79 



MATHEMATICAL APPENDIX 



WAVES INCIDENT UPON THE INFINITE PLANE FACE OF A TARGET 



The only case of wave reflection that can be handled easily is the 

 incidence of waves upon a plane reflecting surface of infinite lateral extent. 

 The waves may be of any type and Incident at any angle, but it must be assumed 

 that they are weak enough to make the linear theory of wave propagation appli- 

 cable. Furthermore, if movement of the surface occurs, its displacement must 

 be small. The surface will be called a target, but it may be wholly or in 

 part merely the free surface of the water. The case thus characterized will 

 be under discussion except as otherwise stated. 



Under these conditions, an expression for the pressure field in the 

 fluid in front of the target can be built up by the method of superposition. 

 Let p- denote the pressure that is added tc the hydrostatic pressure p^ at 

 any point in the fluid by the incident wave or waves; that is, p^ + p Is the 

 pressure that would exist there if the target were replaced by fluid. Let a 

 set of reflected waves be added such as would occur if the target were rigid. 

 These waves are simply the mirror image of the incident waves in the face of 

 the target; together with the Incident waves, they give a pressure field in 

 which, at any point on the target, the excess of pressure is 2p., while the 

 component of the particle velocity perpendicular to the face is zero. 



The target and the fluid must, however, have the same normal compo- 

 nent of velocity. This may be secured by adding further waves such as would 

 be emitted by a suitable distribution of point sources located on the face of 

 the target. In the waves emitted by a point source, the pressure p and the 

 particle velocity t; at a distance s from the element may be written 



^.= 7/'(^-7). ^. =^/t^-f)-^/(^-f) 198] 



where ( is the time, p is the density of the fluid, c is the speed of sound 

 in it, and where f(t - s/c) stands for some function of the variable t - s/c, 

 and /' for the derivative of this function. The fluid emitted by 

 the source will be that which crosses a small hemisphere drawn 

 about the source as a center; see Figure 29. The volume V^ emit- 

 ted per second will be, therefore, Zns^v^. Or, since the first 

 term in Equation [98] becomes negligible in comparison with the 

 second as s -*■ 0, 



'''->'d""'^A'~i)]-T-f''> 



If there are N sources per unit area, the volume emit- 

 ted per second from an element 6S of the surface will be NV^^S. Figure 29 



