Question

Find the intensity of a sound wave whose frequency is 250 Hz. The displacement amplitude of particles of the medium at this position is $1 \times {10^{ - 8}}m$. The density of the medium is $1{\text{ }}kg/{m^3}$, bulk modulus of elasticity of the medium is $400{\text{ }}N/{m^2}$.

Answer 1

Hint: Consider a sound wave whose frequency, displacement amplitude, density of medium is given. Then, we can find velocity of sound in medium by using Bulk Modulus and density of medium. Finally, we can calculate the intensity of the sound wave by using the suitable formula.

Formula used: The formula of intensity of sound wave is expressed as: $I = 2{\pi ^2}{f^2}{\delta ^2}\rho v$
Where: $I$ is intensity of sound wave, f is frequency sound, $\delta $ is amplitude of sound wave, $\rho $ is density of medium in which sound is travelling, $v$ is velocity of sound in medium.
Velocity of sound in medium ($v$) is expressed as: $v = \sqrt {\dfrac{B}{\rho }} $.
Where: B is Bulk modulus of Elasticity, $\rho $ is density of medium in which sound is travelling.

Complete step-by-step answer:
Sound intensity is the amount of energy flowing per unit time through a unit area that is perpendicular to the direction in which the sound waves are travelling. The SI unit of sound intensity is watt per square meter ($W/{m^2}$).
The intensity of sound wave is expressed as: $I = 2{\pi ^2}{f^2}{\delta ^2}\rho v$………………….. (i)
Given, f=250 Hz, $\rho $=$1{\text{ }}kg/{m^3}$,$\delta $=$1 \times {10^{ - 8}}m$
Velocity of sound in medium ($v$) is expressed as :$v = \sqrt {\dfrac{B}{\rho }} $………………………… (ii)
Given: B=$400{\text{ }}N/{m^2}$,$\rho $=$1{\text{ }}kg/{m^3}$
Putting these values in above equation (ii), we have:
$
  v = \sqrt {\dfrac{{400}}{1}} \\
   \Rightarrow v = \sqrt {400} \\
  \therefore v = 20{\text{ m/sec}} \\
 $
So, $f=250 Hz$, $\rho $=$1{\text{ }}kg/{m^3}$, $\delta $=$1 \times {10^{ - 8}}m$,$v$=20 m/sec
 Putting these above values in intensity of sound wave of equation (i), then we have:
\[
\Rightarrow I = 2{\pi ^2} \times {(250)^2} \times {({10^{ - 8}})^2} \times 1 \times 20 \\
\Rightarrow I = 2{\pi ^2} \times 62500 \times {10^{ - 16}} \times 20 \\
\Rightarrow I = 0.25{\pi ^2} \times {10^{ - 9}} \\
 \]
Now, putting the value of $\pi = 3.14$,we have:
$
\Rightarrow I = 0.25 \times {3.14^2} \times {10^{ - 9}} \\
\Rightarrow I = 2.464 \times {10^{ - 9}}{\text{ W/}}{{\text{m}}^2} \\
 $

Thus, the intensity of the sound wave is $I = 2.464 \times {10^{ - 9}}{\text{ W/}}{{\text{m}}^2}$

Note: Bulk Modulus Of Elasticity: It is given by the ratio of pressure applied to the corresponding relative decrease in the volume of the material.
Mathematically, it is represented as follows:
$B= \dfrac{{\vartriangle P}}{{\dfrac{\vartriangle V}{V}}}$
Where:
B: Bulk modulus, $\vartriangle P$: change of the pressure or force applied per unit area on the material
$\vartriangle V$: change of the volume of the material due to the compression, V: Initial volume of the material