- Email: [email protected]
Visualization of acoustic wavefronts by means of acoustic stroboscopy
Volume 6, number 1
OPTICS COMMUNICATIONS
September 1972
VISUALIZATION OF ACOUSTIC WAVEFRONTS BY MEANS OF ACOUSTIC STROBOSCOPY J.LAPIERRE
and SLOWENTHAL
Institut d ‘Optique, Faculte’des Sciences, 91 . Orsay, France Received 27 June 1972
The amplitude and phase distributions of a progressive acoustic wave are visualized simultaneously by means of acoustic stroboscopy. Two methods are described: (i) modulation of a laser beam by a progressive acoustic wave; (ii) modulation by a stationary wave. Spatial filtering of diffracted orders enables the progressive wavefronts to be seen as if they were stationary.
1. Introduction In this article, we describe a method for visualizing acoustic progressive waves whereby spatial variations of both the amplitude and phase of an acoustic field are made visible along the direction of propagation. Progressive wavefronts can be visualized by high speed photography [l] but this method does not permit the evolution of phenomena to be followed. One can alse use classical methods of stroboscopy [2,3] , but those systems are generally complex due to synchronization requirements. We propose a stroboscopic method which makes use of a simple acoustic modulator operating at the same frequency as that of the phenomenon to be observed. This kind of stroboscope had been used by Bar [4] in 1963 and by Fox and Rock [5] in 1939 for wavelength measurements. Nevertheless, it does not seem that these authors gave a correct interpretation of the phenomena involved in their experiment: (i) they were considering that a light beam traversing an acoustiu wave is modulated in intensity, instead of complex amplitude; (ii) their observation plane was not the image plane of the acoustic medium, so that they were performing a spatial filtering due to the limited aperture of the optical system. Our acoustic modulator can be operated in two different ways: with a progressive wave or with a
standing wave, corresponding to the stroboscope of Fox and Rock, and the one of Bar respectively. In both cases, a light beam, modulated by the stroboscope, illuminates the acoustic medium in which the wave to be visualized is propagating. Light diffracted by the acoustic wave is analyzed spatially; we will show that some of the diffracted orders can be used to produce a stationary image of the propagating wave. The progressive wave method of modulation will be shown to yield a contrast factor equal to one, while it is l/2 in the case of the standing wave method. On the other hand, this second method of modulation has a wider application than the first one, as far as the shape of the wavefronts to be observed is concerned.
2. Study of phenomena: wave
diffraction
by an acoustic
Consider a monochromatic plane wave: A = ewiwt, incident normally on a transparent medium in the z direction. A plane acoustic wave of the form p = Ap cos (Sb-Kx)
,
(1)
propagates in the x direction; here,p, a, and K represent the acoustic pressure, angular frequency and wave number, respectively. The index of refraction n 1
Volume 6, number
September 1972
OPTICS COMMUNKATIONS
1
We will show in the following section how the frequency shifts +51.and -R can be eliminated. Then, recombining the +l and ---1 orders. we will get A = a e-lwf cos Kx
+1 A=
e-
iwt
(5)
0 -1
z
Fig. 1. Diffraction of light by an acoustic wave. obeys the same law of variation
The light intensity corresponding to this expression will give an image of the acoustic wavefronts as they could be observed at a given time; this is due to the presence of the time-independent factor COSKX. At the same time, a representation of the spatial distribution of acoustic amplitude is afforded by the parameter a. In the following section, we will examine the problem of visualizing a plane acoustic wave. The case of a more complex wave will be briefly considered in section 4.
(2)
n = n0 + An cos (a2t-Kx).
Following the approach of Raman and Nath [6], we treat the acoustic medium like a pure phase grating so that the complex amplitude of the light beam right after the acoustic medium can be written A = exp i [cpO+ A9 cos (fir-Kx)
- ot]
,
(3)
where 90 and A9 stand for kin, and klAn respectively, with k being the wave number of the light in the acoustic medium, 1 the thickness. Expression (3) can be expanded in a Bessel functions series, of which we will retain the first two terms only because the higher order terms will be eliminated by spatial filtering. Therefore, the light amplitude right after the acoustic tank has the form A = eeiwl[ 1 t iu cos (Rt-Kx)]
,
(4)
where a is a constant, and where we have put 9,-, = 0. The expression in brackets represents the transmittance of the tank. In the filter plane F, we have the Fourier transform of this last expression. The three component amplitudes are
3. The acoustic stroboscope 3.1. Modulation by a progressive wave In fig. 2, tank C2 is a transparent medium where the acoustic wave to be visualized is propagated; we assume it is a plane wave in the present case. It is illuminated with a laser beam modulated by C, where an acoustic plane wave of the same frequency as in C, is propagated. At the end of C,, like in C,, there is an absorber in order to prevent the formation of stationary waves. According to section 2, the light amplitude, right after the modulator Cl, is given by A = eeiwt[ 1 + ia, cos (at-Kx)]
0: -1:
Jiaexp[-i(o+R)t] exp (-iwt)
.
