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Wave Motion overview

Topic 5 of 6

Intensity and spreading

Intensity describes how much power a wave transfers per unit area. It can decrease as the same power spreads over a larger area, even when no energy is absorbed.

Power per area normal to propagation

The intensity I is the time-averaged power transferred per unit area normal to propagation. For power P distributed uniformly across that area A:

I = P/A
Unit: W/m2

Power is an energy-transfer rate, in watts; intensity also depends on the area over which that power is distributed. P/A gives the average over a chosen area when intensity varies across it.

Intensity follows amplitude squared under the same conditions

For the same wave type, medium and relevant frequency conditions:

I ∝ (amplitude)2
I2/I1 = (amplitude2/amplitude1)2

Tripling amplitude multiplies intensity by 9. A fourfold intensity means twice the amplitude, not four times.

Specify which amplitude is meant. Mechanical displacement amplitude is measured in metres, whereas electromagnetic electric-field amplitude is a field quantity. Two different wave types or frequencies need not share the same proportionality constant, so the square-law ratio is not a licence to compare unspecified waves.

Derive inverse-square spreading

Model a point source radiating time-averaged power P equally in all directions, with no absorption or reflection. A spherical surface at centre distance r has area 4πr2. The same total power crosses every complete sphere:

I = P/(4πr2)
I ∝ 1/r2
I2/I1 = (r1/r2)2

The distance is measured from the source centre, and the point-source approximation requires distances large enough compared with the source's size. A collimated beam does not automatically spread over whole spheres. For a directional source, an inverse-square comparison must keep the same angular direction and unchanged radiation pattern.

Estimate the intensity scale

A source of order 50 W spreading uniformly over a sphere of radius about 2 m gives an area of order 50 m2. Its intensity is therefore of order 1 W/m2, assuming negligible loss. This is a rough scale check, not a detector reading.

Worked spherical spreading

Compare 2.0 m with 5.0 m

An isotropic point-source model radiates power P = 16π W without loss.

At r = 2.0 m:
I = 16π/(4π × 2.02) = 1.00 W/m2
At r = 5.0 m:
I = 16π/(4π × 5.02) = 0.160 W/m2

The same total power is spread over a larger area

An isotropic point-source model emits P = 16π W, with no absorption or reflection. The circles below are cross-sections of complete spherical surfaces, on one radius scale.

Equal total power crosses the two spheres while intensity decreasesTwo concentric circles show cross-sections of spheres centred on an isotropic point source O. The radii are 54 and 135 drawing units, representing two and five metres on one distance scale. The inner radius is measured vertically from the centre; the outer radius is measured horizontally. The same total power, sixteen pi watts, crosses each complete sphere in this lossless model. Their spherical areas are sixteen pi and one hundred pi square metres. Intensity is therefore one watt per square metre at two metres and 0.160 watt per square metre at five metres. The drawing represents spherical spreading rather than two flat discs or a collimated beam.O2.0 m5.0 mCross-section through source ORadii measured from the centreEach complete sphere: P = 16π WAt 2.0 m: I = 1.00 W/m2At 5.0 m: I = 0.160 W/m2

The larger sphere has 6.25 times the area, so its intensity is 0.160 times as large. Under the same wave conditions, its amplitude is √0.160 = 0.400 times the amplitude at 2.0 m.

The same total power crosses both complete spherical surfaces. The larger area has less power per unit area. The drawing represents spreading energy, not particles travelling outwards with the wave.

The intensity ratio is (2.0/5.0)2 = 0.160. With the same medium, wave type and frequency, the amplitude ratio is:

amplitude at 5.0 m / amplitude at 2.0 m
= √0.160 = 0.400

This is a dimensionless ratio, not an amplitude of 0.400 m. The source still radiates 16π W; spreading reduces intensity without removing power from the complete spherical account.

Optional check An unchanged isotropic point source radiates without loss. Intensity is 1.00 W/m^2 at 2.0 m. At 5.0 m, what are the intensity and the amplitude relative to its value at 2.0 m, under the same wave conditions?
An unchanged isotropic point source radiates without loss. Intensity is 1.00 W/m^2 at 2.0 m. At 5.0 m, what are the intensity and the amplitude relative to its value at 2.0 m, under the same wave conditions?

Investigate the distance relationship

Use background-corrected intensity readings at measured source-centre distances. Keep source output, detector orientation and the instrument's operating range controlled. Plot intensity against 1/r2; the ideal lossless model predicts a straight line under the stated geometry.

A detector's raw voltage is not automatically intensity. Establish its calibration or supplied proportionality first. If a signal is proportional to amplitude instead, squaring may be needed to compare intensities; use the actual instrument relationship rather than assuming one.

Absorption, reflections and background signals can change the observed distance relation. Measurements too close to an extended source can also violate the point-source approximation. Repeat readings to assess variation while checking these model conditions separately.