The Inverse Square Law

The loudness of a sound is often referred to as sound intensity. The intensity of a sound is the amount of sound energy (measured in Joules, J) passing through a unit area (a square metre) in one second. Intensity has the unit of \cfrac { J }{ s } per square metre. As 1 Joule per second is also the unit for power, watts (W), we get the unit for intensity: W{ m }^{ -2 }  

The energy from a light or sound wave will spread out as it moves away from its source. As it moves away the energy will spread across an increasingly larger area and the intensity of the sound or light will decrease. The intensity of the sound or light will decrease in proportion to the inverse of the square of the distance from the source. We assume that the source of the sound or light acts as a point source and the relationship between intensity and distance is known as the inverse square law.

Mathematically, the inverse square law is: I\propto \cfrac { 1 }{ { r }^{ 2 } }

To compare the intensity of sound or light at two points: { I }_{ 1 }\;{ r }_{ 1 }^{ 2 }=I_{ 2 }\;{ r }_{ 2 }^{ 2 }

Where:

{ I }_{ 1 } = intensity at position 1

{ r }_{ 1 } = distance between the source and position 1

I_{ 2 } = intensity at position 2

{ r }_{ 2 } = distance between the source and position 2 

Example 1:

The sound emitted from a speaker has an intensity of 5.0\times { 10 }^{ -2 }\; W{ m }^{ -2 }. Calculate the intensity of the sound 50m from the speakers:

using: I\propto \cfrac { 1 }{ { r }^{ 2 } }

I=\cfrac { 5\times { 10 }^{ -2 } }{ 50^{ 2 } }

I=2\times { 10 }^{ -5 }\;W{ m }^{ -2 }

Example 2:

The intensity of a sound wave is 2.5\times { 10 }^{ -3 }\; W{ m }^{ -2 } at a distance of 5m from the source. What is the intensity of the sound at a distance of 20m from the source?

using: { I }_{ 1 }\;{ r }_{ 1 }^{ 2 }=I_{ 2 }\;{ r }_{ 2 }^{ 2 }

(2.5\times { 10 }^{ -3 })\times { 5 }^{ 2 }={ I }_{ 2 }\times { 20 }^{ 2 }

{ I }_{ 2 }=\cfrac { (2.5\times { 10 }^{ -3 })\times { 5 }^{ 2 } }{ { 20 }^{ 2 } }

{ I }_{ 2 }=1.5625\times { 10 }^{ -4 }\;W{ m }^{ -2 }