IB Physics HL C.4 Standing waves and resonance questions

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Question 1

Multiple choiceMedium1 mark

Two sinusoidal waves, each of amplitude $a$ and wavelength $\lambda$ travel in opposite directions along a string and form a standing wave. There is a displacement node at $x=0$ Points $P$ and $Q$ are at $x=\lambda/12$ and $x=\lambda/3$ respectively. What is the ratio of the oscillation amplitude at $P$ to that at $Q$ and what is the phase difference between their oscillations?

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Answer: C. For two equal waves forming a standing wave with a node at $x=0$ the amplitude at position $x$ is $2a|\sin(2\pi x/\lambda)|$ Thus $A_P=a$ and $A_Q=\sqrt{3}a$ giving $A_P/A_Q=1/\sqrt{3}$ Both points lie between the same pair of adjacent nodes, so their oscillations are in phase and their phase difference is $0$

A: reverses the requested amplitude ratio.
B: finds the ratio correctly but treats points in the same segment between nodes as oscillating in antiphase.
D: assumes amplitude is proportional to distance from the node, giving $x_P/x_Q=1/4$

Remember: points in the same segment between adjacent nodes oscillate in phase.

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