a system of which some parts are more permeable to the waves and others less permeable. Surfaces of this kind are realized, for example, in the form of diffraction gratings, zone plates, and other systems. Spatially modulated waves also arise as a result of diffraction by individual obstacles.
A very important question is the stability, in further propagation, of a spatially modulated wave produced in one way or another. Although this question has been discussed in the literature\(^{1,2,4}\), I should consider it interesting to examine it by a somewhat different, rather visual method.
If we have an infinite straight tube of rectangular cross section with perfectly rigid walls, whose cross-sectional dimensions are \(a\) and \(b\), filled with a gas or a liquid, and excite wave motion in its initial cross section by prescribing some distribution of velocities in the initial cross section, then we can obtain in the tube spatially modulated waves of various types. Let the axes \(x\) and \(y\) be directed along the edges of the tube, and the axis \(z\) along its length (see Fig. 1). Suppose that in the initial cross section a distribution of velocities normal to the plane \(z=0\) is prescribed, characterized, for example, by a sinusoidal standing wave
\[ \zeta = v_m \cos(k_m x)e^{i\omega t}, \tag{1} \]
where \(k_m=\dfrac{m\pi}{a}\); \(m=0,1,2,3,\ldots\), and \(\omega=2\pi f\) is the angular frequency. In this case, within the segment \(a\) there fit \(m\) half-waves, i.e., the wavelength in the plane \(z=0\) will be equal to \(\Lambda=\dfrac{2a}{m}\).
Solving the wave equation
\[ \Delta \Phi = \frac{1}{c^2}\frac{\partial^2 \Phi}{\partial t^2} \tag{2} \]
under the boundary condition (1) and, taking the velocities normal to the side faces to be equal to zero, we obtain for the velocity potential \(\Phi_m\) the expression
\[ \Phi_m=-\frac{v_m}{ik'}\cos(k_m x)e^{i(\omega t-k'z)}, \tag{3} \]
where
\[ k'=\sqrt{k^2-k_m^2};\qquad k_m=\frac{m\pi}{a}=\frac{2\pi}{\Lambda};\qquad k=\frac{\omega}{c}=\frac{2\pi}{\lambda}. \]
Replacing \(\cos k_m x\) by \(\frac{1}{2}(e^{ik_m x}+e^{-ik_m x})\), we obtain:
\[ \Phi_m=-\frac{v_m}{2ik'}e^{i(\omega t+k_m x-k'z)} +\frac{v_m}{2ik'}e^{i(\omega t-k_m x-k'z)}. \tag{4} \]
If the condition \(k>k_m\) is satisfied, then expression (3) shows that the wave process arising in the tube consists of two plane waves whose wave vectors lie in the \(xz\) plane and make with the \(x\) and \(z\) axes the angles \(\alpha\) and \(\gamma\), determined by the expressions
\[ \cos\alpha=\frac{k_m}{k};\qquad \cos\gamma=\frac{k'}{k}=\frac{\sqrt{k^2-k_m^2}}{k}. \]
Consequently,
\[ \sin\gamma=\pm\cos\alpha=\pm\frac{k_m}{k}=\pm\frac{\lambda}{\Lambda}. \tag{5} \]
The phase velocity of propagation of the plane waves (4) in the directions \(+\gamma\) and \(-\gamma\) is determined from the expression
\[ c=\frac{\omega}{\sqrt{k_m^2+k'^2}}=\frac{\omega}{k}=c, \]
i.e., it is equal to the velocity of sound. The phase velocity along the \(z\) axis will be equal to \(\frac{\omega}{k'}=\frac{c}{\cos\gamma}\), i.e., it is greater than the velocity of sound \(c\).