The complex amplitude according to section 1
transmittance
of tank C2 is,
t] . 7 = 1 + ia2(x, y) cos (fi;2t-Kx) )
2
(7)
,
,
$a exp [-i(o-s2)
(6)
where al is a constant if we assume the modulating acoustic wave to be uniform over the whole cross section of the laser beam. The dc component of eq. (6) is filtered out in plane F, , and then the beam, after expansion, illuminates C,. Calling g the magnification of the afocal system (L1, L,, L3), the incident light amplitude on C2 is A = iehwfa cos (Rt-Kxg-l)
t 1:
,
(8)
September 1972
OPTICS COMMUNICATIONS
Volume 6, number 1
“1
Fig. 2. Acoustical stroboscope. C1 : modulator tank, Cz : test tank, L1 and L4: Fourier transform lenses, Lz and Ls : collector lenses, La: part of telescope with L1, F1 and Fz: spatial filters. Solid lines: pupil imagery, dashed lines: objective imagery. where a2(x, v) specifies information
on the possible variations of acoustic amplitude in the x, y plane. In the following, we will write simply a2 instead of a2(x, JJ). It follows that the light amplitude, right after C,, is [ 1 + ia cos (at-Kx)]
A = eViwf ia, cos (SU-Kxg-l)
- &zla2{cos [Kx( lg-1)
.
(9)
i
In order to get a time independent phenomenon at the acoustic frequency, one must simply pick the last term out of the last expression, that is .
[Kx(l-g-l)]
(10)
This operation is performed through the use of filter F,, having v = f K( 1-g- 1) as transmitted frequencies. The double diffraction system [7] (Ld, F2, L,, I> produces a fitered image of D2 with a distribution of light intensity given by ,
+ cos (!mKx)]
Right after C,, the light amplitude phase constant
- mt]
•t cos [Kx( l-g-l]}
E = IAl = Ccos2[Kx(l-g-1)]
3.2. Modulation by a stationary wave Without an absorber at the end of tank C,, a stationary wave is produced p = apl [cos (cl-Kx)
= emiwt ia, cos (S&Kxg-‘) i
A = e-iwt&zla2cos
in the x, y plane. The term cos2 [Kx( 1-g-l)] gives a scaled image of the acoustic wavefronts, with a contrast equal to one.
(11)
where C, which is proportional to la2(x, y)12, gives a representation of the acoustic amplitude distribution
A = e-iwf (1 + ia, [cos (at-Kx)
.
(12)
is, neglecting a
+ cos (CWKx)]}
= e-iat[ 1 + ia, ~0s ilt(eiJ%2-i~x)]
.
(13)
By the use of filter F,, only one of the two terms eiiKx is kept, yielding after F, A = e-lwrial
cos at e*iKx
(14)
The phase factor edKx relates to the direction of the light beam. It can be omitted if, after F,, the system is conveniently oriented. Thus, the incident amplitude on C, may be written A = emUJtal cosSlt ,
(15)
where we have dropped the imaginary factor i. 3
Volume
6. number
1
September
OPTICS COMMUNICATIONS
Like previously, the amplitude tank C, is given by
transmittance
.
r = 1 + ia*cos (at-Kx)
of
kHz) due to the fact that, because of large wavelengths, the separation of diffraction orders becomes impractical.
(16)
It follows that the amplitude be written
right after C2 may
A = eeiwf{al cos fif + $ala2 [cos (2Rt-Kx)
+ cos Kx]}
(17)
Blocking off the central order, we obtain for the image amplitude
5. Visualization
of waveforms other than plane
Consider the more general problem of a pressure wave with a wave vector having bothx and.)’ components; the z component is assumed to be negligible, that is we consider an almost cylindrical wave. The acoustic pressure field can be written, very generally y)J .
p(x, .v) = Re [eeinff(x, A = $iala2[cos(2LL-Kx)
1972
+ cosKx] e-iwf,
(21)
(18) where we have
and the corresponding
image intensity
is (32)
f(x, y ) = a(x, y) eiq(xJ) E = IAf2 = C[cos2(2L2-Kx)
+ cos2Kx Eq. (21) can thus be written
+ 2cosKx cos (2flt-Kx)]
.
