The Principle of Superposition for Waves — Question 5

Question 5

A fixed endpoint requires an inverted reflected pulse

A string occupies $x<0$ and is fixed at $x=0$. An incident pulse traveling right is $y_i=f(x-vt)$. Seek a leftward reflection in terms of the same function.

1. Construct a reflected field that enforces $y(0,t)=0$ for every time.

2. Find the endpoint slope and power. Does a nonzero force on the support imply energy transfer into an immobile support?

Question 5 - Solution

A fixed endpoint requires an inverted reflected pulse

Method and mathematical working

Construct the reflected argument so that both profiles sample the same value at the boundary. Choose its sign to enforce zero endpoint displacement for every time.$$\begin{aligned}y(0,t)&=f(-vt)-f(-vt)=0,\\y_x(0,t)&=2f^{\prime}(-vt),\qquad y_t(0,t)=0.\\[4pt]y_r&=-f(-x-vt),\qquad P(0,t)=-Fy_x(0,t)\underbrace{y_t(0,t)}_{0}=0.\end{aligned}$$

(a) Calculation and conclusion

Choose $y_r=-f(-x-vt)$. At zero, incident and reflected displacements cancel exactly. The reflected pulse is inverted in displacement and moves toward decreasing $x$.

(b) Interpretation and limits

Total slope at the endpoint is $2f^{\prime}(-vt)$, so a transverse force can be present. But endpoint velocity is zero, and $P=-Fy_xy_t=0$. A stationary ideal support can exchange momentum through force without doing mechanical work.

Physical check

The endpoint can sustain a transverse force but has zero velocity. Their product, the boundary power, is therefore zero despite the nonzero boundary slope.