34 20 



The expression for the pressure can be constructed by using the 

 principle of superposition. 



The waves are first imagined to be reflected from the surface of 

 the target as if it were rigid. This gives a resultant wave field in which, 

 at the surface, the incident pressure is doubled, while the particle velocity 

 has no component normal to the surface. 



A correction is then added to allow for the motion of the surface. 

 This correction is obtained by assuming the existence on the surface of a 

 suitable distribution of simple point sources emitting waves of pressure. 

 Because the surface is plane, each of these waves affects the normal compo- 

 nent of the particle velocity only at the element that emits the wave. For 

 this reason the strength of the point sources is easily ad.iusted so as to 

 satisfy the necessary boundary condition, which is that the surface and the 

 ad.jacent fluid must have a common component of velocity normal to the sur- 

 face. It is found that the pressure emitted by each element of the surface 

 must be proportional to its normal component of acceleration. 



The contributions made by the emitted waves to the pressure at any 

 given point in the fluid will be retarded in time because of the time re- 

 quired for the waves to travel from their point of origin. The following ex- 

 pression is obtained for the pressure at any point Q on the surface at time 



P = 2P. -2^/}^...<iS+p„ [13] 



where p^ is the total hydrostatic pressure, including atmospheric pressure, 

 p. is the incident pressure at the point Q and at the time t, 

 p is the density of the fluid, 

 c is the speed of sound in the fluid, 

 dS is an element of area on the face of the target, 

 z is the component of displacement of dS in a direction perpendicular 



to the initial position of the target, measured positively away from 



the fluid, and 

 s is the distance of dS from Q. 

 z denotes d^z/dt^, and the subscript ( - s/c means that each element 



dS is to be multiplied by the value of its acceleration z not at the 



time t but at the time t - s/c. 



The integration extends over the entire face. See the Appendix, Equation 

 [100], and Figure l4. 



The factor 2 in Equation [13] may be regarded as a reflection ef- 

 fect arising from the mere presence of the target. The term containing the 

 integral represents a relief of pressure, as explained by Butterworth (1 ), or 