Formula (5) is completely analogous to the formula which determines the angle of deflection of first-order spectra when waves pass through an ordinary diffraction grating. The plane waves (4), excited in the tube by oscillations in the initial cross section \((z=0)\), emerge in two beams of width \(a\cos\gamma\) in the directions determined by the angles \(\pm\gamma\), then are reflected successively from the opposite faces of the tube (Fig. 1), and, as a result, fill the entire volume of the tube with a system of two plane waves intersecting one another at an angle \(2\gamma\). In this system of waves there will pass nodal planes corresponding to the values
\[ \cos k_m x=0 \]
or
\[ x_m=\frac{a}{2m}(2n+1), \]
where \(n=0,1,2,\ldots,m-1\); along these planes the oscillatory motion along the \(z\) axis will be abs—
to act. Taking into account that the solid side walls of the tube are mirrors reflecting sound, we may consider that they have been removed, and that the wave pattern in the tube has been supplemented by their mirror reflection in the lateral boundaries; these imaginary mirror waves are shown in Fig. 1 by dashed lines. Since the reflections in the lateral boundaries will be multiple, and the mirror-reflected waves will again be reflected in the opposite boundaries, we obtain as a result the picture of a single unbounded sound field, created by a plane sinusoidal grating extending infinitely along the \(x\)-axis. This sound
Fig. 1.
field fills the entire half-space \(z>0\), while the real field in the tube is the part of it bounded by the dimensions of the tube.
If in the plane \(z=0\) an arbitrary oscillatory motion with velocity \(\zeta_0(x,y)\) is prescribed, then it can be represented in the form of a sum of the form:
\[ \dot{\zeta}=\sum_{m,n=0}^{\infty} v_{mn}\cos(k_m x)\cdot \cos(k_n y)e^{i\omega t}, \tag{6} \]
where
\[ k_m=\frac{m\pi}{a};\qquad k_n=\frac{n\pi}{b};\qquad m,n=0,1,2,3\ldots \]
It is not difficult to see that prescribing the velocity in the initial section by a formula of the form \(v_{mn}\cdot \cos k_m x\cdot \cos k_n y\) is equivalent to prescribing a system of two standing waves of equal amplitude, with wavelength
\[ \Lambda_{mn}=\frac{2\pi}{\sqrt{k_m^2+k_n^2}}, \]
VISUALIZATION OF SOUND WAVES
where the directions of propagation of these waves make angles with the axes \(x\) and \(y\):
\[ \alpha_{mn}=\arccos \frac{\pm k_m}{\sqrt{k_m^2+k_n^2}} \quad \text{and} \quad \beta_{mn}=\arccos \frac{\pm k_n}{\sqrt{k_m^2+k_n^2}} . \]
Each of these standing waves gives a double beam of plane diffraction waves of the first order, traveling at an angle \(\gamma_{mn}\) to the axis \(z\), where
\[ \sin \gamma_{mn}=\frac{\sqrt{k_m^2+k_n^2}}{k}=\frac{\lambda}{\Lambda_{mn}} . \tag{7} \]
Each wave mode \((m,n)\) will be represented by a quadruple beam of plane diffraction waves. These waves, successively reflecting from the faces, fill the whole tube with a quadruple system of plane waves, modulated along the front according to the same law \(\cos k_m x \cos k_n y\) as in the initial section. The spatial modulation in this case reduces to the formation of channels of rectangular cross-section with sides
\[ \frac{\pi}{k_m}=\frac{a}{m} \quad \text{and} \quad \frac{\pi}{k_n}=\frac{b}{n}, \]
extending along the axis \(z\), along which the waves propagate as though the side walls of these channels were rigid. Thus the entire tube is, as it were, divided into a number of rectangular tubes with rigid walls.
We have seen that, when two plane waves of the same frequency and intensity intersect at an angle \(2\gamma\), nodal planes arise in the resultant wave motion; these planes are located along the bisector of the angle \(2\gamma\) and are separated from one another by a distance \(\Delta x\), determined by the condition \(k_m \Delta x=\pi\), which gives
\[ \Delta x=\frac{\lambda}{2\sin \gamma}. \tag{8} \]
Such a pattern of “pseudo-standing” waves is produced, for example, by the superposition of a wave incident on a plane surface and the wave reflected from it, at a grazing angle equal to \(\gamma\) (i.e., an angle of incidence \(90^\circ-\gamma\)).
