EFFECT OF A BOUNDARY 



31 



Equation (119) gives interesting results when it is 



applied to the case of a sound wave in water hitting 



the surface separating water from air. The numerical 



values are (subscripts for air; no subscripts for 



water) : 



c p 



- = 4.3; ^- = 770. 



C\ pi 



By substituting these values into equation (119), 



Ar _ l-3,31lVl +0.95tan^g. 

 ■4 . 1 + 3,31 iVl + 0.95tan2ei 



For perpendicular incidence Bi vanishes, and Ar/Ai 

 differs from — 1 by less than one part in a thousand ; 

 for greater values of di the approximation to —1 is 

 even better. A wave in water reflected by air thus 

 preserves its real amplitude almost exactly, that is, 

 almost all the energy in the incident wave remains in 

 the water. But it reverses its phase; this means tiiat 

 Vt = — Pi at the boundary. This conclusion, that the 

 resulting total pressure at this type of interface 

 should be very nearly zero, was called a boundary 

 condition in Section 2.6.1. The derivation of this 

 section furnishes the justification for assuming this 

 boimdary condition, which was stated without rigor- 

 ous proof in Section 2.6.1. Equation (119) provides 

 an estimate of the error caused by replacing transi- 

 tion conditions at a boundary with the more simple 

 boundary conditions. In the case of the interface 

 separating water and air this error is clearly very 

 slight. 



Another interesting case is the incidence of under- 

 water sound on a hard bottom like solid rock or 

 tightly packed coarse sand. The treatment of sound 

 waves in solids is rather more involved than the 

 treatment of sound waves in fluids because a solid 

 has two different kinds of elastic forces : those which 

 resist changes in volume; and those which resist 

 changes of shape (bulk modulus and shear modulus). 

 Consequently, two different kinds of propagation of 

 3ound are possible in a soUd. The two types are 

 usually referred to as longitudinal waves and trans- 

 verse waves. In the oblique incidence of underwater 

 sound in a water-solid interface both types of waves 

 are generated in the solid and the transition condi- 

 tions are, therefore, more involved than those dis- 

 cussed previously. If, however, the solid is quite 

 rigid — that is, if both bulk modulus and shear 

 modulus are appreciably greater than the bulk 

 modulus of water — then it may be assumed, in good 

 approximation, that the interface will not permit 



displacements perpendicular to itself. In other words, 

 Uy will vanish approximately. If z/„ at the interface is 

 zero, then its time derivative vanishes as well; and 

 by reason of the equations of motion (17), 



dp 



— = at 2/ = 0. 

 dy 



This is the boundary condition which is often as- 

 sumed in the treatment of reflection from a hard 

 bottom. If this boundary condition is realized, it can 

 be shown that the incident and reflected waves will 

 have equal amplitude and the same phase. Thus, 

 when sound is reflected from a rock bottom, almost 

 all the energy of the incident wave will be found in 

 the reflected wave. For a soft bottom like mud, this 

 boundary condition will no longer be satisfied, even 

 in approximation, and considerable sound energy may 

 be lost by transmission through the interface. 



2.6.3 Homogeneous Medium with 

 Single Boundary 



Point Source Near Sea Surface 



We shall now solve the problem of finding the solu- 

 tion of the wave equation which satisfies the bound- 

 ary condition p = at the interface y = 0, and cor- 

 responds to a sound wave radiated by a point source 

 at the depth h. This situation is illustrated in 

 Figure 11. The depth of the ocean is assumed to be 



y 



ORIGIN 



r __^ — 'Plx.y.x) 



o*'ro7-'h,0) 



Figure 11. Pressure produced at location P by sound 

 source O. 



infinite. The initial conditions are specified by the 

 assumption that in the immediate vicinity of the 

 source, that is, for points whose distance from the 



source r, 



r =V x^ + z''+ {y +hy 

 is very small, the pressure satisfies the relationship 

 rp{r,t) = Fit). (120) 



