Principle of superposition of waves and formation of stationary waves

The Mathematics of the First Harmonic

The frequency of the first harmonic depends on the physical properties of the string. A tighter, lighter, or shorter string will vibrate faster (at a higher frequency).

The formula for the frequency of the first harmonic is:

\frac{1}{2l} \sqrt{\frac{T}{\mu}}

Where:

f

is the frequency of the first harmonic in hertz (

Hz\text{Hz}

)

l

is the length of the vibrating string in metres (

m\text{m}

)

T

is the tension in the string in newtons (

N\text{N}

)

μ\mu

is the mass per unit length of the string in kilograms per metre (

m−1\text{kg m}^{-1}

)

Common Mistake

Mass vs. Mass per unit length

A very common mistake is plugging the total mass of the string ( $m$ ) into the formula instead of $μ\mu$ . If a question gives you the mass and the length, you must first calculate the mass per unit length using $μ=ml\mu = \frac{m}{l}$ before using the main equation!

Example

Calculating string tension

A guitar string has a length of $\text{ m}$ and a mass of $3.0 \times 10^{-3} \text{ kg}$ . The frequency of its first harmonic is $\text{ Hz}$ . Calculate the tension in the string.

First, calculate the mass per unit length, $μ\mu$ :

\mu = \frac{m}{l} = \frac{3.0 \times 10^{-3}}{0.65} = 4.615 \times 10^{-3} \text{ kg m}^{-1}

State the formula for the first harmonic:

\frac{1}{2l} \sqrt{\frac{T}{\mu}}

Rearrange the formula to make Tension ( $T$ ) the subject. First, multiply by $2 l$ :

\sqrt{\frac{T}{\mu}}

Square both sides:

(2lf)^2 = \frac{T}{\mu}

4l^2 f^2 = \frac{T}{\mu}

Multiply by $μ\mu$ :

l^2 f^2 \mu

Substitute the values into the rearranged equation:

\times (0.65)^2 \times (330)^2 \times 4.615 \times 10^{-3}

Calculate the final answer:

\text{ N} \approx 850 \text{ N}

Link to Required Practical 1

You will investigate this formula in your first Required Practical. By using a vibration generator and changing weights on a string, you can verify that the frequency

f

is proportional to

1l\frac{1}{l}

(if you plot

f

against

1l\frac{1}{l}

, you get a straight line through the origin) and that

f

is proportional to

T\sqrt{T}

(plotting

f

against

T\sqrt{T}

also yields a straight line through the origin).

Stationary Waves with Other Types of Waves

Stationary waves are not limited to strings; any wave can form them if reflected back on itself.

Microwaves: If you direct a microwave transmitter at a flat metal plate, the microwaves reflect back. The reflected wave superposes with the incoming wave, creating a stationary microwave pattern. If you move a microwave detector probe between the transmitter and the plate, the signal will repeatedly drop to zero (at the nodes) and rise to a maximum (at the antinodes).

Sound Waves: Stationary sound waves can form in pipes. A loudspeaker playing a single tone into a glass tube closed at one end will create a stationary wave. The closed end of the tube forces the air molecules to stop moving, making it a displacement node. You can sometimes see this by sprinkling a fine powder into the tube; the powder is thrown away from the fiercely vibrating antinodes and settles quietly at the nodes.

Exam technique

In the exam

If asked "how a stationary wave is formed", your answer must always include: "two progressive waves", "same frequency/wavelength", "travelling in opposite directions", and "superposition".
Pay strict attention to the units of $μ\mu$ . If mass is given in grams, convert to $kg\text{kg}$ immediately.
Remember that frequency is constant across all points on the stationary wave (except at the exact nodes, where there is zero amplitude). Every point on the wave reaches its maximum displacement at the exact same time.

Self review

Check yourself

What is the distance between a node and the adjacent antinode in terms of wavelength ( $λ\lambda$ )?
How many nodes and antinodes are present in the second harmonic on a fixed string?
If you quadruple the tension in a string ( $\rightarrow 4T$ ), by what factor does the frequency of the first harmonic change?

Harmonics on a string

The Mathematics of the First Harmonic

The frequency of the first harmonic depends on the physical properties of the string. A tighter, lighter, or shorter string will vibrate faster (at a higher frequency).

