The Wave Equation — Question 9

Question 9

The differential equation contains a local energy balance

For a uniform ideal string, define $u=\tfrac12\mu y_t^2+\tfrac12F y_x^2$ and $P=-F y_x y_t$. The motion satisfies $\mu y_{tt}=F y_{xx}$.

1. Derive $u_t+P_x=0$.

2. Integrate over a fixed interval $[a,b]$ and interpret the signs of the endpoint powers.

Question 9 - Solution

The differential equation contains a local energy balance

Method and mathematical working

Differentiate the sum of kinetic and elastic energy densities, then replace acceleration using the wave equation. The product rule identifies the spatial derivative of power.$$\begin{aligned}u_t&=\mu y_t y_{tt}+F y_x y_{xt}\\&=F(y_t y_{xx}+y_x y_{xt})=F\partial_x(y_x y_t),\\u_t+P_x&=0.\\[4pt]\frac{d}{dt}\int_a^b u\,dx&=-\int_a^bP_x\,dx=P(a,t)-P(b,t).\end{aligned}$$

(a) Calculation and conclusion

$u_t=\mu y_ty_{tt}+F y_xy_{xt}=F y_ty_{xx}+F y_xy_{xt}=F\partial_x(y_xy_t)=-P_x$. Thus increasing local stored energy corresponds to a net incoming energy flux.

(b) Interpretation and limits

$d\int_a^b u\,dx/dt=P(a,t)-P(b,t)$. Rightward power at the left endpoint enters, while rightward power at the right endpoint leaves. This conservation identity applies to arbitrary solutions, including overlapping waves, not only one sinusoidal traveler.

Physical check

Integrating over a fixed interval gives $dE/dt=P(a)-P(b)$. Equal incoming and outgoing powers leave the stored energy constant, as required by conservation.