Visualization of the pattern of sound waves can be carried out very clearly with ultrasonic waves in a liquid by using the Schlieren method. By using stroboscopic illumination, it is possible to obtain, visually or photographically, very clear instantaneous patterns of the fronts of ultrasonic waves in various cases of propagation, reflection, refraction, diffraction, etc., as was shown by me in joint works with S. I. Krechmer at the Physics Institute of the Academy of Sciences of the USSR in 1937–1939.^3
When observing without a stroboscope, it is obviously impossible to see the fronts of individual traveling waves, since their velocity is too high.
is large, but it is possible to see those places in the sound field where oscillations are always absent. Such places are, in particular, in the case of pseudo-standing waves under total reflection, the nodal planes of velocity. In Fig. 2 (see insert) a photograph is given of pseudo-standing waves under total reflection, obtained by us (without stroboscopy) in one of our earlier works. In it one can clearly see dark lines, parallel to the reflecting plane surface, corresponding to the pseudo-standing waves. Along the channels formed by the nodal planes, traveling waves propagate with the increased phase velocity \(c/\cos \gamma\). No transverse nodal surfaces are formed along these channels.
The complex wave, spatially modulated along the front according to the law \(\cos k_m x = \cos \dfrac{2\pi}{\Lambda} x\), obtained by the superposition of two plane waves whose wave vectors form an angle \(2\gamma\), is stable, and the nodal planes along the wave front, going in the direction of the bisector of the angle \(2\gamma\), remain unchanged as the wave propagates. Calculating the Umov vector (the vector of energy-flux density) along the \(z\)-axis, we obtain the expression
\[ J=\frac{1}{2}\operatorname{Re}[p\zeta^*] =\frac{1}{2}\operatorname{Re}\left[i\omega\rho\Phi\cdot\left(-\frac{\partial\Phi}{\partial z}\right)^*\right] = \]
\[ =\frac{1}{2}\, \frac{\rho c v_m^2}{\sqrt{1-\dfrac{k_m^2}{k^2}}}\, \cos^3 k_m x . \tag{9} \]
The Umov vector, representing the intensity of sound, is constant along the \(z\)-axis, while along the \(x\)-axis it varies proportionally to \(\cos k_m x\). Along each channel of width \(\dfrac{\pi}{k_m}=\dfrac{\Lambda}{2}=\dfrac{a}{m}\) there will propagate a wave with mean intensity
\[ \frac{1}{4}\, \frac{\rho c v_m^2}{\sqrt{1-\dfrac{k_m^2}{k^2}}} = \frac{1}{4}\, \frac{\rho c v_m^2}{\cos\gamma}. \]
If \(\Lambda < \lambda\) (\(k_m > k\)), i.e., if the wavelength of the sinusoidal grating is smaller than the wavelength of the sound wave (in free space), then the wave number \(k'=\sqrt{k^2-k_m^2}\) will be imaginary and the wave equation (3) is written in the form
\[ \Phi_m=\frac{v_m}{ik'}\cos(k_m x)e^{-\sqrt{k_m^2-k^2}\,z}e^{i\omega t}, \tag{10} \]
Fig. 2.
Fig. 3.
Fig. 4.
Fig. 5.
Fig. 6.
and expression (5) will show that \(\sin\gamma>1\); in this case the angle \(\gamma\) turns out to be complex. From formula (10) it is clear that the oscillations excited in the initial section with frequency \(\omega=kc\) and with period of spatial modulation \(\Lambda=\dfrac{2\pi}{k_m}\) will gradually decay with a damping coefficient equal to \(\sqrt{k_m^2-\dfrac{\omega^2}{c^2}}\). The streamlines of the medium, whose form can be found by calculating the velocity components along the \(x\) and \(z\) axes \(\left(-\dfrac{\partial\Phi}{\partial x}\ \text{and}\ -\dfrac{\partial\Phi}{\partial z}\right)\), will be curves closing over short distances between adjacent antinodes of the standing wave shown in Fig. 3. Thus, in the case \(\Lambda<\lambda\), in the half-space \(z>0\) an oscillatory process arises only in the zone nearest to the region of excitation, while waves propagating away are not formed.
