The Wave Equation — Question 6

Question 6

Initial data influence only a finite interval at a later event

For an infinite string, use the supplied solution $y(x,t)=\tfrac12[h(x-vt)+h(x+vt)]+\tfrac1{2v}\int_{x-vt}^{x+vt}g(s)ds$, where $h$ and $g$ are initial displacement and velocity. Let $v=3\,\mathrm{m/s}$.

1. Identify the initial region that can affect the event $x=5\,\mathrm{m},t=1\,\mathrm{s}$.

2. If both initial fields vanish outside $[-1,1]\,\mathrm{m}$, what is the earliest possible arrival at $x=5\,\mathrm{m}$? Distinguish the boundary arrival from a guaranteed nonzero displacement.

Question 6 - Solution

Initial data influence only a finite interval at a later event

Method and mathematical working

Read the endpoints of the initial-data interval from the two characteristic directions in the supplied solution. Compare that interval with the support of the initial disturbance.$$\begin{aligned}I(x,t)&=[x-vt,x+vt],\\I(5,1)&=[2,8],\qquad t_{\mathrm{first}}=\frac{5-1}{3}.\\[4pt][5-3(1),5+3(1)]&=[2,8],\qquad3t\ge5-1\Rightarrow t\ge4/3\,\mathrm s.\end{aligned}$$

(a) Calculation and conclusion

The characteristic endpoints are $x-vt=2$ and $x+vt=8\,\mathrm{m}$. Only data within $[2,8]$ can influence that event; displacement enters at the endpoints and velocity through the interval integral.

(b) Interpretation and limits

The nearest possible initial disturbance is at $x=1$, a distance $4\,\mathrm{m}$ away. Earliest possible arrival is $4/3\,\mathrm{s}$. Smooth compact data can vanish at their support edge, so equality need not give nonzero displacement exactly at the first causal boundary.

Physical check

The earliest possible influence occurs when a characteristic first meets the nearest edge of the support. Actual displacement can still vanish at that edge, so this causal bound need not imply a nonzero reading at the exact first instant.