(19) p(x, y) = a cos (fit-p)
The average intensity, that is expression tegrated with respect to time, is (E) = C[ 1 + &OS 2Kx] .
(20)
Like in eq. (1 l), the factor C relates to the distribution of acoustic amplitude in the xy plane. The cosine term represents the acoustic wavefronts, with a contrast equal to l/2.
Using this expression to calculate the amplitude transmittance of tank C, similar results are obtained. For instance in the case of the stationary method of modulation, eq. (16) becomes 7 = 1 •t iu, cos (m-q)
,
(24)
that yields, in the same manner as before A = &la2
4. Experimental
(23)
(19) in-
[cos (2Rt-p)
+ cosip] e- iwt ,
(25)
results giving for the average image intensity:
We have observed the wavefronts of a plane acoustic wave propagating in water at 2.3 MHz, with both methods of modulation. Photographs (a) and (b) of fig. 3 show the results. It can be seen that the method of modulation by a progressive wave has the advantage of a contrast equal to unity. On the other hand the first filtering operation is easier to achieve in the case of stationary wave modulation because only one term must be extracted from the spectrum. There is no theoretical limit on the range of frequencies over which either method can be applied. Nevertheless a practical limitation arises at low frequencies (< 100 4
(E) = c(l+~cos
2p) .
(26)
The factor C, proportional to 0$x, y), maps the variations of acoustic amplitude. The equiphase lines, for which cp= constant, appear with a contrast equal to l/2. The experimental conditions that are necessary to make this visualization are the following: (i) la(x, y)l < rr in order that exp [ia cos (C&p)] = 1 + iu cos(fwcp);
Volume 6, number 1
OPTICS COMMUNICATIONS
September 1972
(b) Fig. 3. Photograph of an acoustic wave propagating in water at 2.3 MHz; (a) by the progressive wave method of modulation, (b) by the stationary wave method of modulation. 5
Volume
6, number
Fig. 4. Photograph
of an acoustic
wave diffracted
by a slit in a vertical blade. The two pairs of vertical diffraction of light by the blade.
(ii) the Fourier spectrum of (22) must be sufficiently spread out in order to make the central order separable. But, because a(x, u) varies slowly as compared to cp(x, y), we need only to consider the spectrum of cos cpwhose mean spatial frequency is high. In fig. 4, we show a picture of an acoustic wave diffracted by a slit two acoustic wavelengths wide. It shall be noted that the method of modulation by a progressive wave yields an image intensity: (E) = C{l + cos 2[&x, v) -Kxg-1
}.
(27)
This method is not as reliable as the stationary method of modulation, but the distortion introduced by the term Kxg-l is negligible if g is large enough. On the other hand, this method of modulation by a progressive wave can be achieved by passing the laser beam twice through the test tank. In this way, one can visualize acoustic fields of randomly varying frequency, which is not possible with ordinary stroboscopy.
bright
1972
lines are due to the
6. Conclusion The usefulness of a simple technique for visualizing acoustical wavefronts has already been pointed out by Newman [3]. Let us say in addition that it can be an interesting method for investigating inhomogeneous waves which have only been studied for the case of the asymptotic solution [8]. It is also the case of surface waves in solids, since our analysis is valid for reflection as well as transmission types of experiments. We have studied in this article the working principle of an acoustic stroboscope which allows the visualization of progressive wavefronts. The resulting image gives also a representation of the amplitude distribution along the acoustic beam.
References [l] E. Hausler and W. Stumm, Ultrasonics 7 (1969) 118. [ 21 C. Bachen, E. Hiedemann and H.R. Asbach, Nature 133 (1934)
6
September
OPTICS COMMUNICATIONS
1
176.
Volume 6, number 1
OPTICS COMMUNICATIONS
[ 31 D.R. Newman, M.Sc. Thesis, Washington State University (1971). [4] R. Bar, Helv. Phys. Acta 9 (1936) 678. [S] E.F. Fox and G.D. Rock, Rev. Sci. Instr. 10 (1939) 345.
September 1972
[6] M. Born and E. Wolf, Principles of optics, 3rd Ed. (Pergamon, London) ch. 12. [7] S. Lowenthal and Y. Belvaux, Rev. d’Opt. 46 (1967) 1. [S] S. Lowenthal, Ann. Radioilectricite 19 (1964) 183.
