Newton showed that deep-water waves have a phase velocity of $\sqrt{g \lambda / 2 \pi}$ . Find the group velocity of such waves and discuss the motion.

Strategy We use Equation (5.33) to relate the group and phase velocities, but first we need to find the phase velocity $v_{ ph }$ in terms of k.

$u_{ gr }=\frac{d \omega}{d k}=\frac{d}{d k}\left(v_{ ph } k\right)=v_{ ph }+k \frac{d v_{ ph }}{d k}$ (5.33)

Question

Newton showed that deep-water waves have a phase velocity of [latex]\sqrt{g \lambda / 2 \pi}[/latex]. Find the group velocity of such waves and discuss the motion.

Strategy We use Equation (5.33) to relate the group and phase velocities, but first we need to find the phase velocity [latex]v_{ ph }[/latex] in terms of k.

[latex]u_{ gr }=\frac{d \omega}{d k}=\frac{d}{d k}\left(v_{ ph } k\right)=v_{ ph }+k \frac{d v_{ ph }}{d k}[/latex] (5.33)

Accepted Answer

If we use [latex]\lambda=2 \pi / k[/latex], we have

[latex]v_{ ph }=\sqrt{\frac{g \lambda}{2 \pi}}=\sqrt{\frac{g}{k}}=\sqrt{g k}^{-1 / 2}[/latex]

Now we can take the necessary derivative for Equation (5.33).

[latex]u_{ gr }=\sqrt{\frac{g}{k}}+k \frac{d}{d k}\left(\sqrt{g} k^{-1 / 2}\right)=\sqrt{\frac{g}{k}}+k \sqrt{g}\left(-\frac{1}{2} k^{-3 / 2}\right)[/latex]

[latex]u_{g r}=\sqrt{\frac{g}{k}}-\frac{1}{2} \sqrt{\frac{g}{k}}=\frac{1}{2} \sqrt{\frac{g}{k}}=\frac{1}{2} v_{ ph }[/latex]

The group velocity is determined to be one half of the phase velocity. Such an effect can be observed by throwing a rock in a still pond. As the circular waves move out, the individual waves seem to run right through the wave crests and then disappear (see Figure 5.17).

Question 5.6: Newton showed that deep-water waves have a phase velocity of...

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