The Wave Equation — Question 2

Question 2

An arbitrary smooth translating shape satisfies the same equation

Let $y(x,t)=f(x-ct)$ for a twice-differentiable function $f$. A string requires $y_{tt}=v^2y_{xx}$ with $v>0$.

1. Determine which values of $c$ allow a nontrivial curved profile.

2. Identify the exceptional profiles for which the equation does not constrain $c$. Explain why those exceptions cannot be used to measure a wave speed.

Question 2 - Solution

An arbitrary smooth translating shape satisfies the same equation

Method and mathematical working

Differentiate the traveling profile twice with respect to each independent variable. Compare the coefficients multiplying the same second derivative.$$\begin{aligned}y_{tt}&=c^2f^{\prime\prime}(x-ct),\qquad y_{xx}=f^{\prime\prime}(x-ct),\\y_{tt}-v^2y_{xx}&=(c^2-v^2)f^{\prime\prime}.\\[4pt]f(s)=as+b&\Rightarrow f^{\prime\prime}=0\Rightarrow y_{tt}=y_{xx}=0\quad\text{for any }c.\end{aligned}$$

(a) Calculation and conclusion

Differentiation gives $y_{tt}=c^2f^{\prime\prime}$ and $y_{xx}=f^{\prime\prime}$. Thus $(c^2-v^2)f^{\prime\prime}=0$. If curvature is nonzero anywhere, $c=\pm v$.

(b) Interpretation and limits

If $f^{\prime\prime}=0$ everywhere, then $f(\xi)=a\xi+b$. Both second derivatives vanish for any $c$. A linear ramp has no localized crest or repeating structure to track, and different descriptions reduce to changes in uniform offset velocity. Boundary and initial conditions still constrain physical realizability.

Physical check

For a nontrivial curved profile the coefficient must vanish, giving $c=\pm v$. The exceptional affine profile has zero curvature and does not select a propagation speed through this test.