The Speed of a Traveling Wave — Question 10

Question 10

Amplitude changes do not automatically change phase speed

Two prescribed waves on a linear nondispersive medium are $y_1=A\cos(kx-\omega t)$ and $y_2=3A\cos(kx-\omega t)$ with the same $k,\omega>0$.

1. Compare phase speeds and maximum transverse particle speeds.

2. Explain why this comparison does not prove that wave speed is amplitude-independent in every possible medium.

Question 10 - Solution

Amplitude changes do not automatically change phase speed

Method and mathematical working

Separate the phase trajectory from the partial time derivative at a fixed material position. These describe different motions and use different combinations of the parameters.$$\begin{aligned}v_{\mathrm{phase}}&=\frac{\omega}{k},\\(y_1)_t&=A\omega\sin(kx-\omega t),\qquad (y_2)_t=3(y_1)_t.\\[4pt]\frac{v_{\phi,2}}{v_{\phi,1}}&=1,\qquad\frac{(y_{2t})_{\max}}{(y_{1t})_{\max}}=3.\end{aligned}$$

(a) Calculation and conclusion

Both have phase speed $\omega/k$. Maximum transverse speeds are $A\omega$ and $3A\omega$, respectively. Increasing the displacement amplitude changes particle motion while the prescribed phase evolution is unchanged.

(b) Interpretation and limits

The conclusion uses the linear nondispersive model and the shared $k,\omega$. In a nonlinear medium, amplitude can alter restoring forces or effective parameters, which can alter the dispersion relation. A kinematic comparison within one stated model is not a universal material law.

Physical check

Both speed measures have units of velocity, but only the material speed scales with amplitude in this model. A material-dependent change in the dispersion law would require a separate dynamical analysis.