Intensity and Energy

Overview

Waves transfer energy from one place to another. This page explains how wave power is distributed through space, how intensity is defined, and how amplitude affects energy transport.

Definition

Power is the rate of energy transfer:

$$P=\frac{E}{t}$$

Intensity is power transmitted per unit area normal to the direction of wave travel:

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

where $S$ is area and $I$ has unit $\mathrm{W}{\mathrm{m}}^{-2}$. This avoids using $A$ for both area and amplitude.

Why It Matters

Intensity measures how concentrated wave energy is. It explains why sound and light weaken with distance, why louder sounds involve larger pressure oscillations, and why amplitude changes have a squared effect on energy transfer.

Key Representations

Energy Transfer by Waves

A progressive wave carries energy as it propagates.

Examples:

• sound transfers energy from a loudspeaker to air;
• light transfers energy from the Sun to Earth;
• water waves transfer energy across a surface;
• seismic waves transfer energy through rock.

Energy moves with the wave, but matter does not move overall with the wave.

Power

Power is:

$$P=\frac{E}{t}$$

If a source emits more wave energy each second, it has greater power. A louder speaker has greater sound power, and a brighter lamp has greater radiant power.

Intensity

Intensity is:

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

where $S$ is the area perpendicular to propagation.

The same power spread over a larger area gives lower intensity. If identical power is spread over four times the area:

$$I\to \frac{1}{4}I$$

Point Source Spreading

For a source radiating equally in all directions, energy spreads over a spherical surface:

$$S=4\pi r^2$$

Hence:

$$I=\frac{P}{4\pi r^2}$$

So:

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

This is the inverse-square law.

If distance doubles, intensity becomes one quarter. If distance triples, intensity becomes one ninth.

Worked Example

A lamp emits uniformly with power:

$$P=100\mathrm{W}$$

Find intensity at:

$$r=2.0\mathrm{m}$$

Use:

$$I=\frac{100}{4\pi (2.0{)}^2}$$

Therefore:

$$I=\frac{100}{16\pi }\approx 1.99\mathrm{W}{\mathrm{m}}^{-2}$$

Two-Dimensional Spreading

Some waves spread approximately in circles on a surface, such as water ripples. Energy spreads around circumference:

$$2\pi r$$

so intensity falls roughly as:

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

This is slower decay than spherical spreading.

Intensity and Amplitude

For many wave types:

$$I\propto A_0^2$$

where $A_0$ is amplitude.

Consequences:

• if amplitude doubles, $I\to 4I$;
• if amplitude triples, $I\to 9I$;
• if amplitude halves, $I\to I/4$.

Wave energy often depends on both displacement size and particle speed. Both scale with amplitude, so total energy transfer scales with amplitude squared.

Combining Distance and Amplitude

For spherical spreading:

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

and:

$$I\propto A_0^2$$

therefore:

$$A_0\propto \frac{1}{r}$$

for ideal spherical spreading.

Sound Interpretation

Greater sound intensity means more sound power per unit area. It is usually perceived as louder sound, although human hearing is logarithmic and frequency-dependent.

Physics-wise, greater sound intensity means larger pressure oscillations and greater energy transfer rate.

Common Mistakes

• Using diameter instead of radius in $4\pi r^2$.
• Forgetting the square on distance.
• Confusing power with intensity.
• Confusing amplitude with intensity.
• Writing $I\propto A_0$ instead of $I\propto A_0^2$.

Links

• Main topic: Waves
• Related concept: Wave Basics
• Related concept: Phase Difference
• Related concept: Sound Measurement Practicals