A spatially modulated wave in the case \(\Lambda<\lambda\), or \(f>\dfrac{c}{\Lambda}\), is unstable; the inhomogeneities created at the wave front are not preserved in this case and gradually disappear. In particular, in this case a wave incident on a diffraction grating gives, on the other side of the grating, only a damped wave process; no waves running away and producing diffraction spectra are formed. Under these conditions a wave incident normally on a grating will be completely reflected from it. It is not difficult to show, by calculating the quantities of the mean sound pressure \(\bar p=i\omega\rho\bar\Phi_m\) and of the mean particle velocity along the \(z\)-axis \(\left(\bar v=-\dfrac{d\bar\Phi_m}{dz}\right)\), the averaging being performed over one half-wave \(\dfrac{\Lambda}{2}\), that the mean impedance per unit surface \(\bar Z_1=\dfrac{\bar p}{\bar v}\) will in this case be purely inertial:
\[ \bar Z_1=i\omega M_1=i\omega\,\frac{\rho\Lambda\lambda}{2\pi\sqrt{\lambda^2-\Lambda^2}} =-\frac{i\omega\rho}{\sqrt{\dfrac{1}{\Lambda^2}-\dfrac{1}{\lambda^2}}}. \tag{11} \]
The quantity \(M_1\) characterizes the added mass per unit area. The inertial character of the impedance determines the complete reflection of sound according to the formula for the reflection coefficient
\[ |r|=\left|\frac{\bar Z_1-\rho c}{\bar Z_1+\rho c}\right| =\left|\frac{i\omega M_1-\rho c}{i\omega M_1+\rho c}\right|=1. \tag{12} \]
The wave pattern changes substantially if, in addition to the standing wave along the \(x\)-axis, there is also a plane wave caused by the piston motion of the initial section with velocity \(v_0 e^{i\omega t}\),
which corresponds to the constant term in the expression of the sum of the form (6), i.e. to the wave mode \(m=0,\ n=0\).
Let us consider modulation over the front by means of a grating with transparent and opaque strips; the velocity in the plane \(z=0\) we specify in the form of the series
\[ \dot{\zeta}_0=v_0 e^{i\omega t}+v_m\cos k_m x\cdot e^{i\omega t} +v_{2m}\cos 2k_m x\cdot e^{i\omega t}+\ldots \tag{13} \]
The amplitudes \(v_{mn}\) decrease rapidly with increasing \(n\). The velocity potential, analogously to (2), we write in the following form:
\[ \Phi=\frac{v_0}{ik}e^{i(\omega t-kz)} +\frac{v_m}{i\sqrt{k^2-k_m^2}}\cos k_m x\cdot e^{\,i\left(\omega t-\sqrt{k^2-k_m^2}\cdot z\right)} + \]
\[ +\frac{v_{2m}}{i\sqrt{k^2-(2k_m)^2}}\cos 2k_m x\cdot e^{\,i\left(\omega t-\sqrt{k^2-(2k_m)^2}\cdot z\right)} \tag{14} \]
The Umov vector will have the form:
\[ J=\frac{1}{2}\operatorname{Re}(p\dot{\zeta}^{*}) =\frac{1}{2}\rho c v_0^2+ \]
\[ +\frac{1}{2}\frac{\rho c v_m^2}{\sqrt{1-\dfrac{k_m^2}{k^2}}} +\frac{1}{2}\frac{\rho c v_{2m}^2}{\sqrt{1-\dfrac{(2k_m)^2}{k^2}}} +\ldots \]
\[ +\frac{1}{2}\rho c v_0 v_m \frac{\sqrt{1-\dfrac{k_m^2}{k^2}}-1} {\sqrt{1-\dfrac{k_m^2}{k^2}}} \cos k_m x\cdot \cos\left[k-\sqrt{k^2-k_m^2}\right]z+ \]
\[ +\frac{1}{2}\rho c v_0 v_{2m} \frac{\sqrt{1-\dfrac{(2k_m)^2}{k^2}}-1} {\sqrt{1-\dfrac{(2k_m)^2}{k^2}}} \cos 2k_m x\times \]
\[ \times \cos\left[k-\sqrt{k^2-(2k_m)^2}\right]z+\ldots \tag{15} \]
