The Wave Equation — Question 8

Question 8 – Solution

Strategy. Test sinusoidal components before claiming that an arbitrary composite profile moves without changing shape.

Step 1: Derive the frequency relation. The derivatives give $u_{tt}=-\omega^2u$, $u_{xx}=-k^2u$ and $u_{xxxx}=k^4u$. Therefore $$\boxed{\omega^2=c^2k^2+\beta k^4,\qquad \omega=k\sqrt{c^2+\beta k^2}.}$$ Since $k$ has units $\mathrm{m^{-1}}$ and $\omega$ units $\mathrm{s^{-1}}$, $\beta$ has units $\mathrm{m^4/s^2}$.

Step 2: Compare the two speeds. Differentiating the positive-frequency branch gives $$\boxed{v_p=\sqrt{c^2+\beta k^2},\qquad v_g=\frac{c^2+2\beta k^2}{\sqrt{c^2+\beta k^2}}.}$$ Their difference is $\beta k^2/\sqrt{c^2+\beta k^2}$, strictly positive if $\beta>0$. Both tend to $c$ as $k\downarrow 0$ and both equal $c$ for every $k$ when $\beta=0$.

Step 3: Test rigid translation of two frequencies. A fixed profile $\cos(k_1x+\phi_1)+a\cos(k_2x+\phi_2)$ with $a\ne 0$ translated by speed $v$ gives temporal frequencies $vk_1,vk_2$. Independence of the two spatial frequencies requires $v=\omega(k_1)/k_1=\omega(k_2)/k_2$ for agreement at every $x,t$. For $\beta>0$, $v_p(k)$ is strictly increasing for $k>0$, so distinct wave numbers cannot share this value. Relative phases change in a way that cannot be represented by one common translation.

Step 4: Interpret the comparison. In the ideal equation, all right-moving sinusoidal components travel with the same speed $c$, so a smooth profile $F(x-ct)$ retains its shape. The given extension makes phase speed depend on frequency; a multi-frequency profile generally changes shape. The graph shows this dependence and the strict separation of the two defined speeds for positive $k$. The open point at $k=0$ records their common limit, since the definitions above were made for $k>0$.