14 



WAVE ACOUSTICS 



density, and other properties) suffer a sharp change 

 at the separating surface. However, the mathematical 

 treatment of a medium with strongly variable physi- 

 cal properties is still more difficult than the treatment 

 of boundary conditions. We are free to schematize 

 the physical situation in the most convenient way. 

 Accordingly, we shall start with the consideration of 

 an infinite medium, where the elasticity and density 

 at equilibrium are not strongly variable, and shall 

 later specialize our treatment by the consideration of 

 boundary conditions (Sections 2.6 and 2.7). 



Initial conditions must always be considered since 

 without them no sound could possibly originate. Un- 

 less the initial conditions are themselves very simple, 

 the solution of problems even in an infinite medium is 

 quite involved mathematically. The most practical 

 procedure is to first find solutions under very simple 

 initial conditions and use these solutions as building 

 blocks for constructing solutions of problems with 

 more complicated initial conditions. 



2.3 SIMPLE TYPES OF SOUND WAVES 



2.3.1 



Plane Waves 



It is convenient to start our study of the solution 

 of the wave equation with the assumption that the 

 disturbance is propagated in layers. We assume that 

 at any time t the excess pressure p is a function of x 

 only; that is, p is independent of y and z. With this 

 understanding, the wave equation (27) reduces to 



d^P ^ 2^ 

 a«2 ^ 3x2' 



(30) 



The eighteenth century mathematicians knew that 

 if p is an arbitrary function of either (t — x/c) or 

 {t + x/c), or a sum of two such functions, then p 

 satisfies equation (30). The proof is easy. Assume 

 that p = f{t — x/c) where / is any function. Then," 





=K'-?)^ S=M'-:)- 



The proof is identical for a function of the argument 

 (t + x/c). The sum of two solutions will itself be a 

 solution because of the general theorem that the sum 

 of two solutions of a homogeneous linear differential 

 equation will also satisfy the equation. 



" In accordance with usual calculus notations, f"{t — x/c) 

 represents the second derivative of /(z) evaluated for z = 



t - x/c. 



Also, it can be shown that any function of x and t 

 which is not of the form 



cannot possibly satisfy equation (30). The proof is 

 carried out as follows. Represent the unknown solu- 

 tion of equation (30) in the form 



P = K^,v), ^ = X ~ ct, 1) = X + a. 



If the differential equation (30) is written in terms of 

 the new variables f and tj, it reduces to 



dm,v) 



d^dr, 



= 0. 



This equation implies that the first derivative df/d^ 

 must be a function of f only and independent of ??, for 

 otherwise the second derivative d-f/d^dr] could not 

 vanish. Thus / itself must have the form 



m,v)=fm+f2iv). 



Let us focus attention on all the solutions of equa- 

 tion (30) which have the form 



-<'-> 



There are an infinite number of such solutions cor- 

 responding to the infinite number of possible choices 

 of /. However, no more than one of these solutions 

 can fit the special conditions of a particular physical 

 situation since the actual pressure at a specified point 

 and specified time can have only one value. Suppose 

 that there is one member of the family of functions, 

 denoted by fi(t — x/c), which satisfies the given 

 initial and boundary conditions. Then a fixed value 

 of {t — x/c), say 4.13, will always be associated with 

 some fixed value of the excess pressure, given by 

 /i(4.13). If /i(4.13) is equal to 0.02, then the excess 

 pressure will be 0.02 at those combinations of time 

 and place where (t — x/c) = 4.13, that is, where 



X = ct — 4.13c. 



In other words, as the time increases, any fixed value 

 of the excess pressure travels in the positive x direc- 

 tion with the speed c. This result is clearly true no 

 matter what the form of /i or the particular value of 

 the excess pressure. Such a process, in which a given 

 pressure change travels outward through a medium, 

 is referred to as propagation of progressive waves. 



Similarly, if a function /2(< + x/c) is the sole mem- 

 ber of the family of functions (31) which satisfies the 

 imposed initial and boundary conditions, progressive 

 waves will be propagated with the speed c in the 