The first terms in this expression give the intensity of the sound due to a plane wave, with velocity amplitude \(v_0\), propagating along the \(z\)-axis, and to waves of complex form, with velocity modulated over the front according to the laws \(v_m\cos k_m x\), \(v_{2m}\cos 2k_m x\), etc. (the modulation wavelength over the front is equal to \(\Lambda=\dfrac{2a}{m}\), \(\dfrac{2a}{2m}\), etc.). These first terms do not depend on \(z\). The subsequent
terms (the third and fourth lines) reveal a periodic dependence on \(z\), the lengths of the spatial period being equal to:
\[ \Lambda'=\frac{2\pi}{k-\sqrt{k^2-k_m^2}} =\frac{\lambda}{1-\sqrt{1-\frac{\lambda^2}{\Lambda^2}}}, \tag{16} \]
\[ \Lambda''=\frac{2\pi}{k-\sqrt{k^2-(2k_m)^2}} =\frac{\lambda}{1-\sqrt{1-\left(\frac{2\lambda}{\Lambda}\right)^2}} \quad \text{and so on,} \]
while the intensities are proportional to \(v_0v_m\), \(v_0v_{2m}\), and so on. The period of the change in intensity along the \(z\)-axis, for small \(\frac{\lambda}{\Lambda}\), will be approximately equal to
\[ \Lambda' \simeq \frac{\lambda}{1-\left(1-\frac{1}{2}\frac{\lambda^2}{\Lambda^2}\right)} =\frac{2\Lambda^2}{\lambda}. \tag{17} \]
The higher terms of series (14), with wave numbers \(2k_m, 3k_m,\ldots\), etc., under the condition \(\lambda \ll \Lambda\), will give periodicity along the \(z\)-axis with a period \(2^2, 3^2\), etc. times smaller, and the whole process as a whole will be approximately periodic in the direction \(z\). If the condition \(\lambda \ll \Lambda\) is not satisfied, then in this case the higher-order terms will no longer have a period an integral number of times smaller than the fundamental period, and the change in intensity along the \(z\)-axis (for a given value of \(x\) and \(y\)) will not have a strictly periodic character.
The minimum intensity will occur at those points where \(\cos \frac{2\pi}{\Lambda'}z\) has the greatest negative value, i.e. when
\[ \frac{2\pi}{\Lambda'}z=(2n+1)\pi \]
or
\[ z_{\min}=\left(n+\frac{1}{2}\right)\Lambda'; \]
the maximum intensity will occur when
\[ \frac{2\pi}{\Lambda'}z=2n\pi \]
or
\[ z_{\max}=n\Lambda'. \]
Rayleigh based on reasoning of this kind the theory of copying diffraction gratings\(^2\). It is clear that at distances \(z=n\Lambda'\) from the grating, under the condition \(\lambda \ll \Lambda\), according to (15), there will be obtained a distribution of intensities corresponding to the “light” and “dark” lines of the grating; by placing a photographic plate at such distances, one can obtain a copy of the grating. For
\[ z_2=\left(n+\frac{1}{2}\right)\frac{\Lambda'}{2} \]
we shall obtain a reversal of the intensities—the light lines of the grating will correspond to intensity minima, and the dark ones to maxima; at these distances, obviously, one can also photograph a copy of the grating. For any \(n\), copies of the grating will also theoretically be obtained; however, owing to the increasing inaccuracy in the periodicity of the terms of series (15), the sharpness of the minima and maxima
of intensity will gradually decrease as the plate is moved away from the grating.
