Intensity: the Energy and Power Carried by a Wave
A wave carries power proportional to the square of its amplitude, and intensity is that power spread over area. Spreading in three dimensions is why intensity falls with the square of the distance.
What a learner can do afterwards
- Relates the power carried by a wave to its amplitude and frequency
- Applies the inverse square law to a source radiating into space
- Converts between intensity and the decibel scale for sound
1 · Read
For a string or sound wave, power grows as the square of amplitude times the square of frequency. Double the amplitude and power quadruples. Intensity is that power spread over area, measured in watts per square metre.
A siren pours out fixed power that spreads over an ever growing sphere. Intensity therefore falls as one over distance squared. Double the distance and the intensity drops to a quarter.
Stand twice as far from a siren and each patch of air gets one quarter of the power, with no absorption needed. The sound fades because the same power spreads thin.
Sound loudness uses decibels: a rise of 10 dB means ten times the intensity. Two such rises in a row mean one hundred times.
Power follows amplitude squared times frequency squared, and distance spreads it over one over r squared.
Readings written for this dataset.
More to read
How waves transfer energy
A wave's power measures its rate of energy transfer. For a sinusoidal wave on the same string, mean power scales with amplitude squared and frequency squared. Doubling amplitude gives four times the power. Higher frequency can offset smaller amplitude to give the same mean power.
Intensity measures power per area. Spreading unchanged power across more area lowers intensity. Sound can therefore weaken with distance even without absorption.
Sound Levels and Distance
Hearing spans whispers to sirens, so intensity levels use logarithmic decibels. Adding 10 dB means tenfold intensity. A 60 dB conversation has a million times the intensity at the 0 dB reference, near the hearing threshold. It does not sound a million times louder.
For a source spreading sound evenly in open space without absorption, intensity follows 1/distance². Twice as far away, you receive one quarter. The speaker need not lose power; its sound covers more area.
Comparing Wave Intensities
Divide wave power by the area it crosses to find intensity, in watts per square metre. This quantity applies to earthquakes, ultrasound scans and concert sound.
For sinusoidal mechanical waves with unchanged medium and wave speed, intensity scales with displacement amplitude squared and frequency squared. A source spreading uniformly in three dimensions without losses gives intensity proportional to 1/distance². Combine these relationships to estimate how weak and strong waves compare.
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