HYDRODYNAMICAL NOTES
length is U, where U is the group-velocity, equal to dcr/dk, where k — 2-7T/X*. Hence in our problem we must take
height varies as U~^, ......................
which includes the former result, since in a non-dispersive medium For waves in water of depth I,
cr2 = gk tanh kl, .........................
whence 2cr Ujg ~ tanh kl + kl (1 — tanh2 kl)..............
As the wave progresses, cr remains constant, (3) determines k of I, and U follows from (4). If we write
(3) becomes kl. tanh kl-l', .........................
and (4) may be written
2<rU/ff = kl + (I'- l'*)/kl....................
By (6), (7) U is determined as a function of I' or by (5) of I.
If kl, and therefore I', is very great, kl = I', and then by (7) if corresponding value of U,
20-00/0 = 1, ............................
and in general
Equations (2), (5), (6), (9) may be regarded as giving the solutic problem in terms of a known cr. It is perhaps more practical to rep (5) by X0; the corresponding wave-length in a great depth. The between cr and X0 being cr2 = 27r0/X,0, we find in place of (5)
Starting in (10) from \0 and I we may obtain I', whence (6) give (9) gives U/UQ. But in calculating results by means of tables of tl bolic functions it is more convenient to start from kl. We find
Id
oo
10
5
2
1-5 1-0
kl kl
4-999
1-928
1-358
•762
•531
•423
1-000 1-000 1-001 1-105 1-176 1-182 1-110 1-048
kl
•6 •5 •4 •3
•2 •1 kl
•322 •231 •152 •087 •039 •010
•964 •855 •722 •566 •390 •200
Proc. Land. Math. Soc. Vol. ix. 1877 ; Scientific Papers, Vol. i. p. 32isponding lengths along the direction of propagation.