Standing Waves in Strings (HSC SSCE Physics): Revision Notes

Standing Waves in Strings

Introduction to standing waves in strings

When a string is fixed at both ends and set into vibration, it can create patterns called standing waves or stationary waves. These standing waves form the foundation of how string musical instruments produce sound. The string must have specific dimensions to create resonance, which allows the standing waves to form and produce clear musical notes.

Modes of vibration in strings

Understanding harmonics

When a string fixed at both ends vibrates, it can form different patterns called harmonics or modes of vibration. Each end of the string must be a node (a point of zero displacement) for all vibration modes because the ends are fixed and cannot move.

The fundamental mode of vibration is the simplest pattern and is also called the first harmonic. This produces the lowest frequency and is what you hear as the basic pitch of the string. The other vibration patterns are called the second harmonic, third harmonic, fourth harmonic, and so on.

Harmonics and overtones

Overtones have higher frequencies than the fundamental frequency and typically have smaller amplitudes. They contribute to the quality or timbre of the sound produced by the instrument.

The five basic harmonic patterns

The table below shows how the first five harmonics appear in a string of length $l$ fixed at both ends:

Vibration Mode	Wavelength ($\lambda$)	Frequency ($f$)	Description
Fundamental (1st harmonic)	$\lambda_1 = 2l$	$f_1 = \frac{v}{2l}$	Half wavelength fits in string length
1st overtone (2nd harmonic)	$\lambda_2 = l$	$f_2 = 2f_1$	One complete wavelength fits in string length
2nd overtone (3rd harmonic)	$\lambda_3 = \frac{2l}{3}$	$f_3 = 3f_1$	One and a half wavelengths fit in string length
3rd overtone (4th harmonic)	$\lambda_4 = \frac{l}{2}$	$f_4 = 4f_1$	Two complete wavelengths fit in string length
4th overtone (5th harmonic)	$\lambda_5 = \frac{2l}{5}$	$f_5 = 5f_1$	Two and a half wavelengths fit in string length

How to generate different harmonics

Mathematical relationships for standing waves

Frequency relationships

The frequency of any harmonic is directly related to the fundamental frequency by a simple relationship:

$f_n = nf_1$

where:

• $f_n$ is the frequency of the $n$th harmonic
• $n$ is the harmonic number (1, 2, 3, ...)
• $f_1$ is the fundamental frequency

Wavelength relationships

For a string of length $l$ fixed at both ends, the wavelength of each harmonic relates to the string length by:

$l = n\frac{\lambda_n}{2}$

where:

• $l$ is the length of the string
• $n$ is the harmonic number
• $\lambda_n$ is the wavelength of the $n$th harmonic

Rearranging this gives:

$\lambda_n = \frac{2l}{n}$

This formula tells us:

• For the 1st harmonic: $\lambda_1 = 2l$ (half a wave fits)
• For the 2nd harmonic: $\lambda_2 = l$ (one complete wave fits)
• For the 3rd harmonic: $\lambda_3 = \frac{2l}{3}$ (one and a half waves fit)

General formula for the nth harmonic

Combining the wave equation $v = f\lambda$ with the wavelength relationship above, we can derive a general formula for any harmonic frequency.

Starting with the fundamental frequency: $f_1 = \frac{v}{\lambda_1} = \frac{v}{2l}$

For the second harmonic: $f_2 = \frac{v}{\lambda_2} = \frac{v}{l} = \frac{2v}{2l} = 2f_1$

Generalising for the $n$th harmonic: $f_n = \frac{v}{\lambda_n} = \frac{nv}{2l} = nf_1$

This confirms our earlier relationship and shows that each harmonic frequency is a whole-number multiple of the fundamental frequency.

Worked example: Calculating overtones and wave properties

Velocity of waves in strings

Factors affecting wave velocity

The speed at which waves travel along a stretched string depends on two key factors:

1. Tension ($T$) in the string (measured in newtons)
2. Mass per unit length ($m \cdot l^{-1}$) of the string (measured in kilograms per metre)

Wave velocity formula

The speed of a wave travelling in a stretched string is given by:

$v = \sqrt{\frac{T}{m \cdot l^{-1}}}$

where:

• $v$ is the wave speed (m $\cdot$ s$^{-1}$)
• $T$ is the tension in the string (N)
• $m \cdot l^{-1}$ is the mass per unit length (kg $\cdot$ m$^{-1}$)

Fundamental frequency with tension

We can combine the wave velocity formula with the fundamental frequency relationship to get a complete expression for fundamental frequency:

Starting with: $f_1 = \frac{v}{2l}$

Substitute: $v = \sqrt{\frac{T}{m \cdot l^{-1}}}$

This gives:

$f_1 = \frac{1}{2l}\sqrt{\frac{T}{m \cdot l^{-1}}}$

Or equivalently:

$f_1 = \frac{1}{2l}\sqrt{\frac{T}{m/l}}$

This formula shows clearly that:

• Longer strings vibrate at lower frequencies
• More massive strings vibrate at lower frequencies
• Greater tension produces higher frequencies

These relationships explain the design of string instruments and how musicians adjust pitch by changing string tension, length, or choosing strings of different thickness.

Worked example: Calculating fundamental frequency from tension

Remember!