Question

Waves on deep water with surface tension T and density 𝜌 are governed by the dispersion relation 𝜔2=𝑔𝑘+𝑇 𝜌𝑘3. Calculate the phase and group velocities of the waves. Find the wave number kc at which phase velocity reaches a minimum. What is the group velocity for this wave number?

Answer 1

The phase speed is given by

$\frac{ω^2}{k^2}=\frac gk+\frac Tρk⟹c_p^2=\frac gk+\frac Tρk$

so Phase speed is

$c_p=±\sqrt{\frac gk+\frac Tρk}$

The group speed is given by

$2ω\frac{∂ω}{∂k}=g+\frac{3T}{ρ}k^2$

so we have

$v_g=\frac{g+\frac{3T}{ρ}k^2}{2ω}$

To determine the minimum phase speed we are essentially compute

$\frac{∂c_p}{∂k}=0$

so we use the first relation

$2c_p\frac{∂c_p}{∂k}=−\frac{g}{k^2}+\frac Tρ$

This is zero for a trial case, $c_p=0$ , but in general we require

$−\frac{g}{k^2}+\frac Tρ=0⟹k_c^2=\frac{gρ}{T}$

so the minimum phase speed occurs for

$k_c=\sqrt{\frac{gρ}{T}}$