296 - 2 - 



Ave rag e Pressure on the Pisto n. 



Let p(t) De the incident wave at any time t 



v(t) De the velocity of the piston at any time t away from the water 



R, be the ratJFus of the piston 



Rj be the radius of the baffle 



pc be the acojst tc impedance of water 



Then the average pressure on the piston is 



p = 2 p(t) - pc v(t) + 



i£c J^^l.U.I) /yr Y ,,.2 J7':p„.r, , ..—^ 



i£E / ''' V (t - I) / -(S Y dr - _i. 

 77- Rj ^ / I 2Rj y 77 R^ 



V^t ,,, . r, /:. i:!l« ^ dr 



(irR,)-^ (1) 



For an infinite piston, i..^. putting R^ = R^ and letting Rj^-oo, (l) reduces to the Taylor 

 expression for an infinite plate provided the displacement of the piston and impulse remain finite: 



p = 2 p(t) - pc v(t) (2) 



For an infinite baffle, i.e. R^ - °j (l) reduces to 



Butterworth's expression 



p = 2 p(t) - pciy(t) +^^^ J"^ v(t-I) /l -(j^ ) dr (3) 



^'^'^^ ^?^'*-^V^-(^t) 



pi*ovided p (-Of) is zero 



If it is assumed that thi 

 function of time, (l) reduces to 



2R 

 If it is assumed that the velocity of the piston in the interval t - 1 to t is a linear 



-p - 2p(t)-_^L v(t) - __ ; 2 1 p(t-:) /i-i_A_i_. dr M 



8 pR. ., , 2 , R/R, , r, / (r'+Rf-Rv 

 _L v(t) J 2 1 p(t-i) /i - L_L 



3 77 77 R^ Rj-Rj (2r R^) ^ 



The non-compress I ve approximation, in which time retardations are assumed negligibly small, is 

 obtained by tettinj c -■ 00 in (l) and gives 



8 pR, 

 p = p(t) - L v{t} for the finite baffle (5) 



3 77 



8 pR, 

 and p = 2 p(t) - i i^[t) for the infinite baffle (6) 



377 



It is seen that the terms due to the motion of the piston in (l) reduce to the same factor in 

 («) as in (5) and (6). 



Deflection of the Piston . 



The equation of motion of the piston is 



my + ky = ■? (7) 



where y is the deflection of the piston 



m is the mass/unit area of the piston 



k is the stiffness of the spring/unit area of the piston. 



Putting .,... 



