The Principle of Superposition for Waves — Question 3

Question 3

Constructive pulse overlap can make all energy momentarily elastic

Two equal-sign localized pulses are $y=f(x-vt)+f(x+vt)$ on an ideal string, with $F=\mu v^2$. They fully overlap at $t=0$.

1. Find the displacement and velocity at overlap.

2. Explain how double displacement is compatible with only twice the energy of one pulse, rather than four times its total energy.

Question 3 - Solution

Constructive pulse overlap can make all energy momentarily elastic

Method and mathematical working

At full overlap, add the spatial derivatives and subtract the time derivatives generated by opposite travel directions. The instantaneous energy is purely elastic.$$\begin{aligned}y(x,0)&=2f(x),\quad y_x(x,0)=2f^{\prime}(x),\quad y_t(x,0)=0,\\u(x,0)&=\tfrac12F(2f^{\prime})^2=2F(f^{\prime})^2.\\[4pt]K_{\mathrm{overlap}}&=0,\qquad U_{\mathrm{overlap}}=\tfrac12F\int(2f^{\prime})^2dx=E_1+E_2.\end{aligned}$$

(a) Calculation and conclusion

At overlap $y=2f(x)$, $y_x=2f^{\prime}(x)$ and $y_t=0$. Thus the total energy is elastic: $E=\tfrac12F\int(2f^{\prime})^2dx=2F\int(f^{\prime})^2dx$.

(b) Interpretation and limits

A single traveling pulse has equal kinetic and elastic energies, each $F\int(f^{\prime})^2dx/2$. The overlap elastic energy is four times its elastic part, but the kinetic part is zero. Relative to its total energy the factor is two, exactly matching two separated pulses.

Physical check

The doubled shape is not itself one traveling pulse. Applying the traveling-wave total-energy formula to it would incorrectly assign an equal kinetic energy that is absent at this instant.