III-l 



III. STRONG INHOMOGENEITIES 



A. SCATTERING OF SOUND FROM A FLUID SPHERE 



Before we examine the actual strong inhomogeneities of the ocean, we 

 would like to specify the equations of sound propagation in water and study the 

 very idealized model of a plane sound wave scattering from a fluid sphere. This 

 will help to give us some insight into the effect of the scatterer's physical prop- 

 erties (size, hardness, density) on the characteristics of the resulting scattered 

 sound (scattered power, directivity). 



The passage of an acoustic wave through ocean water changes slightly 

 the local state of the pressure, density and water velocity. We shall designate 

 the unperturbed state by (po , Po , Uq ) and denote the incremental fluctuations 

 of pressure, density and velocity caused by the sound wave by (p, p, u). The lat- 

 ter are in general functions of position x and time t, e.g., p = p(x, t). Through- 

 out this chapter the unperturbed state variables (po , Po . Uq ) will be regarded as 

 constant in the region of space filled with water, i.e. , outside the strong inhomo- 

 geneities. In fact, the unperturbed velocity Uo is invariably set to zero; all analy- 

 sis is performed in a frame of reference in which the unperturbed water is stagnant. 



The equations of motion of the medium for acoustic (very small) disturb- 

 ances are well known and consist of an equation of mass conservation (continuity) 

 and three equations of momentum conservation: 



P + Pq u. . = (mass conservation) (III-l) 



o u. + p . = , j = 1,2,3 (momentum conservation) (III-2) 



In addition to these equations of motion, there is an equation of state which relates 

 the local pressure in the medium to the local density. * Since the square of the local 

 sound velocity, Cq , is just the derivative of the pressure with respect to the density 

 (at constant entropy), and since p and p are small variations of the unperturbed 

 values Po and Po, we may write at once: 



p = c^ p (equation of state) (HI- 3) 



*A11 motions are of sufficiently low amplitude and frequency that strictly adiabatic 

 compression and expansion occurs when a sound wave passes. 



:arthur a.ltittleJnc. 



S-7001-0307 