7
OPTICS COMMUNICATIONS
September 1972
VISUALIZATION OF ACOUSTIC WAVEFRONTS BY MEANS OF ACOUSTIC STROBOSCOPY J.LAPIERRE
and SLOWENTHAL
Institut d ‘Optique, Faculte’des Sciences, 91 . Orsay, France Received 27 June 1972
The amplitude and phase distributions of a progressive acoustic wave are visualized simultaneously by means of acoustic stroboscopy. Two methods are described: (i) modulation of a laser beam by a progressive acoustic wave; (ii) modulation by a stationary wave. Spatial filtering of diffracted orders enables the progressive wavefronts to be seen as if they were stationary.
1. Introduction In this article, we describe a method for visualizing acoustic progressive waves whereby spatial variations of both the amplitude and phase of an acoustic field are made visible along the direction of propagation. Progressive wavefronts can be visualized by high speed photography [l] but this method does not permit the evolution of phenomena to be followed. One can alse use classical methods of stroboscopy [2,3] , but those systems are generally complex due to synchronization requirements. We propose a stroboscopic method which makes use of a simple acoustic modulator operating at the same frequency as that of the phenomenon to be observed. This kind of stroboscope had been used by Bar [4] in 1963 and by Fox and Rock [5] in 1939 for wavelength measurements. Nevertheless, it does not seem that these authors gave a correct interpretation of the phenomena involved in their experiment: (i) they were considering that a light beam traversing an acoustiu wave is modulated in intensity, instead of complex amplitude; (ii) their observation plane was not the image plane of the acoustic medium, so that they were performing a spatial filtering due to the limited aperture of the optical system. Our acoustic modulator can be operated in two different ways: with a progressive wave or with a
standing wave, corresponding to the stroboscope of Fox and Rock, and the one of Bar respectively. In both cases, a light beam, modulated by the stroboscope, illuminates the acoustic medium in which the wave to be visualized is propagating. Light diffracted by the acoustic wave is analyzed spatially; we will show that some of the diffracted orders can be used to produce a stationary image of the propagating wave. The progressive wave method of modulation will be shown to yield a contrast factor equal to one, while it is l/2 in the case of the standing wave method. On the other hand, this second method of modulation has a wider application than the first one, as far as the shape of the wavefronts to be observed is concerned.
2. Study of phenomena: wave
diffraction
by an acoustic
Consider a monochromatic plane wave: A = ewiwt, incident normally on a transparent medium in the z direction. A plane acoustic wave of the form p = Ap cos (Sb-Kx)
,
(1)
propagates in the x direction; here,p, a, and K represent the acoustic pressure, angular frequency and wave number, respectively. The index of refraction n 1
Volume 6, number
September 1972
OPTICS COMMUNKATIONS
1
We will show in the following section how the frequency shifts +51.and -R can be eliminated. Then, recombining the +l and ---1 orders. we will get A = a e-lwf cos Kx
+1 A=
e-
iwt
(5)
0 -1
z
Fig. 1. Diffraction of light by an acoustic wave. obeys the same law of variation
The light intensity corresponding to this expression will give an image of the acoustic wavefronts as they could be observed at a given time; this is due to the presence of the time-independent factor COSKX. At the same time, a representation of the spatial distribution of acoustic amplitude is afforded by the parameter a. In the following section, we will examine the problem of visualizing a plane acoustic wave. The case of a more complex wave will be briefly considered in section 4.
(2)
n = n0 + An cos (a2t-Kx).
Following the approach of Raman and Nath [6], we treat the acoustic medium like a pure phase grating so that the complex amplitude of the light beam right after the acoustic medium can be written A = exp i [cpO+ A9 cos (fir-Kx)
- ot]
,
(3)
where 90 and A9 stand for kin, and klAn respectively, with k being the wave number of the light in the acoustic medium, 1 the thickness. Expression (3) can be expanded in a Bessel functions series, of which we will retain the first two terms only because the higher order terms will be eliminated by spatial filtering. Therefore, the light amplitude right after the acoustic tank has the form A = eeiwl[ 1 t iu cos (Rt-Kx)]
,
(4)
where a is a constant, and where we have put 9,-, = 0. The expression in brackets represents the transmittance of the tank. In the filter plane F, we have the Fourier transform of this last expression. The three component amplitudes are
3. The acoustic stroboscope 3.1. Modulation by a progressive wave In fig. 2, tank C2 is a transparent medium where the acoustic wave to be visualized is propagated; we assume it is a plane wave in the present case. It is illuminated with a laser beam modulated by C, where an acoustic plane wave of the same frequency as in C, is propagated. At the end of C,, like in C,, there is an absorber in order to prevent the formation of stationary waves. According to section 2, the light amplitude, right after the modulator Cl, is given by A = eeiwt[ 1 + ia, cos (at-Kx)]
0: -1:
Jiaexp[-i(o+R)t] exp (-iwt)
.
