Waves on deep water with surface tension T and density π are governed by the dispersion relation π2=ππ+π ππ3. 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?
The phase speed is given by
so Phase speed is
The group speed is given by
so we have
To determine the minimum phase speed we are essentially compute
so we use the first relation
This is zero for a trial case, , but in general we require
so the minimum phase speed occurs for
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