## Mastering Physics Solutions: The Decibel Scale

The Decibel Scale

Part A = 10 dB
Part B = 20 dB
Part C = 3,6,9 dB

Solution Below:

The decibel scale is a logarithmic scale for measuring the sound intensity level. Because the decibel scale is logarithmic, it changes by an additive constant when the intensity as measured in W/m2 changes by a multiplicative factor. The number of decibels increases by 10 for a factor of 10 increase in intensity. The general formula for the sound intensity level, in decibels, corresponding to intensity I is

Β = 10log(I / I0) dB,

where I0 is a reference intensity. For sound waves, I0 is taken to be 10-12 W/m2. Note that log refers to the logarithm to the base 10.

Part A

What is the sound intensity level Β, in decibels, of a sound wave whose intensity is 10 times the reference intensity (i.e., I = 10I0)?
Express the sound intensity numerically to the nearest integer.

Just follow the formula:

Β = 10log(I / I0)
Β = 10log(10 / 1)
Β = 10log(10)

Β = 10 dB

10 dB

Part B

What is the sound intensity level Β, in decibels, of a sound wave whose intensity is 100 times the reference intensity (i.e., I = 100I0)?
Express the sound intensity numerically to the nearest integer.

Just follow the formula:

Β = 10log(I / I0)
Β = 10log(100 / 1)
Β = 10log(100)

Β = 20 dB

20 dB

Part C

One often needs to compute the change in decibels corresponding to a change in the physical intensity measured in units of power per unit area. Take f to be the factor of increase of the physical intensity.

Calculate the change in decibels (ΔΒ2, ΔΒ4, and ΔΒ8) corresponding to f = 2, f = 4, and f = 8.

To find the increase take I0 in the general formula to be the initial intensity and then take I to be the factor of increase: I=fI0. Then, log(I / I0) = log(f) and the change in intensity measured in decibels is 10log(f).