# PHYS 2211 Module 14.6

## Standing Waves and Resonance

14.6 Standing Waves and Resonance

### Learning Objectives

By the end of this section, you will be able to:

• Describe standing waves and explain how they are produced
• Describe the modes of a standing wave on a string
• Provide examples of standing waves beyond the waves on a string

Practice!

For a standing wave, the particles at a node do not move at all. In a standing wave, the wave pattern stays in the same position on the string and the amplitude of the wave fluctuates. A node is defined as a point that does not move.

For a standing wave on a string, the antinodes are defined as points that move perpendicular to the length of the string with maximum amplitude. The particles at an antinode move in simple harmonic motion about the equilibrium position of the string.

For a standing wave, the particles at an antinode oscillate back and forth with maximum amplitude perpendicular to the length of the string. In a standing wave, the wave pattern stays in the same position on the string and the amplitude of the wave fluctuates. For a standing wave on a string, the antinodes are defined as points that move perpendicular to the length of the string with maximum amplitude. The particles at an antinode move in simple harmonic motion about the equilibrium position of the string.

A node is defined as a point that does not move.

The wavelength of the standing wave shown is equal to 2L. This standing wave pattern exhibits two nodes and one antinode. The distance between two successive nodes is equal to half a wavelength. So this string of length L contains half a wavelength: L = /2 and therefore = 2L.

The wavelength of the standing wave shown is equal to L. This standing wave pattern exhibits three nodes and two antinodes. The distance between two successive nodes is equal to half a wavelength. So this string of length L contains two halves of a wavelength: L = 2/2 and therefore = L.

The wavelength of the standing wave shown is equal to 2L/3. This standing wave pattern exhibits four nodes and three antinodes. The distance between two successive nodes is equal to half a wavelength. So this string of length L contains three halves of a wavelength: L = 3/2 and therefore = 2L/3.

The distance between successive antinodes on this standing wave is /2, where is the wavelength of the standing wave. The distance between two successive nodes is equal to half a wavelength and the distance between two successive antinodes is also equal to half a wavelength. This is true for any standing wave.

The frequency of the third harmonic, or second overtone, is 3f1. The frequencies of standing waves, which are also called harmonics, are determined by multiplying the fundamental frequency, f1, by the number of antinodes, n: fn = nf1. The third harmonic contains three antinodes, which gives n = 3.

If a weight with half the mass was used, the frequency of the standing wave would decrease by a factor of . The frequency of the standing wave depends directly on the wave speed, v, according to f = v/2L, where L is the length of the wire. The wave speed is calculated with so the wave speed will change with changing tension in the wire. Using a lighter block will reduce the tension in the wire by a factor of 2. Therefore the wave speed will decrease by according to: .

Discuss!

Middle C on a piano has a fundamental frequency of 262 Hz, and the first A above middle C has a fundamental frequency of 440 Hz.

(a) Calculate the frequencies of the next two harmonics of the C string.

(b) If the A and C strings have the same linear mass density μ and length L, determine the ratio of tensions in the two strings.