1176 
ae 
and thus from (3) and (4) the boundary condition may be written in the form 
1 Op > p= xX =) 0 (5) 
B Ox 
where 
B= 2@ (6) 
m 
In order to solve the equation (1), (2) and (5) let us introduce a pressure p" defined by 
De (7) 
BimoKx 
then p' satisfies the wave equation, vanishes at x = 0 and contains the incident wave terms 
pt o[ coen|, f_(ct - R) (a) 
B Ox R R 
The problem for p’ is thus that of a wave source and doublet at A with a free surface at 
x= 0, Hence by the addition of terms corresponding to a virtual source and doublet of 
appropriate signs at the imaje point A" the solution for p’ is 
ee gS [tea ) , tena 
B Ox R R 
AReeOls (tin (cth—nR)|) mn ta(cte=ake) 
i B ox k' R' (9) 
From (7) and (9) we have 
r) =| (ec f (ct — R) ft (ct - R') 
Sy wee [&+ 4] [ R : R 
228 eel (10) 
and the solution for p is obtained in the form 
* = B(x -X) 
Deta(ctteiR)i ee faction) Ay t(ct - Rp) or 
p — ————— - 28 —————______ 
R R Ry 
Xo 
o2x2 Xo oe 
it 
ct >R'DR 
where 
Re = r? + (x + a)? 
Beene es (x <a)? (12) 
Ragem rete. (ara)? 
Xo aad = V ct -rfr 
It can be verified by direct substitution in (1) and (5) that (11) satisfied both these 
equations, . For ct < R' the solution for p is of course given solely by the incident wave (2). 
By eevee 
