since λ is nearly equal, by assumption, to λ′. Hence we have—
1 / L = 1 / λ – 1 / λ′ ; and also V = NL
Accordingly—
V = / N / 1 / L = v / λ – v′ / λ′ / 1 / λ – 1 / λ′
Let us write 2π / k instead of λ, and 2π / k′ instead of λ′; then we have—
V = vk – v′k′ / k – k′ (i.)
And since k and k′, v and v′ are nearly equal, we may write the above expression as a differential coefficient; thus—
V = d(vk) / d(k) (ii.)
Suppose, then, that, as in the case of deep-sea waves, the wave-velocity varies as the square root of the wave-length. Then if C is a constant, which in the case of gravitation waves is equal to g/2π , where g is the acceleration due to gravity, we have—
v² = Cλ, or v² = g / 2π λ