Wave Intensity

Progressive waves transfer energy
The amount of energy passing through a unit area per unit time is the intensity of the wave
Therefore, the intensity is defined as power per unit area

$I = \frac{P}{A}$

Where
- I is intensity in W m⁻²
- P is power in W
- A is area in m²

The area the wave passes through is perpendicular to the direction of its velocity
The intensity of a progressive wave is also proportional to its amplitude squared and frequency squared

$I \propto A^{2}$

$I \propto f^{2}$

Where
- A is amplitude in m
- f is frequency in Hz
Recall that ∝ means "proportional to"

This means that if the frequency or the amplitude is doubled, the intensity increases by a factor of 4 (2²)

Spherical waves

A spherical wave is a wave from a point source which spreads out equally in all directions
The area the wave passes through is the surface area of a sphere: 4πr²
As the wave travels further from the source, the energy it carries passes through increasingly larger areas as shown in the diagram below:

Inverse square law for intensity

Intensity of a spherical wave, downloadable AS & A Level Physics revision notes

Intensity is proportional to the amplitude squared

Assuming none of the wave energy is absorbed, the intensity I decreases with increasing distance from the source
The inverse square law states that

$I \propto \frac{1}{r^{2}}$

- The inverse square law means when the source is twice as far away, the intensity is 4 times smaller

Worked example

The intensity of a progressive wave is proportional to the square of the amplitude of the wave. It is also proportional to the square of the frequency. The variation with time t of displacement x of particles when two progressive waves Q and P pass separately through a medium, are shown on the graphs.

The intensity of wave Q is I₀.What is the intensity of wave P?

Answer:

Step 1: Compare waves Q and P:

The amplitude of wave Q is twice the amplitude of wave P
The frequency of wave P is double the frequency of wave Q

Step 2: Evaluate how amplitude affects the intensities:

Q has twice the amplitude of P

$\frac{A_{Q}}{A_{P}} = 2$

The ratio of amplitudes squared is equal to the ratio of intensities, because intensity and amplitude squared are proportional

$\frac{{A_{Q}}^{2}}{{A_{P}}^{2}} = 4 = \frac{I_{Q}}{I_{P}}$

$I_{Q} = 4 I_{P}$

So Q's larger amplitude increases its intensity by a factor of 4

Step 3: Evaluate how frequency affects the intensities:

Q has half the frequency of P

$\frac{f_{Q}}{f_{P}} = \frac{1}{2}$

The ratio of frequencies squared is equal to the ratio of intensities because intensity and frequency squared are proportional

$\frac{{f_{Q}}^{2}}{{f_{P}}^{2}} = \frac{1}{4} = \frac{I_{Q}}{I_{P}}$

$I_{Q} = \frac{1}{4} I_{P}$

So Q's lower frequency decreases its intensity by a factor of 4

Step 4: Determine the intensity of wave P in terms of I₀ :

P has a lower amplitude which reduces its intensity by a factor of 4 compared to Q
However, P has a higher frequency which increases its intensity by a factor of 4 compared to Q
These effects cancel out, so wave P has an intensity of I₀ , which is equal to that of wave Q

Examiner Tip

The key concept with intensity is that it has an inverse square relationship with distance (not a linear one). This means the energy of a wave decreases very rapidly with increasing distance

Wave Intensity (CIE A Level Physics)

Revision Note

Wave intensity

Spherical waves

Inverse square law for intensity

Worked example

Examiner Tip

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Author: Katie M