Question

A woman stands a distance d from a loud motor that emits sound uniformly in all directions. The sound intensity at her position is an uncomfortable 4.0×10^-3 W/m². At a distance 3.2 times as far from the motor, what are (a) the sound intensity and (b) the sound intensity level relative to the threshold of hearing?

Answer 1

Sound Intensity is given by:

I = P/A

P = Power of Source, in Watts

A = Area, in m^2 = 4*pi*R^2

R = distance from Source

I = P/(4*pi*R^2)

Since here power of source remains constant, So Intensity will be inversely proportional to the square of R. So

I2/I1 = (R1/R2)^2

I1 = 4.0*10^-3 W/m^2

R2 = 3.2*R1

So,

I2 = I1*(R2/R1)^2

Using known values:

I2 = 4.0*10^-3*(R1/(3.2*R1))^2

I2 = 4.0*10^-3*(1/3.2)^2

I2 = 3.9*10^-4 W/m^2

Part B.

Sound Intensity level is given by:

SL = 10*log (I2/I0)

I0 = threshold of hearing = 10^-12 W/m^2

I2 = 3.9*10^-4 W/m^2

So,

SL = 10*log (3.9*10^-4/10^-12)

SL = 85.91 dB = 86 dB

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