Question

To understand the decibel scale.

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(II0)dB,

where I0 is a reference intensity. For sound waves, I0 is taken to be 10−12W/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)?

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.

β =		dB

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 m to be the factor of increase of the physical intensity (i.e., I=mI0).

Part C

Calculate the change in decibels ( Δβ2, Δβ4, and Δβ8) corresponding to m=2, m=4, and m=8.

Give your answers, separated by commas, to the nearest integer--this will give an accuracy of 20%, which is good enough for sound.

Δβ2, Δβ4, Δβ8=

Answer 1

Concepts and reason

The concept required to solve this problem is decibel scale of sound intensity.

Use the formula of sound intensity level in decibels and substitute the value of intensity to calculate decibels for all the parts.

Fundamentals

The formula for the sound intensity level $\beta$ in decibels is,

$\beta = 10\log \left( {\frac{I}{{{I_0}}}} \right){\rm{ dB}}$

Here, $I$ is the intensity for which decibels is to be calculated, and ${I_0}$ is the reference intensity with value ${10^{ - 12}}{\rm{ W/}}{{\rm{m}}^2}.$

(A)

Use the formula of sound intensity level in decibels.

Substitute $10{I_0}$ for $I$ in the equation $\beta = 10\log \left( {\frac{I}{{{I_0}}}} \right){\rm{ dB}}$ .

$\begin{array}{c}\\\beta = 10\log \left( {\frac{{10{I_0}}}{{{I_0}}}} \right){\rm{ dB}}\\\\ = 10\log \left( {10} \right){\rm{ dB}}\\\\ = 10{\rm{ dB}}\\\end{array}$

(B)

Use the formula of sound intensity level in decibels.

Substitute $100{I_0}$ for $I$ in the equation $\beta = 10\log \left( {\frac{I}{{{I_0}}}} \right){\rm{ dB}}$ .

$\begin{array}{c}\\\beta = 10\log \left( {\frac{{100{I_0}}}{{{I_0}}}} \right){\rm{ dB}}\\\\ = 10\log \left( {100} \right){\rm{ dB}}\\\\ = 20{\rm{ dB}}\\\end{array}$

(C)

Use the formula of sound intensity level in decibels.

Substitute $2{I_0}$ for $I$ , and $\Delta {\beta _2}$ for $\beta$ in the equation $\beta = 10\log \left( {\frac{I}{{{I_0}}}} \right){\rm{ dB}}$ to calculate $\Delta {\beta _2}$ that is change in intensity for m = 2.

$\begin{array}{c}\\\Delta {\beta _2} = 10\log \left( {\frac{{2{I_0}}}{{{I_0}}}} \right){\rm{ dB}}\\\\ = 10\log \left( 2 \right){\rm{ dB}}\\\\ = 3{\rm{ dB}}\\\end{array}$

Substitute $4{I_0}$ for $I$ , and $\Delta {\beta _4}$ for $\beta$ in the equation $\beta = 10\log \left( {\frac{I}{{{I_0}}}} \right){\rm{ dB}}$ to calculate $\Delta {\beta _4}$ that is change in intensity for m = 4.

$\begin{array}{c}\\\Delta {\beta _4} = 10\log \left( {\frac{{4{I_0}}}{{{I_0}}}} \right){\rm{ dB}}\\\\ = 10\log \left( 4 \right){\rm{ dB}}\\\\ = 6{\rm{ dB}}\\\end{array}$

Substitute $8{I_0}$ for $I$ , and $\Delta {\beta _8}$ for $\beta$ in the equation $\beta = 10\log \left( {\frac{I}{{{I_0}}}} \right){\rm{ dB}}$ to calculate $\Delta {\beta _8}$ that is change in intensity for m = 8.

$\begin{array}{c}\\\Delta {\beta _8} = 10\log \left( {\frac{{8{I_0}}}{{{I_0}}}} \right){\rm{ dB}}\\\\ = 10\log \left( 8 \right){\rm{ dB}}\\\\ = 9{\rm{ dB}}\\\end{array}$

Ans: Part A

The sound intensity level β, in decibels, of a sound wave whose intensity is 10 times the reference intensity is $10\,{\rm{dB}}{\rm{.}}$

Part B

The sound intensity level β, in decibels, of a sound wave whose intensity is 100 times the reference intensity is $20\,{\rm{dB}}{\rm{.}}$

Part C

The change in decibels $\Delta {\beta _2},\Delta {\beta _4},{\rm{ and }}\Delta {\beta _8}$ corresponding to m = 2, m = 4, and m = 8 are $3\,{\rm{dB, 6}}\,{\rm{dB, and 9 dB respectively}}{\rm{.}}$