IV-6 



the first- order correction term giving the first approximation to the scattered 

 pressure remains a good approximation also for large distances of propagation. * 

 This is done as follows: In a medium without inhomogeneities, the waves of 

 interest (plane, spherical, cylindrical) may be represented in terms of their 

 amplitude and phase by: 



iS (x) -tnA +iS it (x) 



p (x) - A (x)e ° - e ° ° = e ° (IV- 16a) 



o — o 



where 



t (x) = S (x) - i ^nA (x) (IV- 16b) 



o — o — o — 



The functions Aq and S^ are both real functions representing, respectively, the 

 amplitude and phase of the wave as a function of position. We observe that a sur- 

 face on which S^ is constant is a surface of constant phase; such a surface can be 

 regarded as a wave front for our purposes. If a source of wave motion, which 

 would have resulted in a pressure distribution (IV- 16a) in a homogeneous medium, 

 is placed in an inhomogeneous medium, the resulting wave will have the same 

 general form but will differ in detail. The resulting wave can always be represented 

 as: 



/ \ A / X iS(x) iiKx) ,^^, ,^ . 



p(x) == A(x)e - = e - (IV- 17a) 



where 



t (x) = S (x) - i ^n A (x) (IV- 17b) 



Tlie function t|; represents the complex phase of the wave and will turn out to be 

 very useful in the analysis. We can always represent the complex phase of the 

 perturbed wave in terms of the complex phase function of the original homogeneous 

 wave plus a perturbation term: 



t = t + ti (IV- 18) 



o 



The perturbation ti is now not necessarily small compared to to- ^^ other words, 

 we permit the inhomogeneous wave to have a large difference in amplitude and phase 

 from the unperturbed homogeneous wave. To compare this approach with the 

 approach used for very small perturbations, we observe that if ti is small: 



it iti 



P = e e =p (l + iti + ---) (IV-19) 



o 



'Chernov - Chapter V, Section 16. 



artbur ai.littlcjnt. 



S-7001-0307 



