Waves and ripples in water, air, and æther

by Sir J. A. Fleming

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