Question

For surface waves in shallow water, the frequency and wavelength are connected through
the relation
3
2
ρλ
π
=
S
f
where S and ρ respectively denote surface tension and density of water. Calculate the
group velocity of waves. How is it related to phase velocity?

Answer 1

Answer on Question #54840-Physics-Other

For surface waves in shallow water, the frequency and wavelength are connected through the relation

$$f = \sqrt {\frac {2 \pi S}{\rho \lambda^ {3}}}$$

where $S$ and $\rho$ respectively denote surface tension and density of water. Calculate the group velocity of waves. How is it related to phase velocity?

Solution

$$v _ {g} = \frac {d \omega}{d k}.$$ $$\omega = 2 \pi f, k = \frac {2 \pi}{\lambda} \rightarrow \lambda = \frac {2 \pi}{k}$$ $$v _ {g} = \frac {d}{d k} \left(\sqrt {\frac {S k ^ {3}}{\rho}}\right) = \frac {3}{2} \sqrt {\frac {S k}{\rho}} = \frac {3}{2} \sqrt {\frac {2 \pi S}{\rho \lambda}} = \frac {3}{2} \lambda f.$$

The phase velocity is

$$v _ {p} = \lambda f.$$

Thus,

$$v _ {g} = \frac {3}{2} v _ {p}.$$

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