In other words, the beats travel forward with the same speed as the constituent waves. And in this case there is no difference between the velocity of the wave-train and the velocity of the individual wave. The above proof may be generalized as follows:—
Let the wave-velocity vary as the nth root of the wave-length, or let vⁿ = Cλ; and let λ = 2π / k as before.
Then—
vⁿ = 2πC / k , and vk = 2πC / v ⁿ ⁻¹ = 2πCv ⁻ ⁽ ⁿ ⁻¹⁾
also k = 2π / λ = 2πC / v ⁿ = 2πCv ⁻ ⁿ
Hence d(vk) / d(k) = n – 1v ⁻ ⁽ⁿ ⁻¹⁾ ⁻¹ / nv ⁻ ⁿ ⁻¹ = n – 1 / n v
or V = n – 1 / n v
That is, the wave-train velocity is equal to n – 1 / n times the wave-velocity.
In the case of sea waves n = 2, and in the case of air waves n = infinity.
If n were 3, then V = 2 / 3 v, or the group-velocity would be two-thirds the wave-velocity.