Intensity of Wave Formula: I = P/A, Inverse Square Law

The intensity of wave formula is $I = \frac{P}{A}$ : the power carried by the wave divided by the area it passes through at right angles. Its SI unit is the watt per square metre (W/m²). Bright midday sunlight delivers roughly 1000 W/m² at the ground, while the faintest sound a healthy ear can detect is about 10⁻¹² W/m².

Two consequences follow from that one definition. Waves spreading out from a point source share their power across a spherical surface of area $4\pi r^2$ , so intensity falls as $\frac{1}{r^2}$ . And for a given wave the energy carried climbs as the amplitude squared, so $I \propto A^2$ . These two proportionalities answer most Class 11 and AP Physics questions on loudness and brightness.

Quick facts

Definition
$I = \frac{P}{A}$
Point source
$I = \frac{P}{4\pi r^2}$
Amplitude rule
$I \propto A^2$ (and $I \propto \omega^2$ )
SI unit
watt per square metre (W/m²)
Dimensional formula
[M¹ L⁰ T⁻³]
Type
Scalar
Sound level
$\beta = 10\log_{10}\frac{I}{I_0}$ dB, with I₀ = 10⁻¹² W/m²

Formula of intensity of wave, symbol by symbol

Quantities used for wave intensity
Symbol	Quantity	SI unit
$I$	Intensity: power per unit area	W/m²
$P$	Power carried by the wave	W
$A$	Area perpendicular to the direction of travel	m²
$r$	Distance from a point source	m

Do not confuse the area A in the definition with the amplitude, which some books also write as A. When both appear in one problem, call the amplitude $y_0$ or $a$ to keep them apart.

Intensity of a wave formula for mechanical and light waves

For a sinusoidal wave on a string or in a fluid, the energy per unit volume is $\frac{1}{2}\rho\omega^2 a^2$ , and that energy moves along at the wave speed v. Multiplying gives

$I = \frac{1}{2}\rho v \omega^2 a^2 = 2\pi^2 \rho v f^2 a^2$

where ρ is the density of the medium and a the amplitude. For a light wave the same idea gives $I = \frac{1}{2}c\varepsilon_0 E_0^2$ , with $E_0$ the peak electric field. Both show the square law: double the amplitude and the intensity rises four times. See the formula of amplitude for how amplitude is read off a wave equation.

Worked examples using the intensity of wave formula

Example 1: sunlight on a solar panel

Sunlight of intensity 1000 W/m² falls straight onto a panel of area 1.6 m². How much power reaches the panel?

$P = IA = 1000 \times 1.6 = 1600\ \text{W} = 1.6\ \text{kW}$

Example 2: a loudspeaker outdoors

A small speaker sends out 2.0 W of sound evenly in all directions. Find the intensity 3.0 m away and the sound level in decibels.

$I = \frac{P}{4\pi r^2} = \frac{2.0}{4\pi(3.0)^2} = \frac{2.0}{113.1} = 0.0177\ \text{W/m}^2$

$\beta = 10\log_{10}\frac{0.0177}{10^{-12}} = 10 \times 10.25 \approx 102\ \text{dB}$

That is very loud, about the level of a power tool close by.

Example 3: moving away from the speaker

How does the intensity change if you walk from 3.0 m to 9.0 m from the same speaker?

The distance triples, so the intensity drops by 3² = 9. New intensity = 0.0177 ÷ 9 ≈ 0.0020 W/m², about 9.5 dB quieter.

Example 4: a bigger amplitude

Two sound waves of the same frequency travel through the same air. One has three times the amplitude of the other. Compare their intensities.

Since I ∝ a², the ratio is 3² = 9. The larger wave carries nine times the power per square metre.

Units and dimensions

A watt is a joule per second, which works out as kg m² s⁻³. Divide by m² and the metres disappear, leaving kg s⁻³. So intensity has dimensions [M¹ L⁰ T⁻³]. This is a useful check on the mechanical formula: $\rho v \omega^2 a^2$ has units (kg/m³)(m/s)(s⁻²)(m²) = kg s⁻³, as it should.

Sound intensities cover a huge range, from 10⁻¹² W/m² up to about 1 W/m² at the threshold of pain. That is why the decibel scale is logarithmic: every factor of 10 in intensity adds 10 dB.

Mistakes that cost marks

Using $\pi r^2$ instead of $4\pi r^2$ for a point source. The power spreads over a whole sphere, not a disc.
Assuming intensity halves when distance doubles. It quarters.
Saying intensity is proportional to amplitude. It goes with the square.
Treating decibels as if they add like watts. Two 60 dB sources together give about 63 dB, not 120 dB.

Intensity tells you how much energy a wave delivers, but not how fast it travels. For that, use the wave velocity formula or the speed of sound formula. The frequency of a sound sets its pitch, which is a separate property from its loudness.

Frequently asked questions

What is the formula of intensity of wave?

Intensity is I = P/A, power divided by the area at right angles to the wave. For a point source radiating evenly, I = P/(4πr²).

Which unit is wave intensity measured in?

The watt per square metre, W/m². In mass, length and time the bracket reads [M¹ L⁰ T⁻³], with no length left over.

How is intensity related to amplitude?

Intensity is proportional to the square of the amplitude. Doubling the amplitude makes the wave four times as intense.

Why does intensity follow an inverse square law?

A point source spreads its power over the surface of a sphere, whose area is 4πr². Doubling the distance makes that area four times larger, so each square metre receives a quarter of the power.

What is the difference between intensity and loudness?

Intensity is a physical quantity in W/m². Loudness is how strong a sound seems to a listener, and is measured on the decibel scale, which grows with the logarithm of intensity.