The Speed of a Traveling Wave — Question 8

Question 8

Different component speeds distort a composite signal

A medium has the supplied dispersion law $\omega=\alpha k^2$, with $\alpha>0$. A signal contains sinusoidal components with wave numbers $k$ and $2k$, both traveling right.

1. Find each component phase speed.

2. Can their sum retain one rigidly translated shape for all time? Compare with a nondispersive law $\omega=ck$.

Question 8 - Solution

Different component speeds distort a composite signal

Method and mathematical working

Phase speed is the slope of a line from the origin to a point on the dispersion curve. Compare this ratio for each nonzero Fourier component.$$\begin{aligned}v_{\phi,1}&=\frac{\alpha k^2}{k}=\alpha k,\\v_{\phi,2}&=\frac{\alpha(2k)^2}{2k}=2\alpha k.\\[4pt]\omega(2k)-2\omega(k)&=4\alpha k^2-2\alpha k^2=2\alpha k^2\ne0.\end{aligned}$$

(a) Calculation and conclusion

The phase speed is $\omega/k$. The first component has speed $\alpha k$, and the second $2\alpha k$. Their ratio is two, despite propagation in the same medium.

(b) Interpretation and limits

For rigid translation at speed $v$, every component must satisfy $\omega_j=k_jv$. The two ratios here differ, so their relative spatial alignment changes and the composite shape generally distorts. For $\omega=ck$, every ratio equals $c$, permitting undistorted translation.

Physical check

A rigidly translating sum requires a common ratio $\omega_j/k_j$, not merely a common direction. This argument concerns phase speeds and does not require identifying a packet or group speed.