Harmonics

​​In a nutshell

Stationary waves can be formed by fixing one end of a stretched string and attaching a driving oscillator to the other end. At certain resonant frequencies, stationary waves known as harmonics will occur. Different harmonics will form in tubes depending on whether the ends of the tubes are open or closed.

Equations

Variable definitions

Stationary waves on a string

One way of forming stationary waves is to fix one end of a stretched string and attach the other end to a driving oscillator. A driving oscillator will produce progressive waves which reflect off the fixed end and superpose with the waves travelling in the opposite direction. This forms a stationary wave, as both waves have the same frequency and amplitude.

These stationary waves will only occur when the oscillator creates progressive waves of certain frequencies. The stationary wave pattern will change depending on the frequency of the superposing progressive waves.

The simplest wave pattern is formed at the fundamental frequency $$f_0$$ and is called the first harmonic. This is the lowest possible frequency that creates a stationary wave. It consists of two nodes and an antinode.

1. Retort stand 2. Signal generator 3. Driving oscillator (vibration generator) 4. Wooden block 5. Pulley 6. Masses (providing tension $$T$$)​ ​$$L$$​​ Length of oscillating string

The wavelength of the progressive wave is equal to twice the length of the string.

Harmonics are produced at every integer multiple of the fundamental frequency ($$2f_0, 3f_0,...$$). The wavelength can be written as a multiple of the length of the string $$L$$.​

Harmonic	​​Pattern	Frequency	​​Wavelength
1		​$$f_0$$​​	$$2L$$​​
2		​$$2f_0$$​​	$$L$$​​
3		​$$3f_0$$​​	$$\frac{2}{3}L$$​​
4		​$$4f_0$$​​	$$\frac{1}{2}L$$​​

Speed of a wave on a string

The frequency of the first harmonic can be calculated using the equation

​$$f_0 = \dfrac{1}{2L}\sqrt{\dfrac{T}{\mu}}$$​​

The speed of a wave on a string can be determined using this equation. The wavelength of the first harmonic is equal to $$2L$$. Using the wave speed equation ($$v = f\lambda$$) and substituting the terms in gives an equation for the speed of a wave on a string.

​$$v = \sqrt{\dfrac{T}{\mu}}$$

​​

Note: This is only a required equation for AQA and Edexcel, but is useful to know for any exam board!

Example

The frequency of a harmonic created on a length of string (see below) is $$435\,Hz$$​. Determine the speed of the wave on the string.

First, write out the values given in the question and check they are in the correct form:

$$f = 435\,Hz \\ L = 0.950 \,m \space \text{(from figure)}$$​​

The harmonic shown in the figure corresponds to the third harmonic. This means the frequency is equal to $$3f_0$$:

$$3f_0 = f \to f_0 = \dfrac{f}{3}$$​​

Next, write down the equation needed:

$$f_0 = \dfrac{1}{2L}\sqrt{\dfrac{T}{\mu}} \\ v = \sqrt{\dfrac{T}{\mu}} \\ \ \\ f_0 = \dfrac{v}{2L} \\ \\\ v = 2Lf_0 \to v = 2L\dfrac{f}{3}$$​​

Then, substitute the values into the equation:

​$$v = 2 \times 0.95 \times \dfrac{435}{3} = 275.5\,ms^{-1}$$​​

Make sure to include units and round to the lowest number of significant figures of the values given in the question:

$$v = 276\,ms^{-1}$$​​

$$\underline{The \space speed \space of \space the \space wave \space on \space the\space string \space is \space 276\,ms^{-1}}$$

​

Stationary waves using microwaves

Stationary waves can also be formed using microwaves. A source emitting microwaves placed in line with a metal reflecting plate will form stationary waves. This is because the reflector reflects the microwave in the opposite direction and the two waves superpose.

A probe can be placed between the microwave source and the metal reflecting plate. The probe is then moved between the source and the plate to detect the nodes (minima) and antinodes (maxima) of the stationary wave pattern.

Stationary waves in air columns

Stationary waves can also be formed by longitudinal waves. These can be made in tubes and the pattern formed depends on whether the ends are open or closed.

Example

A flute is a tube with both ends upon that uses stationary waves to form music notes.

A clarinet is a tube with one end open, the other closed that also uses stationary waves to form musical notes.

Tube closed at one end

The stationary waves formed in a tube with one closed end and one open end have to have an antinode at the open end and a node at the closed end. This is because the air at the closed end cannot move so forms a node. The air at the open end has the largest oscillations so forms an antinode.

The fundamental mode of vibration (at the fundamental frequency) consists only of a node and an antinode. The harmonics only occurs for odd multiples of the fundamental frequency ($$3f_0,5f_0,...$$)

Harmonic	​​Pattern	Frequency	​​Wavelength
1		​$$f_0$$​​	$$4L$$​​
3		​$$3f_0$$​​	$$\dfrac{4}{3}L$$​​
5		​$$5f_0$$​​	$$\dfrac{4}{5}L$$​​
7		​$$7f_0$$​​	$$\dfrac{7}{9}L$$​​
9		​$$9f_0$$​​	​$$\dfrac{4}9L$$​​

Tube with open ends

The stationary waves formed in a tube with both ends open consist of an antinode at each end.

The fundamental mode of vibration consists of an antinode at each end and a node in the middle. The harmonics can occur for all integer multiples of the fundamental frequency ($$2f_0,3f_0,...$$)

Harmonic	​​Pattern	Frequency	​​Wavelength
1		​$$f_0$$​​	$$2L$$​​
2		​$$2f_0$$​​	$$L$$​​
3		​$$3f_0$$​​	$$\dfrac{2}{3}L$$​​
4		​$$4f_0$$​​	$$\dfrac{1}{2}L$$​​
5		​$$5f_0$$​​	​$$\dfrac{2}{5}L$$​​

Example

The speed of sound can be found using a tuning fork and a tube of water. 

1. Clamp stand 2. Tube 3. Water 4. Beaker 5. Tuning fork 6. Length of air in tube, $$L$$​​

The air around the tuning fork will vibrate at the same frequency as the fork vibrates. If this frequency, $$f$$, matches the fundamental frequency, $$f_0$$, for the tube closed at one end, the sound will become louder. 

The length of the tube, $$L$$, can be changed by raising or lowering the tube. This changes the fundamental frequency as the length of the tube is equal to $$\frac{1}{4}\lambda$$. 

The speed of sound in air can be calculated by using:

$$v = f\lambda = f \times 4L$$

By measuring the length $$L$$ for a variety of tuning forks, an average value of the speed of sound in air can be calculated using a graph of $$L$$ against $$\frac{1}{f}$$.