32 18 



upon a rigid wall ought not to include a term of magnitude pv^ or pv^Z. The 

 answer furnished by analysis Is In the negative. 



Even the exact theory of Rlemann for the propagation of plane waves 

 of finite amplitude leads to no direct contribution from the particle veloc- 

 ity V to the pressure on a rigid wall. The pressure should be a little more 

 than twice the incident pressure, but the excess is due entirely to depar- 

 tures from Hooke's law of elasticity; see the Appendix. It can be said that 

 the entire Increase in pressure arises from the arrest of the particle mo- 

 tion by the wall. No further Increase corresponding to pv^ should, therefore, 

 be expected. 



As an example, when water is compressed adiabatically from zero 

 pressure and a temperature of 20 degrees Centigrade, its pressure, up to 

 10,000 pounds per square inch, is approximately given by the formula 



p= 309000s (i + 75I00) ^''/^"^ f^^^ 



75000 



where s is the fractional compression or the decrease in volume divided by 

 the original volume; see the Appendix, Equation [l84]. The term p/75000 rep- 

 resents the departure from Hooke's law. Because of this term, the pressure 

 on a rigid wall due to the incidence of a wave of pressure of magnitude p^ 

 pounds per square inch is raised from 2p^ to 



2p. (l+^3^) [12] 



See the Appendix, Equation [185]. For an Incident wave having a pressure of 

 5000 pounds per square inch, the Increase is 3 per cent. 



In the reflection of spherical waves, also, the usual linear theory 

 leads to the conclusion that the pressure against a rigid wall is simply 

 doubled; the afterflow velocity* gives rise to no additional term in the 

 pressure. 



The familiar Bernoulli term in the pressure formula thus puts in 

 its appearance only when (a) the pressure field is two- or three-dimensional, 

 and (b) terms of the second order in the velocity are included. A small pi- 

 tot tube, for example, turned with its mouth toward the oncoming wave, will 

 register a pressure equal to p + pv^/2 where p Is the pressure and v is the 

 particle velocity caused by the wave In unbroken water, whereas with its 

 mouth turned at right angles to the direction of propagation it registers 

 Just the pressure p. The motion around the tube is three-dimensional; and 

 the increase in pressure Is of order v^. Similarly, the pressure at the 



See TMB Report ^80 (10), page 39. 