The complex amplitude according to section 1
transmittance
of tank C2 is,
t] . 7 = 1 + ia2(x, y) cos (fi;2t-Kx) )
2
(7)
,
,
$a exp [-i(o-s2)
(6)
where al is a constant if we assume the modulating acoustic wave to be uniform over the whole cross section of the laser beam. The dc component of eq. (6) is filtered out in plane F, , and then the beam, after expansion, illuminates C,. Calling g the magnification of the afocal system (L1, L,, L3), the incident light amplitude on C2 is A = iehwfa cos (Rt-Kxg-l)
t 1:
,
(8)
September 1972
OPTICS COMMUNICATIONS
Volume 6, number 1
“1
Fig. 2. Acoustical stroboscope. C1 : modulator tank, Cz : test tank, L1 and L4: Fourier transform lenses, Lz and Ls : collector lenses, La: part of telescope with L1, F1 and Fz: spatial filters. Solid lines: pupil imagery, dashed lines: objective imagery. where a2(x, v) specifies information
on the possible variations of acoustic amplitude in the x, y plane. In the following, we will write simply a2 instead of a2(x, JJ). It follows that the light amplitude, right after C,, is [ 1 + ia cos (at-Kx)]
A = eViwf ia, cos (SU-Kxg-l)
- &zla2{cos [Kx( lg-1)
.
(9)
i
In order to get a time independent phenomenon at the acoustic frequency, one must simply pick the last term out of the last expression, that is .
[Kx(l-g-l)]
(10)
This operation is performed through the use of filter F,, having v = f K( 1-g- 1) as transmitted frequencies. The double diffraction system [7] (Ld, F2, L,, I> produces a fitered image of D2 with a distribution of light intensity given by ,
+ cos (!mKx)]
Right after C,, the light amplitude phase constant
- mt]
•t cos [Kx( l-g-l]}
E = IAl = Ccos2[Kx(l-g-1)]
3.2. Modulation by a stationary wave Without an absorber at the end of tank C,, a stationary wave is produced p = apl [cos (cl-Kx)
= emiwt ia, cos (S&Kxg-‘) i
A = e-iwt&zla2cos
in the x, y plane. The term cos2 [Kx( 1-g-l)] gives a scaled image of the acoustic wavefronts, with a contrast equal to one.
(11)
where C, which is proportional to la2(x, y)12, gives a representation of the acoustic amplitude distribution
A = e-iwf (1 + ia, [cos (at-Kx)
.
(12)
is, neglecting a
+ cos (CWKx)]}
= e-iat[ 1 + ia, ~0s ilt(eiJ%2-i~x)]
.
(13)
By the use of filter F,, only one of the two terms eiiKx is kept, yielding after F, A = e-lwrial
cos at e*iKx
(14)
The phase factor edKx relates to the direction of the light beam. It can be omitted if, after F,, the system is conveniently oriented. Thus, the incident amplitude on C, may be written A = emUJtal cosSlt ,
(15)
where we have dropped the imaginary factor i. 3
Volume
6. number
1
September
OPTICS COMMUNICATIONS
Like previously, the amplitude tank C, is given by
transmittance
.
r = 1 + ia*cos (at-Kx)
of
kHz) due to the fact that, because of large wavelengths, the separation of diffraction orders becomes impractical.
(16)
It follows that the amplitude be written
right after C2 may
A = eeiwf{al cos fif + $ala2 [cos (2Rt-Kx)
+ cos Kx]}
(17)
Blocking off the central order, we obtain for the image amplitude
5. Visualization
of waveforms other than plane
Consider the more general problem of a pressure wave with a wave vector having bothx and.)’ components; the z component is assumed to be negligible, that is we consider an almost cylindrical wave. The acoustic pressure field can be written, very generally y)J .
p(x, .v) = Re [eeinff(x, A = $iala2[cos(2LL-Kx)
1972
+ cosKx] e-iwf,
(21)
(18) where we have
and the corresponding
image intensity
is (32)
f(x, y ) = a(x, y) eiq(xJ) E = IAf2 = C[cos2(2L2-Kx)
+ cos2Kx Eq. (21) can thus be written
+ 2cosKx cos (2flt-Kx)]
.
