Question

A string is made to vibrate at its third harmonic. The diagram shows two points P and Q at a particular instant in time. Which of the following compares the period of vibration of P and Q?

(The string is closed at both ends; P is just before the first maximum antinode and Q is at the next minimum antinode).
A. Period of P and Q is the same
B. Period of P and Q is different

Why is it A not B??

Answer 1

A is correct

Explanation:

A standing wave in a string is described by the equation

`A cos(kx) sin(omega t)`

As a result, a point labeled by a fixed position (say `x=xi`) undergoes a simple harmonic oscillation with amplitude `A cos (k xi)`. The time dependence of the oscillation for all points on the string is given by `sin(omega t)` - they all execute oscillations with the same angular frequency `omega`, and hence the same time period `T=(2pi)/omega`. The only thing that differs is the amplitude!

The only exception to this, if you can call it that, are the nodes - points of the string for which `cos(k xi)=0`. These points do not move at all . Even for these, however, you could simply argue that they do have the same period as all other points, but has an amplitude of zero!