WAVE EQUATION IN A PERFECT FLUID 



13 



It can be shown that the velocity components 

 Ux,Uy,Ui also satisfy a differential equation of the 

 form of (23), provided the motion of the disturbed 

 fluid is irrotational. That is, if sound is propagated in 

 a perfect fluid in such a manner that no eddies are 

 produced, 



dC " poVaa:^ "•" dy- dz- ) 



with similar equations for it^ and Uz. 



Equations (23), (24), and (25) are equivalent; that 

 is, the fundamental laws of sound propagation can be 

 deduced from any one of them by using the known 

 relationship between a, u^, and p. In the following 

 sections, the most frequent reference is to equation 

 (24). Variation in the excess pressure is the most 

 familiar and probably the most intuitive change in 

 the disturbed fluid; also, the majority of hydrophones 

 used in the reception of underwater sound respond 

 directly to variations in excess pressure rather than 

 to variations in particle velocity or condensation. 



It is convenient, in equations (23) to (25), to set 



Po 



so that the wave equation becomes 



= cV 





(26) 



(27) 



It will be pointed out in Section 2.3.1 that c, defined 

 by equation (26), has the general significance of soimd 

 velocity. 



The wave equation (27) gives the relationship be- 

 tween the time derivatives and the space derivatives 

 of the pressure in the fluid through which the sound 

 is passing. Relationships of this sort have been used 

 by generations of mathematicians as a starting point 

 for the development of physical theory. In the field 

 of sound, these mathematicians have explored the 

 methods by which the future course of pressure in a 

 fluid can be calculated if only the initial distribution 

 of pressure is given. Mathematically, this amounts to 

 solving the wave equation (27) with given initial and 

 boundary conditions. Once the distribution of pres- 

 sure is known, the sound intensity at any point and 

 time can be calculated by the methods of acoustics. 



2.2.2 Initial and Boundary 



Conditions 



The differential equation of a physical process gives 

 a dynamical description of the process relating the 



various temporal and spatial rates of change, but does 

 not of itself tell all we want to know. In the case of 

 the wave equation, we desire knowledge of how the ex- 

 cess pressure varies in space and time. This informa- 

 tion is obtainable, not from the wave equation itself, 

 but from its mathematical solution. The general solu- 

 tion of a partial differential equation Uke the wave 

 equation always contains arbitrary constants and 

 even arbitrary functions. These arbitrary constants 

 and functions are, in any individual problem, ad- 

 justed to make the solution fit the special condi- 

 tions of the problem. 



These special conditions are of two kinds : boundary 

 conditions and initial conditions. In the problem of 

 sound propagation, the two types of conditions can 

 be defined as follows. Boundary conditions are fixed 

 by the geometry of the medium itself. If the medium 

 is finite, boundary conditions must always be con- 

 sidered. The excess pressure must fulfill certain con- 

 ditions at a boundary such as the sea surface, sea 

 bottom, or internal obstacle. The pressure may have 

 to be zero at one boundary, or a maximum at some 

 other boundary, or may satisfy some other condi- 

 tion.'' 



Initial conditions are concerned not with the fixed 

 geometry of the fluid and its surroundings, but with 

 the special disturbances which cause soimd to be 

 propagated. One type of initial conditions specifies 

 the pressure distribution at a certain instant of time, 

 t = to, over the whole fluid. That is, we are given a 

 function p(x,y,z), and are told that 



pix,y,z,ta) = p{x,y,z). (28) 



Another type of initial conditions specifies the pres- 

 sure as a function of time at a fixed point (a;o,2/o,Zo) 

 of the fluid. That is, we are given a function p{t) and 

 are told that 



p(xo,yo,zo,t) = p{t). (29) 



Every actual case of soimd propagation involves 

 both initial conditions and boundary conditions. 

 However, in our mathematical approximations to 

 reahty it is best to start with the simplest case, that 

 is, where sound is propagated into a medium that is 

 infinite. Of course, in theory every problem can be 

 regarded as a problem in an infinite medium. We can 

 consider the sea and air together as one medium, 

 whose physical properties at equilibrium (elasticity, 



■■It will be seen in Section 2.6.1 that the excess pressure 

 must be nearly zero at the boundary separating water and air 

 and a maximum at the soUd bottom of the sea. 



