Intensity of Sound

One factor that influences loudness can be analyzed objectively, This factor is called sound intensity. In this post, we will study the definition, concepts, and formula or equation of sound intensity. Then we will solve a sample numerical problem based on the sound intensity equation.

Definition

Sound intensity is a measure of the power of a sound per unit area. Since power is the rate at which energy is received, intensity could also be defined as (energy/time)/(unit area).
The SI unit of sound intensity is watts per square meter or W/m².

Concepts of Sound Intensity

As we get farther from the source of a sound, the loudness we perceive becomes less. To relate this to sound intensity, consider an “ideal” situation in which a point source of sound is sending out sound waves in all directions and there are no barriers from which these waves can reflect.

At a certain distance, r, from the source, the energy is spread out over a sphere of surface area 4 Π r². If this distance is doubled to 2r, the surface area of the sphere becomes 4 Π (2r)², which is 4 times as large.

Thus, the power per unit area reaching the larger sphere is 1/4 as much as at the smaller sphere. Similarly, if the distance becomes 3r, the area is 4 Π (3r)², which is 9 times as large. Thus, the power per unit area reaching the larger sphere is 1/9 as much as at the smaller sphere.

Sound Intensity Equation

This typical inverse square relationship between sound intensity and distance can be expressed mathematically as follows:
I ∝ (1/r²)
Intensity I is directly proportional to (1/r²) where I is the intensity in watts per square meter and r is the distance in meters from the source of the sound

Sample Numerical Problem

Question: At a distance of 59 m from a jet taking off, the intensity of the sound is 0.10 W/m². At what distance will the intensity be 1/10 of this value? (Neglect the effects of the reflection of sound off the ground.)

Solution:

Using this relation: I ∝ (1/r²)
I₁ ∝ (1/r₁²)
I₂ ∝ (1/r₂²)

I₂ /I₁ = (r₁²/r₂²)
=>r₂ =r₁ (I₁/I₂)^1/2
=>r₂ = (59) (0.1/0.01)^1/2 = 59x 10^1/2 = 59×3.16 = 186 m

At 186 m distance, the intensity will be 1/10 of the initial value. (Answer)