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A

Wave Amplitude
and Energy

Explore how amplitude determines the energy a wave carries, and why doubling the amplitude quadruples the energy.

High School Physics

What Is a Wave?

A wave is a disturbance that transfers energy from one place to another without permanently displacing the medium it travels through. Waves are everywhere: sound, light, ocean swells, and even seismic vibrations.

Mechanical
Require a medium: sound waves, water waves, seismic waves
Electromagnetic
No medium needed: light, radio, X-rays travel through vacuum

Key Wave Properties

equilibrium A λ (wavelength) crest trough
Amplitude (A)
Maximum displacement from equilibrium
Wavelength (λ)
Distance for one full cycle
Frequency (f)
Cycles per second (Hz)

Defining Amplitude

Amplitude is the maximum displacement of a point on the wave from its rest (equilibrium) position. It is always a positive value and is measured in the same units as the displacement: metres for mechanical waves, volts per metre for EM waves, etc.

A₁ A₂ Small Amplitude Large Amplitude Low energy High energy

Energy Carried by a Wave

A wave's energy depends on how vigorously the medium is displaced. In physics, the energy transported by a wave is directly related to the square of its amplitude. This is one of the most important relationships in wave physics.

Proportionality
\(E \propto A^2\): energy is proportional to the square of amplitude
Consequence
Double the amplitude means 4 times the energy

The Core Rule: \(E \propto A^2\)

Amplitude (A) Energy (E) A=1, E=1 A=2, E=4 1 2 3 1 4

Why Is It Squared?

The squared relationship comes directly from the physics of simple harmonic motion. Each small element of the medium oscillates like a mass on a spring. The energy stored in a spring is:

1
Each particle in the medium acts like a tiny oscillator with energy \(\tfrac{1}{2}k A^2\).
2
Summing over all particles gives the total wave energy, which keeps the \(A^2\) dependence.
3
The result applies universally to mechanical waves, sound waves, and even electromagnetic waves (where energy density \(\propto E_0^2\)).

Amplitude vs. Frequency and Wavelength

Students often confuse how different wave properties affect energy. Let's clarify the distinctions:

Amplitude Frequency Wavelength
Symbol A f λ
Unit metres (m) hertz (Hz) metres (m)
Energy relation \(E \propto A^2\) \(E \propto f^2\) Indirect (via \(v = f\lambda\))
Independent? Yes: set by source Yes: set by source No: depends on \(f\) and \(v\)

How to Apply the Relationship

1
Identify the amplitudes. Read or measure the amplitude before and after the change. Call them \(A_1\) and \(A_2\).
2
Set up the ratio. Since \(E \propto A^2\), write \(\dfrac{E_2}{E_1} = \dfrac{A_2^2}{A_1^2}\).
3
Compute the factor. Calculate \(\left(\dfrac{A_2}{A_1}\right)^2\) to find how many times the energy changes.
4
Solve. If the original energy \(E_1\) is known, multiply: \(E_2 = E_1 \times \left(\dfrac{A_2}{A_1}\right)^2\).

Example 1: Doubling Amplitude

A wave on a string has amplitude \(A_1 = 0.05\,\text{m}\) and carries energy \(E_1\). The amplitude is increased to \(A_2 = 0.10\,\text{m}\). By what factor does the energy change?

1
Amplitude ratio: \(\dfrac{A_2}{A_1} = \dfrac{0.10}{0.05} = 2\)
2
Energy ratio: \(\dfrac{E_2}{E_1} = 2^2 = 4\)
3
Result: The energy increases by a factor of 4.

Example 2: Tripling Amplitude

A sound wave has an initial energy of \(E_1 = 20\,\text{J}\) at amplitude \(A_1\). If the amplitude is tripled (\(A_2 = 3A_1\)), what is the new energy?

1
Amplitude ratio: \(\dfrac{A_2}{A_1} = 3\)
2
Energy ratio: \(\dfrac{E_2}{E_1} = 3^2 = 9\)
3
Calculate: \(E_2 = 9 \times 20\,\text{J} = \) 180 J

Example 3: Finding Amplitude from Energy

A seismograph records that an earthquake's energy increased by a factor of 25 compared to a smaller quake. By what factor did the wave amplitude increase?

1
Energy ratio given: \(\dfrac{E_2}{E_1} = 25\)
2
Use the relationship backwards: \(\dfrac{A_2^2}{A_1^2} = 25\), so \(\dfrac{A_2}{A_1} = \sqrt{25} = 5\)
3
Result: The amplitude increased by a factor of 5.

Visualising the Squared Relationship

Amplitude Relative Energy A 1x 2A 2x 3A 3x 1 1x 4 4x 9 9x →

Real-World Applications

Sound and Volume
Louder sounds have greater amplitude. Doubling loudness requires roughly 10 times the power because our perception is logarithmic (decibels).
Earthquakes
Seismic wave amplitude determines destructive power. The Richter scale is logarithmic: each whole number increase means ~31.6 times more energy.
Light Intensity
The intensity of an EM wave is proportional to the square of the electric field amplitude: \(I \propto E_0^2\).

Common Mistakes

✕ Assuming Linear
Thinking that doubling amplitude doubles energy. It actually quadruples it because \(E \propto A^2\), not \(E \propto A\).
✕ Confusing A with λ
Amplitude (vertical displacement) is not wavelength (horizontal distance). They are independent properties.
✕ Forgetting the Square Root
When given an energy ratio and solving for amplitude, students forget to take the square root: \(A_2/A_1 = \sqrt{E_2/E_1}\).

Summary: Amplitude and Energy

Amplitude Factor Energy Factor Calculation
\(\times 1\) \(\times 1\) \(1^2 = 1\)
\(\times 2\) \(\times 4\) \(2^2 = 4\)
\(\times 3\) \(\times 9\) \(3^2 = 9\)
\(\times \tfrac{1}{2}\) \(\times \tfrac{1}{4}\) \(\left(\tfrac{1}{2}\right)^2 = \tfrac{1}{4}\)
\(\times 5\) \(\times 25\) \(5^2 = 25\)
\(\times 10\) \(\times 100\) \(10^2 = 100\)

Question 1 of 5

If the amplitude of a wave is tripled, by what factor does its energy change?

Question 2 of 5

A wave carries \(E_1 = 50\,\text{J}\) of energy. If the amplitude is halved, what is the new energy?

Question 3 of 5

The energy of a wave increases by a factor of 16. By what factor did the amplitude increase?

Question 4 of 5

Two waves have the same frequency. Wave X has amplitude \(A\) and Wave Y has amplitude \(4A\). What is the ratio \(\frac{E_Y}{E_X}\)?

Question 5 of 5

Which statement best describes the relationship between wave amplitude and energy?