The Principle of Superposition for Waves — Question 6

Question 6

A force-free endpoint requires an upright reflected pulse

A string occupies $x<0$. Its endpoint slides vertically without friction on a massless guide, so the transverse force at $x=0$ vanishes. In the small-slope model, $y_x(0,t)=0$. The incident pulse is $f(x-vt)$.

1. Construct the reflected pulse and find the endpoint displacement.

2. Find endpoint power and compare the physical reason for zero flux with a fixed endpoint.

Question 6 - Solution

A force-free endpoint requires an upright reflected pulse

Method and mathematical working

Use an uninverted reflected profile and differentiate it carefully: its spatial argument has derivative $-1$. This cancels the slope at the free end.$$\begin{aligned}y(0,t)&=2f(-vt),\\y_x(0,t)&=f^{\prime}(-vt)-f^{\prime}(-vt)=0.\\[4pt]y_r&=+f(-x-vt),\qquad P(0,t)=-F\underbrace{y_x(0,t)}_{0}y_t(0,t)=0.\end{aligned}$$

(a) Calculation and conclusion

Choose $y_r=f(-x-vt)$. Its derivative is $-f^{\prime}(-x-vt)$, canceling the incident slope at zero. The reflected pulse is upright, and endpoint displacement is $2f(-vt)$.

(b) Interpretation and limits

$P=-Fy_xy_t=0$ because the slope, and hence transverse force, is zero. The endpoint can move. At a fixed end, zero velocity gives zero flux instead. These different boundary conditions explain the different reflection signs.

Physical check

The end may move with nonzero velocity, but its transverse force is zero. Thus the boundary still performs no work, and doubling endpoint displacement does not violate energy conservation.