573 Prof. T. H. Havelock on 
Suppose that at c=0 we are given 
fo) 
z ot “(QE ACE 4 6 « » CD 
Then from (20) and (21) we obtain an expression for ¢> 
valid for positive values of 2, and adjusted so that it 
represents progressive waves at large values of « ; we find 
Ge “Eten | da aa{"/(, B) sin (ot —2 Va? =n) x 
x e-“*%eosKn (y= B) dk! as 2h noe cos ot a dB x 
7 os) 
(x? —x«'?)5 
x (6 f (a, B)e-2(©?—0?)}— Koa = (y=B) dx' a ,cosat X 
Ko-)* 
a i(? icf apr f(a, B)e-#+*" cos «' (y—B) 
sg MICOS eee eo SIM) 0 08 es SITES) Fa sail (95 
(x? + Ko”) (2 + «/?)h 
A particular case which would illustrate the spreading of 
plane waves emerging from a canal into an infinite sea is 
obtained by taking 
FY 2)=(gar/aje“*cosct, . . . (24) 
over the whole range for z and between the limits +b for y, 
the function being zero outside these limits for y. Substi- 
tuting in (23), the third term disappears, and also the 
integrations with respect to 8 can be effected in the 
remaining terms. We find that the surface elevation for 
this case is given by 
c— 
al sin «'b cos K’y cos ee Cay —n'?)! ee 
0 '(ko — K')a 
22 
Qua “sin «bcos Kye 2("?—"o? 
— — cos ot Sune COS ese RELY ND . (24) 
7 Ko 
Tv 
kK! («'*? — K?)2 
The form of the surface could be studied by approximate 
evaluation of these ee as in similar diffraction 
problems. 
5. We return to plane waves, and suppose now that the 
water is of finite depth A. We have the additional condition 
oe = Oi tOr GHRo 6 6 6 oo | (SS) 
