ON THE DIFFRACTION OF LIGHT BY PROGRESSIVE AND STANDING SUPERSONIC \VAVES

BY ROBERT MERTENS (Seminarie voor Wiskundige Natuurktmde, Rijksuniversiteit Gent, Belgium)

Received September 27, 1955 (Communicated by Sir C. V. Raman, r.R.s., N.L.)

1. INTRODUCTION

USING a method based on series expansions in Mathieu functions, we have shown the following property concerning the intensities of the lines of the spectra caused by the diffraction of light by supersonic waves at normal incidence of the light ~ : if only the zero and first orders are visible, the values of the intensities of the orders -k 1 and -- 1 for standing supersonic waves ate half the values of the intensities of the same orders in the case of pro- gressive sound waves.

A confirmation of this property was found in the experiments of Sanders, z which give a systematic comparison between the diffraction spectra caused by progressive and standing supersonic waves.

The purpose of the present paper is to deduce this property f rom the generalised theory of Raman and Natho 3

2. THE RESULTS OF NAGENDRA NATH FOR PROGRESSIVE SOUND WAVES 4

According to the generalised theory of Raman and Nath the amplitudes of the diffracted light waves satisfy differential equations

d•Ÿ191 2 d~ - - - ( ( ~ n - 1 - ~ , ~ § = in2 p r

with the

where

the following system of difference

boundary conditions

r 1, e n ( 0 ) = 0

= 2zrt~z )t '

(n = -- e o , . . . . , -k co), (1)

(n # O) ; (2)

z = the width of the sound field along the direction of the incident light,

p -- q191 , 195

196 ROBERT MERTENS

;~ = the wave-length of the incident light in vacuum,

~*= the wave-length of the supersonic waves in the considered medium,

iz o = the refractive index of the undisturbed medium,

iz ----the maximum variation of the refractive index.

Nagendra Nath has further shown, 4 that if the parameter p >~ 1, only a small number of orders will appear. In the case that only zero and first orders are visible, the system (1) becomes,

I ~.~~O + r = 0 (r = - - r (3 a)

2 (3 b)

and its solution satisfying the boundary conditions (2) is

r 1 p2+8s inZ 4 x

O (pz+ 8)�89 )}3 (4a) e x p [ i {~{ - - t an -i ((pz_t_g~~ tan 4

2 {(pZ + 8)~ r 1 8 9 sin 4 ~} exp { ip~}. (4b)

The first order intensity is then

4 sinO{ -(oz+8)�89 e} ~ 4 p~ + 8 . 4 f i s i n 2 Px /1 �91 (5)

3. THE RESULTS FOR STANDING SOUND WAVES

In the case of standing sound waves the system determining the ampli- tudes of the different diffracted orders is 3

2 d~__n _ sin E (~bn_i--r = in~pCn with

e = 2rrv*t,

v* = the frequency of the sound waves.

The boundary conditŸ still are

r (0) = l , ~~ (0) - - 0.

( n = - - o % . . . . , + o o ) (6)

(6 a)

(7)

Diffraction of Light by Progressive and Standing Supersonic Waves 197

I f orders higher than the first are to be neglected, the system (6) becomes explicitly,

(8 a) I dd~~~ ~ + r sin ~ = 0

_dr -- r sin E = iPr 2

Maldng the substitution

Ÿ = �91 sin E

= p/sin E,

we can write this system as

(8 b)

(9 a) (9 b)

I d~ -o + r = 0 (10 a)

2 dr _ r = iar (10 b) d~

which has the same forro as the system (3 a, b) and, of course, the same solution, ir one respectively replaces �91 and p by ~ and ~r. Formula (4 b) giving the first order amplitude thus becomes,

I li~, 2 sin (e~ + 8)�89 g e r = (cr2 + 8)�89 4

2 s i n e { ( p ~ + 8 s i n ~ e)�89 } i,t = (p2 _~ gsi¡ ep sin 4 ~ e

or taking into account, that p >~ 1

2 s i n e (_~ ) ipt ei~ -- e-i~ (~ ) ip~ r ~ - - sin �91 e-5- - - - - sin ~ e ~ - (11) p lP "

This wave is thus splitted in two waves, having frequency-changes of the amount + v* and -- v*. The amplitude of the wave undergoing a frequency- change + v * is ~ + = (1~lp) sin (M/4) exp (ipf/4), while the ampli tude of the wave having a frequency-change --v* is ~ _ = (-- 1~lp) sin (.o�91 exp (iM/4). So the total intensity of the ¡ order becomes

2 p~ (12) h (~) = g~-+~+*+ ~)7._~._*= p~ sin 2 ~ - .

The property stated in the introduction now foUows immediately f rom comparison of the fo rmule (5) and (12).

198 ROBERT MERTENS

4. P~MARKS

(1) I f sin E = 0 (t = m/2v*), it is seen f rom the system (8 a, b), that 4o = constant = 1 and 41 = constant x exp (ip~/2) = 0, this last constant being zero according to the boundary condit ions (7).

(2) The formula (12) may also be obtained in another way, which does not take account of the spectral character of the different orders. F rom the formula (5) the first order intensity in the case of standing sound waves may be written

4 {(a~q-8)�89 } 4sin2~ {(p2q-8sin~~)�89 } Ii (~) = a-~-+~8 s in~ 4 ~ -- p~ + 8 sin% sin~ 4

4 si~12E sin~ ~ ~ (p >~ 1). p2

This intensity is depending periodically on time, so that only the mean intensity will be observed experimentally. This mean value is given by

4 (~)1 /1 (~) = / 1 (~) = ~ sin ~ ~ ~-, f

0

2 sin 2 2~rv*t dt = sin z ~ 4

( r * = the period of the sound waves), which is the value given by (12).

5. SUMMARY

It has been shown that the following property of the spectrum of the diffraction of light by supersonic waves, in which orders higher than the first are missing, can easily be deduced from the generalised theory of Raman and Nath ir p >2> 1 : the intensity of the first orders in the case of standing sound waves has half the value of the intensity of the same orders in the case of progressive sound waves.

The author wishes to express his thanks to Professor Dr. M. Nuyens for the valuable discussions he had with hito.

REFERENCES

1. Mertens, R. .. Simon Stevin, 1950/51, 28, 1.

2. Sanders, F. .. Canad. J. Res., 1936, 14, 158. 3. Raman, C.V. and Proc. lnd. Acad. Sci., 1936, 3A, 119,459.

Nagendra Nath, N. S. 4. Nagendra Nath, N.S. .. lbid., 1938, 8 A, 499.