IiI-70 



which is incident on the ocean surface 



z = S(x,y) 



(III- 46b) 



The ocean surface supposedly changes shape only very slightly during one cycle 

 of the incident sound wave, so that the motion of the surface may be ignored in 

 the scattering analysis. The pressure distribution of the resulting scattered 

 wave Pgj,(x,y,z,t) must satisfy the wave equation. Furthermore, the total 

 acoustic pressure p = p.^ + p^^ must satisfy an appropriate boundary con- 

 dition on the surface S(x,y). Because of the very poor impedance match between 

 water and air, the boundary condition is approximated very closely* by requiring 

 the total overpressure at the surface to vanish, i.e. , letting the surface be a 

 pressure release and allowing no sound to be transmitted to the air. Finally, 

 p must be an "outgoing" wave, i.e., its energy must flow from the surface 



downward, and p must remain finite as z-»- " . These three conditions 

 sc 



1. 



V^ p 

 ^sc 



1 

 c^ 







d^Psc 



dt^ 





2. 



p. + 



mc 



Psc ^ 



on z = 



= S(x,y) 



3. p is outgoing and remains finite (the so-called radia- 

 tion condition). 



completely define the theoretical problem. The question of interest is: how does 



p depend on the characteristics of S(x,y)? 



"This may be seen by studying reflection of a sound wave p^^r ~ e '^''■'^ Yz)- itSJt 

 from a plane interface between water and air. We recall from Section III-B that 

 the ratios of density and sound velocity for water and air are approximately 



P' _ c' 



g - — "^ 1.3x10 , h - — ^0.2. Tlie incident wave causes a reflected and a 



transmitted wave, p^^^ = p^ eik(ax-Yz)-iuJt ^nd Pjrans'Pt^^'^'^"^''''^'^^'^^^- 

 If we require the total pressure and normal velocity to be continuous at the inter- 



face, we find readily that p = - 



ll-h" 



1+gh^P^ 



::. Since gh =- 0.26x10''^, the 



reflected wave has a pressure which lies between 99.948% and 100% of the incident 

 pressure, depending on a . 



artbur B.lUtleJnir. 



S-7001-0307 



