34 



WAVE ACOUSTICS 



It turns out that a solution synthesized as was equa- 

 tion (124), but with a plus sign in equation (122) in- 

 stead of a minus sign, will satisfy this boundary con- 

 dition. This solution is 



V = 



' / 



(134) 



We verify that equation (134) satisfies dp/dy = by 

 differentiating equation (134) with respect to y, and 

 noting that dr/dy = —dr'/dy aX y = 0. 



We now examine the possibility of stationary 

 states for the case where the boundary is a hard sea 

 bottom. Again, we assume a harmonic source so far 

 from the bottom that waves reaching the bottom are 

 plane and we assume perpendicular incidence. Then 

 equation (134) must be replaced by 



V = al cos 2TrJ\t - -) + cos 2ir/f< + - j (135) 



which, by trigonometry, reduces to 



p = 2a cos 2 J- sin lirft. (136) 



A 



We easily see that equation (136) also represents a 

 stationary state of our fluid. The nodes of utter 

 silence are situated where cos 2ir(y/X) disappears, 

 that is, at ?/ = X/4, 3X/4, 5X/4, ■ • • ; the loops 

 of maximum sound intensity are located where 

 cos 2ir(y/\) equals +1, that is, at y = 0, X/2, X,- ■ •. 



2.7 



NORMAL MODE THEORY 



2.7.1 Plane Waves in a Medium 

 with Parallel Plane Boundaries 



The problem of sound propagation in a medium 

 bounded on two sides is extremely complicated and 

 cannot be solved in general. The difficulty lies in the 

 fact that the solution must satisfy not only the wave 

 equation, but also the initial conditions and the 

 boimdary conditions at each boundary. 



In Section 2.6 it was shown that certain definite 

 and instructive results could be obtained for the case 

 of a single boundary by considering the case of plane 

 waves and assuming (1) perpendicular incidence and 

 (2) an infinite change in density at the boundary. 

 The result was a standing wave pattern whose geo- 

 metrical properties depended on the wavelength and 

 whose maximum amplitude depended on the energy 

 in the incident wave. 



We shall keep these two assumptions in this sec- 

 tion and shall first find out under what conditions a 

 stationary wave pattern of any sort can be set up in 

 our bounded medium. The general expression for a 

 standing wave pattern is 



p = ^(2/) cos 2Trf{t - e) (137) 



where ^ is any function of y. In other words, equa- 

 tion (137) means that all points of the fluid perform 

 vibrations with the same frequency / and phase con- 

 stant e, but with amplitude ^(j/) depending on the 

 position coordinate y. The immediate problem is to 

 find out what sort of functions ^(j/) are necessary to 

 make equation (137) a solution of the plane wave 

 equation 



df dy^ 



(138) 



and also a solution of the boundary conditions, on 



the boundaries y = and y = L. These boundary 



conditions are either equation (139a), (139b), or 



(139c). 



p = at both y = Oandy = L (139a) 



dp 

 p = 0at2/ = 0; — = 0at2/ = L (139b) 

 dy 



dp 



dy 



= at both y = and y = L. 



(139c) 



It is immediately apparent that the condition for 

 equation (137) to satisfy the plane wave equation 

 (138) is 



dy^ X''^ 



0. 



(140) 



First consider the case of the boundary conditions 

 (139a). The boundary conditions (139a) can be re- 

 stated as 



vKO) = 0, HL) = 0. (141) 



The problem is thus reduced to the case of finding the 

 solution of an ordinary differential equation with 

 boundary conditions on both ends of an interval 

 ^y ^L. 



Equation (140) is a simple differential equation 

 whose general solution is well known to be 



2ir 2x 



yj/iy) =Asin—y + B cos —y (142) 



X A 



where A and B are arbitrary constants. 



The condition ^(0) = implies that B = and 



fiy) 



A sm — w. 



(143) 