The considerations set forth can be successfully illustrated by photographs of spatially modulated ultrasonic waves as they pass through a diffraction grating. The diffraction grating for these experiments was made of steel rods 2 mm in diameter. Ultrasounds of different wavelengths were obtained from a piezoquartz radiator; the grating and the radiator were placed in a bath of vaseline oil, having plane-parallel walls of mirror glass.
In the photographs of Figs. 4 and 5 (see the insert) two cases of the formation of diffraction copies of the grating are presented.* Measurement gives, in the first photograph, \(\lambda = 1.80\) mm, the grating spacing \(\Lambda = 3.20\) mm, and the half-period along the \(z\)-axis is equal to \(\dfrac{\Lambda'}{2} \simeq 10\) mm, i.e. \(\Lambda' \simeq 20\) mm. Calculation by formula (17) gives \(\dfrac{\Lambda'}{2'} = 11.3\) mm, i.e. there is a rather substantial discrepancy with the value 10 mm obtained from the experiment (in the present case the measurement of \(\Lambda'\) is very inaccurate). However, taking into account that \(\dfrac{\lambda}{\Lambda} = 0.562\), by the exact formula (16) we obtain \(\dfrac{\Lambda'}{2} = 10.4\) mm, which is already in satisfactory agreement with experiment. It is clear that in the present case the higher-order terms in formula (15) will not give a periodicity coinciding with the periodicity of the fundamental (first) term, and as one moves away from the grating the sharpness of the “copies” of the grating will be greatly reduced. We see in Fig. 4 that only the first, reversed, row of “images” of the rods of the grating at the distance \(\dfrac{\Lambda'}{2}\) is visible quite clearly; the subsequent images are completely blurred. In the photograph of Fig. 4, the diffraction waves of the 1st order and the dark “rays” of pseudo-standing waves, running along the bisector between the spectra of the first and zeroth orders, are clearly visible.
Figure 5 gives a photograph of the diffraction of waves for a grating with spacing \(\Lambda = 3.2\) mm at wavelength \(\lambda = 0.9\) mm, i.e. \(\dfrac{\lambda}{\Lambda} = 0.28\). In the photograph at least three rows of images or copies of the grating are visible.
In this case, calculation by the approximate formula (17) gives \(\Lambda' = 22.6\), and by the exact formula (16) \(\Lambda' = 22.5\); the difference in hundredths of a millimeter is difficult to verify by measurements on the photograph. Measurement from the photograph gives in this case \(\Lambda' = 22.8\) mm, which agrees with the theoretical value to an accuracy of 1.3%.
* The photographs shown in Figs. 4, 5, and 6 were obtained in 1937–1938 in the acoustics laboratory of FIAN; however, at that time they did not receive a proper interpretation, and their description was not given in previously published works.
Figure 6 (see insert) shows the pattern of the ultrasonic field when the wave is reflected from a surface with parallel grooves at an oblique angle of incidence; the cross-section of the grooves is clearly visible in the photograph. It is evident from the photograph that in this case as well the reflected waves are spatially modulated, with a series of points of increased sound intensity and a series of points of decreased intensity being obtained; that is, in this case intensity modulation occurs both along the front of the reflected wave and along the direction of wave propagation.
The latter case is of interest from the standpoint of architectural acoustics; it shows that the sound field, upon reflection from a row of columns or grooves on walls, will be nonuniform in space. It will contain points of increased intensity and points of decreased intensity.
References
- Rayleigh, The Wave Theory of Light, GTTI, pp. 82–89 (1940).
- Rayleigh, Phil. Mag. (1881), p. 504.
- S. I. Krechmer and S. N. Rzhevkin, UFN, 18, 1 (1937); Techn. Phys. (USSR), 4, 1 (1937); Proceedings of the Lebedev Physical Institute, vol. I, issue 4 (1939).
- S. M. Rytov, UFN, 41, 425 (1950).