Question

Question: (II) If the amplitude of a sound wave is made 3.5 times greater, (a) by what factor will the intensity increase? (b) By how many dB will the sound level increase?

Short answer

(a) The factor of the sound intensity increases is \(12.25\). (b) The increase in sound level is \(10.88\;{\rm{dB}}\).

Step-by-step solution

Understanding the relation among intensity and amplitude of sound

The fractional relation between the intensity and threshold intensity of sound is equal to the square of the amplitude.

Given data

The change in amplitude of sound is \(a = 3.5\).

Evaluating the increase in sound intensity

The factor for the sound intensity increases is calculated below:

\(\frac{I}{{{I_0}}} = {a^2}\)

Substitute the values in the above equation.

\(\begin{array}{c}\frac{I}{{{I_0}}} = {\left( {3.5} \right)^2}\\\frac{I}{{{I_0}}} = 12.25\end{array}\)

Hence, the factor of the sound intensity increases is \(12.25\).

Evaluating the sound level by increasing amplitude

The increase in sound level is calculated below:

\(\beta = 10\log \left( {\frac{{{I_1}}}{{{I_0}}}} \right)\)

Substitute the values in the above equation.

\(\begin{array}{c}\beta = 10\log \left( {12.25} \right)\\\beta = 10.88\;{\rm{dB}}\end{array}\)

Hence, the increase in sound level is \(10.88\;{\rm{dB}}\).