Where are the nodes and antinodes in the simplest standing wave on a string fixed at both ends?

There is a node at each fixed end and a single antinode at the midpoint of the string.

Give an expression for the lowest frequency at which a string of length L fixed at both ends will vibrate, in terms of the wave speed v.

f = v/2L, where v is the speed of the progressive waves along the string in m s⁻¹ and L is its length in metres.

Describe the next possible mode of vibration of a string fixed at both ends after the simplest one.

There are nodes at both ends and a third node at the midpoint, with two antinodes at the quarter and three-quarter positions, and the wavelength equals the length of the string.

Give a general expression for the wavelengths at which a string of length L fixed at both ends can vibrate.

λ = 2L/n, where n is a whole number counting the loops, so the possible frequencies are f = nv/2L.

CCEA A-Level Physics 1210

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Superposition, Interference and Diffraction

Standing waves on stretched strings

3 things to know
Unit AS 2

Overview

Recall questions on the positions of the nodes and antinodes in each mode, the wavelength of the simplest mode in terms of the string length and the general expression, the lowest frequency, worked calculations on strings of given length and wave speed, and the effects of tension, thickness and fretting.

What you need to know

Answer each one before you reveal it.

Your recallNot started

Numbers

A string 0.80 m long carries waves at 240 m s⁻¹. Calculate its lowest vibration frequency.

Key term

Express the wavelength of the simplest standing wave on a string of length L fixed at both ends.

Key term

Express the wavelength of the third possible mode of vibration of a string of length L fixed at both ends.

Questions from this subtopic

Each one with the model answer underneath.

1
Where are the nodes and antinodes in the simplest standing wave on a string fixed at both ends?
Model answer
There is a node at each fixed end and a single antinode at the midpoint of the string.
2
Express the wavelength of the simplest standing wave on a string of length L fixed at both ends.
Model answer
The wavelength is 2L, because the string holds exactly half a wavelength between the two end nodes.
3
Give an expression for the lowest frequency at which a string of length L fixed at both ends will vibrate, in terms of the wave speed v.
Model answer
f = v/2L, where v is the speed of the progressive waves along the string in m s⁻¹ and L is its length in metres.
4
Describe the next possible mode of vibration of a string fixed at both ends after the simplest one.
Model answer
There are nodes at both ends and a third node at the midpoint, with two antinodes at the quarter and three-quarter positions, and the wavelength equals the length of the string.
5
Express the wavelength of the third possible mode of vibration of a string of length L fixed at both ends.
Model answer
The wavelength is 2L/3, because the string then holds three half-wavelength loops.
6
Give a general expression for the wavelengths at which a string of length L fixed at both ends can vibrate.
Model answer
λ = 2L/n, where n is a whole number counting the loops, so the possible frequencies are f = nv/2L.

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