64 



RAY ACOUSTICS 



3.6.1 Eikonal Wave Fronts versus 

 General Wave Fronts 



It will be remembered that the entire method of 

 rays was based on the eikonal equation (13), which 

 in turn was based on the assumption that the wave 

 fronts (7) "grow" perpendicularly to themselves. 

 That is, the eikonal equation was derived by assum- 

 ing that the wave front at time t -\- dt is found from 

 the wave front at time t by moving each point on the 

 latter a distance cdt along the outward normal. We 

 shall now show that wave fronts ordinarily do not 

 obey this law of propagation rigorously, but that the 

 assimaption often provides a good approximation. 



It is intuitively apparent that wave fronts, defined 

 purely as surfaces of constant phase without refer- 

 ence to the way they grow, exist in the exact case, at 

 least when the dependence on time is harmonic. We 

 shall define these wave fronts in the rigorous case by 



Vix,y,z) = Co{t - to) (91) 



reserving the expressions W for those cases where the 

 wave fronts grow perpendicularly to themselves, and 

 where W therefore satisfies the eikonal equation. We 

 shall call surfaces (91) general wave fronts, and sur- 

 faces defined by similar equations, with V replaced 

 by W, eikonal wave fronts. 



We know that in instances where the sound source 

 vibrates harmonically with a single frequency / the 

 solution of the wave equation can be expressed as the 

 real part of the complex expression 



p = ^(a;,2/,3)e^"^''-'^^<"'«'^'^^'^°i (92) 



This expression is identical with equation (6), except 

 that we assume that the expression (92) with the 

 fimction V{x,y,z) is a rigorous solution of the wave 

 equation, while the expression (6) with the function 

 WXx,y,z) was obtained by means of a Huyghens con- 

 struction so that W{x,y,z) would satisfy the eikonal 

 equation. 



We now shall see under what conditions the ex- 

 pression (92) can satisfy the wave equation and, 

 simultaneously, V{x,y,z) satisfy the eikonal equation. 

 Suppose p satisfies equation (27) of Chapter 2, and 

 simultaneously V satisfies the eikonal equation (13). 

 The latter condition is 



/dFV /aFV /sfV , „ 



The former condition may be simply calculated by 

 noting that equation (92) may be written as 



(94) 



Substitution of the expression (94) into the wave 

 equation, performance of the indicated differentia- 

 tions, and collection of terms is a straightforward 

 calculation which will not be reproduced here. The 

 real and imaginary parts must vanish separately; 

 these parts are 



\U^(}ogA) 

 .dy/ ■ \dz/ " 4x1 dx^ 



{dV\ (dV\ {dV\ 

 \Vx) + \Ty) + \Tz) 



a^iog^) a''(iogA) Vd{\ogA) 



dy' dz^ "^ L dx 



] 



I dy 



and 

 dx^ "^ 



dy^ dz^ 



VdVdilogA) 



Li 



0. (95) 



.dx dx 



dV 3(log A) dV djlogA ) 

 dy dy dz dz 



] = o. 



(96) 



Clearly, V will satisfy condition (93) only if 



d^^ogA) d'dogA) ^ a^log^) 



"I dx^ 



+ 



pdogA) ^ 



+ 



dy' 



L dy . 



dz' 

 2 , VdilogA) 



+ 



L dz 



]■} = »■ 



(97) 



)-. 



(98) 



p = e 



log A 



-2jrif(.V/co)^2nft 



This can happen if Xo is zero, or if 



l/d'A d'A dU 



Bs- 1 1 



A\dx^ dy' dz' 



since the expression in braces in (97) easily reduces 

 to the above. This condition (98) is usually not 

 satisfied. While it happens to be satisfied by the pres- 

 sure wave of a point source in a homogeneous 

 medium, it does not hold, for instance, for the radia- 

 tion of a double source. In general, equations (93) 

 and (95) will be rigorously equivalent only if the 

 wavelength Xo vanishes. 



3.6.2 Conditions for Nearly 



Eikonal Wave Fronts 



We derived in Section 3.6.1 the conditions under 

 which wave fronts, defined as expanding surfaces of 

 constant phase of condensation, expand perpendicu- 

 larly to themselves. It is more useful to know how 

 large the frequency must be, relative to the other 

 parameters of the problem, before the function 

 V(x,y,z) of equation (92) very nearly satisfies the 

 eikonal equation; we will then know under what con- 

 ditions the wave fronts are very nearly perpendicu- 

 larly expanding. 



