1908. ] Groups of Waves in Dispersive Media, ete. 416 
x,y be in the undisturbed surface and the axis of z vertically upwards; we 
write w for ,/(z2+ y?). Then, corresponding to (25), the surface elevation £ 
due to an initial displacement J(@), set free without initia] velocity, is 
given by 
ee | Dea ea GAMacyas, (62) 
where Aine ( Ow Gaia (63) 
For an initial point-elevation we may take for simplicity $(«) equal to 
1/27; then we have 
1 
Soe 
{ Wh (xa) cos (kV) «dk 
0 
1/2 fe 
= 3 “a| cos («aw cos B) cos (« Vt) Kd 
0 0 
1/2 oo) 
cos cos B— Vt) xd, 
27? [ag \, © ji j /2 c} 
ats a {ap [cos k(acosB+Vt) «de. (64) 
For deep water we separate a real principal group from the first integral, 
with respect to x, around the value of « given by 
wos 3/8 
t meas K 
This is replaced by the equivalent form 
343 pm/2 d 2 
— B URES ) 65 
Sai Sarita? i cos B cos ( % Cos 8 sa 2) 
Considering now the range for 8, we can again select the principal group of 
oscillations from (65); it occurs at @ equal to zero, so we take one-half the 
result given by the expression (14) and obtain the known result 
oh ea pet 66 
Sea ) 
Similarly, for an initial point impulse we have, instead of (64), the 
expression 
ar /2. Lo - 
t= itl ae | dg | eV {sin « (w cos B—Vt)—sin«(wcos8+Vt)} «de, (67) 
2gpT? J 0 
leading in the same way to the result 
Be Gas NOt 68 
om 2arput ay 4a C2) 
19 