(19) p(x, y) = a cos (fit-p)
The average intensity, that is expression tegrated with respect to time, is (E) = C[ 1 + &OS 2Kx] .
(20)
Like in eq. (1 l), the factor C relates to the distribution of acoustic amplitude in the xy plane. The cosine term represents the acoustic wavefronts, with a contrast equal to l/2.
Using this expression to calculate the amplitude transmittance of tank C, similar results are obtained. For instance in the case of the stationary method of modulation, eq. (16) becomes 7 = 1 •t iu, cos (m-q)
,
(24)
that yields, in the same manner as before A = &la2
4. Experimental
(23)
(19) in-
[cos (2Rt-p)
+ cosip] e- iwt ,
(25)
results giving for the average image intensity:
We have observed the wavefronts of a plane acoustic wave propagating in water at 2.3 MHz, with both methods of modulation. Photographs (a) and (b) of fig. 3 show the results. It can be seen that the method of modulation by a progressive wave has the advantage of a contrast equal to unity. On the other hand the first filtering operation is easier to achieve in the case of stationary wave modulation because only one term must be extracted from the spectrum. There is no theoretical limit on the range of frequencies over which either method can be applied. Nevertheless a practical limitation arises at low frequencies (< 100 4
(E) = c(l+~cos
2p) .
(26)
The factor C, proportional to 0$x, y), maps the variations of acoustic amplitude. The equiphase lines, for which cp= constant, appear with a contrast equal to l/2. The experimental conditions that are necessary to make this visualization are the following: (i) la(x, y)l < rr in order that exp [ia cos (C&p)] = 1 + iu cos(fwcp);
Volume 6, number 1
OPTICS COMMUNICATIONS
September 1972
(b) Fig. 3. Photograph of an acoustic wave propagating in water at 2.3 MHz; (a) by the progressive wave method of modulation, (b) by the stationary wave method of modulation. 5
Volume
6, number
Fig. 4. Photograph
of an acoustic
wave diffracted
by a slit in a vertical blade. The two pairs of vertical diffraction of light by the blade.
(ii) the Fourier spectrum of (22) must be sufficiently spread out in order to make the central order separable. But, because a(x, u) varies slowly as compared to cp(x, y), we need only to consider the spectrum of cos cpwhose mean spatial frequency is high. In fig. 4, we show a picture of an acoustic wave diffracted by a slit two acoustic wavelengths wide. It shall be noted that the method of modulation by a progressive wave yields an image intensity: (E) = C{l + cos 2[&x, v) -Kxg-1
}.
(27)
This method is not as reliable as the stationary method of modulation, but the distortion introduced by the term Kxg-l is negligible if g is large enough. On the other hand, this method of modulation by a progressive wave can be achieved by passing the laser beam twice through the test tank. In this way, one can visualize acoustic fields of randomly varying frequency, which is not possible with ordinary stroboscopy.
bright
1972
lines are due to the
6. Conclusion The usefulness of a simple technique for visualizing acoustical wavefronts has already been pointed out by Newman [3]. Let us say in addition that it can be an interesting method for investigating inhomogeneous waves which have only been studied for the case of the asymptotic solution [8]. It is also the case of surface waves in solids, since our analysis is valid for reflection as well as transmission types of experiments. We have studied in this article the working principle of an acoustic stroboscope which allows the visualization of progressive wavefronts. The resulting image gives also a representation of the amplitude distribution along the acoustic beam.
References [l] E. Hausler and W. Stumm, Ultrasonics 7 (1969) 118. [ 21 C. Bachen, E. Hiedemann and H.R. Asbach, Nature 133 (1934)
6
September
OPTICS COMMUNICATIONS
1
176.
Volume 6, number 1
OPTICS COMMUNICATIONS
[ 31 D.R. Newman, M.Sc. Thesis, Washington State University (1971). [4] R. Bar, Helv. Phys. Acta 9 (1936) 678. [S] E.F. Fox and G.D. Rock, Rev. Sci. Instr. 10 (1939) 345.
September 1972
[6] M. Born and E. Wolf, Principles of optics, 3rd Ed. (Pergamon, London) ch. 12. [7] S. Lowenthal and Y. Belvaux, Rev. d’Opt. 46 (1967) 1. [S] S. Lowenthal, Ann. Radioilectricite 19 (1964) 183.
7