417 Dr. T. H. Havelock. The Propagation of [Aug. 26, 
§ 13. Point Impulse Travelling over Deep Water. 
Let the impulse be moving along Ow with constant velocity c; let B be 
the position at time ¢, A at any previous time ¢,, and suppose the system to 
have been moving for an indefinitely long time. 
/ P[z.4) 
Fie. 3. 
We have OA = chy; OB = ct. 
PB = o = {(ot—a) +9}; 
cos a = (ct—2)/o. 
Then in (67) we have to substitute {@?- 2c w(¢—t,)cos a+ c(t - ty) a\h 
for &,¢—¢, for ¢,and integrate with respect to 2) from —o to ¢; we obtain 
= TPIS ye 2y2\h— 
c= cS se [a w {eV [sin «(cos 8 {ew 2cum cos a + c7u?}t— Vw) 
—sin «(cos B {w?—2cum cosa+c?u?}t4+Vu)] «de. (69) 
With V = (9/«)}, we select the group around the value of « given by 
«1 = 4 cos? B (w?— 2cum cos a + ¢7u?)/ gu. (70) 
By using the formula (17) we find 
pao Mala 
l6p7? J » cost B (a?— 2cua cos «+ cu?) 
cos caren SUSU, oe ean in| > (al) 
4 cos B (w?— 2cus cos a+ c?u")t 
Selecting from this the chief group which occurs near 8 equal to zero, we 
find 
eas WG wdu ge (72) 
~ — Barp ),, (w?—2cwm cos eee "(ae — 2cum cos «+ c?u?)t 
Finally we choose the chief groups of terms in w from the condition 
g tgu? (w?— 2cus cos a+ c?u?)-t = 0; (738) 
2 
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