All Dr. T. H. Havelock. The Propagation of [Aug. 26, 
If we assume ¢ («) equal to 1, so that it is confined to an indefinitely narrow 
strip of impulse (cf. § 5), we obtain the result corresponding to (29) for 
initial displacement by multiplying that expression by the value of «V; thus 
we find 
al AS (£ 1 ) Ad 
Lie Snipat Zag ie) 
For comparison with the previous results, suppose that 
ca? f ere 
WO)VS ra b (Kk) = mea *. 
Then we find the surface form as an aggregate of groups, each of them 
cumulative and so prominent only in a limited region, given by 
+ 242 2 2 
— mcag*t? — xe (g ) 5 
= Cos 5 42 
a Apai © 4a +a C2) 
For a given place the maxima are given by 
£ (Go-stt) = 0, that is, by 7 = 4,/(ga). 
Thus the maximum moves with velocity 4 ./(ga), and consists of nearly 
simple waves of wave-length 27a. Comparing with the result in §6 for an 
initial displacement of the same character, we see that the maximum is pro- 
pagated outwards with slower velocity, the wave-length at the maximum 
being one-half the corresponding value in the former case. 
§ 9. Moving Line Impulse on Deep Water. 
Suppose that the line impulse of the previous section is moving over the 
surface of deep water at right angles to its length with uniform velocity c, 
having started at some time practically infinitely remote. Then we may 
regard the effect at (x, t) as the summation of the effects due to all the con- 
secutive elements of impulse, and we can obtain an expression by modifying 
(40) and integrating with respect to the time. We measure z from a fixed 
origin which the line impulse passes at zero time; then we substitute x—cto 
for z and¢—ty for ¢ in (40), and integrate with respect to ¢ for all the time 
the impulse has been moving. Thus we obtain 
t ce) 
Tgpn = 2 ato | «KV sink {w—cty— V (t—to)} dk 
0 
t co 
—4t f ato) KV sin « {t©—cto+ V (t—to)} de 
—ao #0 
= 1( au KV sin x {o+(c—V) wu} de 
0 Jo 
= i [ae KV sink {o+(c+V)u}dn, (43) 
0 0 
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