﻿630 Dr. C. V. Burton on the Scattering and 



them contains 27r<rpdp = 27rasds secondary vibrators, whose 



contribution to the velocity potential of emitted plane waves 



is, for x positive, 



C 

 dyfr" = 2ttg sds . — cos (pt — vs — y), 



whence 



2iraO s=Jl 



* 



sin (pt — vs — y) 



The upper limit is written as s = R instead of 5=00 for the 

 sake of definiteness, the sheet of vibrators being limited to a 

 circle of such great radius p x (say), that R or p x V (l + x 2 [p-^) 

 is as nearly as we please constant over those values of x with 

 which we concern ourselves. Thus, w T ithin a constant, 



,„ 27TO-0 / . 7T\ 



■f"= — C08[pt-VX-ry+-j. . . . (7) 



12. In particular, if the vibrators are tuned to resound to 

 the frequency of the primary weaves, y = ^ir and 



2tto-C 



^r"— — ~ cos (pt — vx) (S) 



For points in the y^-plane, the total regular disturbance is 

 represented by 



(^ + ^'0z=0 = (A-27TO-C/u) COS^, 



and for the amplitude-constant C of the disturbance ema- 

 nating from each resonator, we have by § 3 (since the ampli- 

 tude of the exciting weaves is no longer A but A — 2iraC/v), 



C=A/v-27r<rG/v 2 , 

 that is 



-;/(>+ 50 ■ 



so that finally 



ylr // = — - cos (pt — vx) 



Y l + « U J I (9) 



where w=27rcr/v 2 



Similarly for negative values of x, that is to say in the 

 reflected wave, 



V'^-^^cosipt + vx) (9 a) 



13. The energy per unit area of the primary waves y 

 (arbitrarily represented by unity), diminished by" the energy 