The formula for the frequency of the first harmonic is:

\frac{1}{2l} \sqrt{\frac{T}{\mu}}

Where:

$f$ is the frequency of the first harmonic in hertz ( $Hz\text{Hz}$ )
$l$ is the length of the vibrating string in metres ( $m\text{m}$ )
$T$ is the tension in the string in newtons ( $N\text{N}$ )
$μ\mu$ is the mass per unit length of the string in kilograms per metre ( $m−1\text{kg m}^{-1}$ )

Common Mistake

Mass vs. Mass per unit length

Example

Calculating string tension

A guitar string has a length of $\text{ m}$ and a mass of $3.0 \times 10^{-3} \text{ kg}$ . The frequency of its first harmonic is $\text{ Hz}$ . Calculate the tension in the string.

First, calculate the mass per unit length, $μ\mu$ :

\mu = \frac{m}{l} = \frac{3.0 \times 10^{-3}}{0.65} = 4.615 \times 10^{-3} \text{ kg m}^{-1}

State the formula for the first harmonic:

\frac{1}{2l} \sqrt{\frac{T}{\mu}}

Rearrange the formula to make Tension ( $T$ ) the subject. First, multiply by $2 l$ :

\sqrt{\frac{T}{\mu}}

Square both sides:

(2lf)^2 = \frac{T}{\mu}

4l^2 f^2 = \frac{T}{\mu}

Multiply by $μ\mu$ :

l^2 f^2 \mu

Substitute the values into the rearranged equation:

\times (0.65)^2 \times (330)^2 \times 4.615 \times 10^{-3}

Calculate the final answer:

\text{ N} \approx 850 \text{ N}

Link to Required Practical 1

You will investigate this formula in your first Required Practical. By using a vibration generator and changing weights on a string, you can verify that the frequency $f$ is proportional to $1l\frac{1}{l}$ (if you plot $f$ against $1l\frac{1}{l}$ , you get a straight line through the origin) and that $f$ is proportional to $T\sqrt{T}$ (plotting $f$ against $T\sqrt{T}$ also yields a straight line through the origin).

Stationary Waves with Other Types of Waves

Stationary waves are not limited to strings; any wave can form them if reflected back on itself.

Exam technique

In the exam

If asked "how a stationary wave is formed", your answer must always include: "two progressive waves", "same frequency/wavelength", "travelling in opposite directions", and "superposition".
Pay strict attention to the units of $μ\mu$ . If mass is given in grams, convert to $kg\text{kg}$ immediately.
Remember that frequency is constant across all points on the stationary wave (except at the exact nodes, where there is zero amplitude). Every point on the wave reaches its maximum displacement at the exact same time.

Self review

Check yourself

What is the distance between a node and the adjacent antinode in terms of wavelength ( $λ\lambda$ )?
How many nodes and antinodes are present in the second harmonic on a fixed string?
If you quadruple the tension in a string ( $\rightarrow 4T$ ), by what factor does the frequency of the first harmonic change?

The Mathematics of the First Harmonic

Link to Required Practical 1

Stationary Waves with Other Types of Waves

Review Principle of superposition of waves and formation of stationary waves by teaching Genie

Recap your knowledge with an interactive lesson

Remember key concepts with flashcards

Principle of superposition of waves and formation of stationary waves

The Mathematics of the First Harmonic

Link to Required Practical 1

Stationary Waves with Other Types of Waves

Review Principle of superposition of waves and formation of stationary waves by teaching Genie

Recap your knowledge with an interactive lesson

Remember key concepts with flashcards

Principle of superposition of waves and formation of stationary waves

The Mathematics of the First Harmonic

Link to Required Practical 1

Stationary Waves with Other Types of Waves

Review Principle of superposition of waves and formation of stationary waves by teaching Genie

Recap your knowledge with an interactive lesson

Remember key concepts with flashcards

Principle of superposition of waves and formation of stationary waves Revision Guide

Principle of superposition of waves and formation of stationary waves

The Mathematics of the First Harmonic

Link to Required Practical 1

Stationary Waves with Other Types of Waves

Review Principle of superposition of waves and formation of stationary waves by teaching Genie

Recap your knowledge with an interactive lesson

Remember key concepts with flashcards

Principle of superposition of waves and formation of stationary waves Revision Guide