The Principle of Superposition for Waves — Question 2

Question 2

Complete displacement cancellation can leave nonzero kinetic energy

Two opposite-sign pulses on an ideal string are $y=f(x-vt)-f(x+vt)$, with localized smooth $f$ and $F=\mu v^2$. Their centers coincide at $t=0$.

1. Find displacement, slope and transverse velocity at complete overlap.

2. Calculate total energy at that instant and compare with two separated pulses.

Question 2 - Solution

Complete displacement cancellation can leave nonzero kinetic energy

Method and mathematical working

Evaluate displacement, slope, and velocity separately at the overlap instant. Cancellation of the first quantity does not imply cancellation of its time derivative.$$\begin{aligned}y(x,0)&=0,\quad y_x(x,0)=0,\quad y_t(x,0)=-2vf^{\prime}(x),\\u(x,0)&=\tfrac12\mu(2vf^{\prime})^2=2F(f^{\prime})^2.\\[4pt]E_{\mathrm{overlap}}&=2F\int(f^{\prime})^2dx=E_1+E_2,\qquad U_{\mathrm{overlap}}=0.\end{aligned}$$

(a) Calculation and conclusion

At $t=0$, $y=0$ and $y_x=0$, but $y_t=-2vf^{\prime}(x)$. The string is momentarily straight while material elements generally move transversely.

(b) Interpretation and limits

Energy is entirely kinetic: $E=\int\tfrac12\mu(2vf^{\prime})^2dx=2F\int(f^{\prime})^2dx$. Each separated traveling pulse has energy $F\int(f^{\prime})^2dx$, so total energy is conserved. Destructive displacement overlap does not destroy energy.

Physical check

All energy is kinetic at full cancellation. The integrated value equals the energy of the two original pulses, so a flat instantaneous string has not lost their energy.