m SURFACE AND OTHER EFFECTS 



if Ui and U2 are the particle velocities in the incident and reflected waves. 

 For plane waves, Ui = Pi/poCo, U2 = —Pi/poCo, and hence we have 



(10.6) u = ^'~ ^' (x = 0) 



PoCo 



= {x = 0,t = 0) 



The incident and reflected plane waves must be of the form Pi{t + x/co) 

 and Poit — x/co), and a specified function Pi determines the values of 

 P2 and u from Eqs. (10.5) and (10.6). Assuming an exponential wave 

 Pi = P7/ze~(^ + -^/^°)/^ and solving for P2{x = 0) and u on the boundary sur- 

 face (assumed not to move appreciably) by standard methods gives 



(10.7) u = ^^^"^ ^ [e-^^/^ - e-V^] {t > 0) 



m{l - /3) 



P2{x = 0) = j^ [(1 + ^)e-^/o - 2^e-^^/o] {t > 0) 



where ^ = poCoQjm. 



The displacement s of the plate in the direction of the incident wave 

 obtained by integrating Eq. (10.7) is 



^ _2P^^ri 

 m{\ - ^) L/3 



(1 - e-^</^) - (1 - e-'/^) 



and the final displacement for ^ = 00 is therefore 



2P^^2 2PmB 



(10.8) 



W/3 poCo 



The final displacement is thus proportional to the impulse P^B of the 

 incident wave, and is in fact just twice the particle displacement of 

 the water by this wave. This result is true regardless of the mass of 

 the plate, because the motion in this case is limited only by the inertia 

 of the plate. If the plate is subject to other constraints, as it must be in 

 nearly any practical case, their effect must restrict the motion, and only 

 for a structure of relatively slow response compared to the duration of 

 the incident wave will the impulse be the controlling factor. A further 

 restriction is the fact that the analysis leading to Eq. (10.8) requires in 

 most cases that the water in front of and in contact with the plate with- 

 stand large negative pressures (tensions). 



The reflected pressure P2 at points in the water ahead of the plate 

 must be a function of {i — x/co), and from Eq. (10.7) the resultant pres- 

 sure P in the water is therefore 



