The Wave Equation — Question 4

Question 4

A snapshot determines acceleration but not initial velocity

At $t=0$, a uniform string of speed $v=4\,\mathrm{m/s}$ has profile $y=A\cos(kx)$ with $A=.010\,\mathrm{m}$ and $k=3\,\mathrm{rad/m}$. No velocity data are provided.

1. Find the acceleration field at that instant.

2. Give two different future motions with this same initial profile, and identify the missing initial data.

Question 4 - Solution

A snapshot determines acceleration but not initial velocity

Method and mathematical working

Use the wave equation directly on the measured snapshot to find acceleration. Initial velocity remains independent data for a second-order time-evolution equation.$$\begin{aligned}y_{xx}(x,0)&=-Ak^2\cos kx,\\y_{tt}(x,0)&=-v^2Ak^2\cos kx.\\[4pt]y_{tt}(x,0)&=-(4)^2(0.010)(3)^2\cos3x=-1.44\cos3x\,\mathrm{m/s^2},\\y_{\pm}(x,t)&=0.010\cos(3x\mp12t).\end{aligned}$$

(a) Calculation and conclusion

The wave equation gives $y_{tt}(x,0)=v^2y_{xx}=-v^2k^2A\cos kx=-1.44\cos(3x)\,\mathrm{m/s^2}$.

(b) Interpretation and limits

Both $A\cos(3x-12t)$ and $A\cos(3x+12t)$ share the snapshot and acceleration but have opposite initial velocities $\pm.12\sin(3x)\,\mathrm{m/s}$. A second-order time equation requires both displacement and velocity fields to select the evolution.

Physical check

The acceleration points back toward equilibrium for this sinusoidal snapshot. Two waves traveling in opposite directions have the same displacement and acceleration at the chosen instant but opposite velocities.