Standing Waves — Question 8

Question 8

A relative reflection phase translates the node pattern

On an unbounded uniform string, equal waves are $A\cos(kx-\omega t)$ and $A\cos(kx+\omega t+\phi)$.

1. Find the standing-wave form and node positions.

2. For a change $\phi:0\to\pi/2$, find the shift of each continuously tracked node. Could this phase change be arbitrary if a node must remain at a fixed support?

Question 8 - Solution

A relative reflection phase translates the node pattern

Method and mathematical working

Use the sum identity with the phase carried through both factors. Solve the spatial-zero condition for position before taking its change with phase.$$\begin{aligned}y&=2A\cos(kx+\phi/2)\cos(\omega t+\phi/2),\\x_n&=\frac{(n+1/2)\pi-\phi/2}{k},\qquad \Delta x_n=-\frac{\Delta\phi}{2k}.\\[4pt]\Delta x&=-\frac{\pi/2}{2(2\pi/\lambda)}=-\lambda/8.\end{aligned}$$

(a) Calculation and conclusion

The sum is $2A\cos(kx+\phi/2)\cos(\omega t+\phi/2)$. Nodes satisfy $kx+\phi/2=(n+1/2)\pi$, so $x_n=[(n+1/2)\pi-\phi/2]/k$.

(b) Interpretation and limits

The shift is $\Delta x=-\Delta\phi/(2k)=-\lambda/8$. If a fixed support must remain a node, its boundary condition constrains the allowed relative phase; one cannot change that phase independently while preserving the same standing pattern and boundary.

Physical check

For $\Delta\phi=\pi/2$, the shift is $-\lambda/8$. On a bounded string, an arbitrary imposed phase shift may violate an endpoint condition, so the free superposition result must be checked against the supports.