For wavelengths less than about 1 cm, the dispersion relation for waves on the surface of water is $\mathbf{\omega }\mathbf{=}\sqrt{\left(\gamma /p\right){\mathbf{k}}^{\mathbf{3}}}$, whereandare the surface tension and density of water. Given$\mathbf{\gamma }\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{072}\mathbf{N}\mathbf{/}\mathbf{m}$and$\mathbf{p}\mathbf{=}{\mathbf{10}}^{\mathbf{3}}\mathbf{kg}\mathbf{/}{\mathbf{m}}^{\mathbf{3}}$, calculate the phase and group velocities for a wave of 5mm wavelength.

Waves of different wavelengths travel at different phase speeds. Water waves on the surface propagate with gravity and surface tension as the restoring forces.

The dispersion relation for waves on the surface of water is

$\omega =\sqrt{\left(\gamma /p\right)k^3}$

Where,$\gamma$and p are the surface tension and density of water

The phase velocity is,

$v_{phase}=\frac{\sqrt{\left(\gamma /p\right)k^3}}{k}$

The group velocity is,

$v_{group}={\overline{)\frac{d\omega }{dk}}}_{k_0}$

The wave number, k is given by

$k=\frac{2\mathrm{\pi }}{\lambda }$

Where, $\lambda$= wavelength

$$\begin{gathered} \mathrm{k}=\frac{2\mathrm{\pi }}{5\times {10}^{-3}\mathrm{m}} \\ =1.265\times {10}^3{\mathrm{m}}^{-1} \end{gathered}$$

Calculate the angular frequency$\omega$, using the wave number k, surface tension of water$\gamma$, and the density of water p

$\omega =\sqrt{\left(\frac{\gamma }{p}\right)k^3}$

Substitute the values of $\gamma$, pand k to get $\omega$

$$\begin{gathered} \mathrm{\omega }=\sqrt{\left(\frac{0.072\mathrm{kg}/{\mathrm{s}}^2}{{10}^3\mathrm{kg}/{\mathrm{m}}^3}\right){\left(1.265\times {10}^{-3}{\mathrm{m}}^{-1}\right)}^3} \\ =377.99\mathrm{rad}/\mathrm{s} \end{gathered}$$

$$\begin{gathered} v_{phase}=\sqrt{\frac{\left(\gamma /p\right)k^3}{k}} \\ {v}_{phase}=\sqrt{\left(\frac{\gamma }{p}\right)k} \end{gathered}$$

Substitute the values of $\gamma$, pand k

$$\begin{gathered} {\mathrm{v}}_{\mathrm{phase}}=\sqrt{\left(\frac{0.072\mathrm{kg}/{\mathrm{s}}^2}{{10}^3\mathrm{kg}/{\mathrm{m}}^3}\right)\left(1.265\times {10}^{-3}{\mathrm{m}}^{-1}\right)} \\ =0.3\mathrm{m}/\mathrm{s} \end{gathered}$$

Group velocity is calculated by taking a derivation of the angular frequency$\omega$with respect to the wave number k

$$\begin{gathered} {\mathrm{v}}_{\mathrm{group}}={\overline{)\frac{\mathrm{d\omega }}{\mathrm{dk}}}}_{{\mathrm{k}}_0} \\ =\frac{1}{2}{\left(\frac{{\mathrm{\gamma k}}^3}{\mathrm{p}}\right)}^{-1/2}{\overline{)\left(\frac{3{\mathrm{\gamma k}}^2}{\mathrm{p}}\right)}}_{{\mathrm{k}}_0} \\ =\frac{3}{2}\sqrt{\frac{{\mathrm{\gamma k}}_0}{\mathrm{p}}} \end{gathered}$$

Use the wave number k as the median wave number $k_0$

$$\begin{gathered} {\mathrm{v}}_{\mathrm{group}}=\frac{3}{2}\sqrt{\frac{(0.072\mathrm{kg}/{\mathrm{s}}^2)(1.265\times {10}^{-3}{\mathrm{m}}^{-1})}{{10}^3\mathrm{kg}/{\mathrm{m}}^3}} \\ =0.45\mathrm{m}/\mathrm{s} \end{gathered}$$

Therefore, phase velocity for a wave is 0.3 m/s

The group velocity for a wave is 0.45 m/s