Energy and Power of a Wave Traveling Along a String — Question 7

Question 7

Narrower triangular pulses store more energy at fixed height

A small-slope traveling triangular pulse has height $A$, base width $2a$ and slopes of magnitude $A/a$. The string tension is $F$. Use $E=F\int (\partial y/\partial x)^2dx$ for a single traveling pulse. Corners may be regarded as slightly smoothed.

1. Find its total energy.

2. Compare with a pulse of the same height and half the width. Explain why a jump-discontinuous rectangular profile is outside the finite-energy small-slope idealization.

Question 7 - Solution

Narrower triangular pulses store more energy at fixed height

Method and mathematical working

For a single traveling pulse, the kinetic and elastic densities are equal. Integrate their sum over the two straight flanks; isolated corner points do not contribute finite area.$$\begin{aligned}E&=F\int_{-\infty}^{\infty}(f^{\prime})^2\,dx,\\|f^{\prime}|&=A/a\quad\text{on a total length }2a,\\E&=F\frac{A^2}{a^2}(2a)=\frac{2FA^2}{a}.\\[4pt]E(a/2)&=\frac{2FA^2}{a/2}=2E(a),\qquad \left|f^{\prime}\right|_{a/2}=2A/a.\end{aligned}$$

(a) Calculation and conclusion

The squared slope is $A^2/a^2$ over total length $2a$, so $E=2FA^2/a$. Both kinetic and elastic energy are included in the supplied traveling-pulse expression.

(b) Interpretation and limits

Halving $a$ doubles the energy and doubles the slope magnitude. The latter must still be small. A vertical jump would require an unbounded slope and does not define a finite squared-slope integral in this model; an edge of finite width must be specified.

Physical check

The dimensions are force times length. Compressing the width increases slope faster than it reduces the occupied length; a discontinuous jump is outside this finite-slope calculation.