Question

Suppose a spherical loudspeaker emits sound isotropically at$\mathbf{10}\mathbf{}\mathit{W}$ into a room with completely absorbent walls, floor, and ceiling (an anechoic chamber). (a) What is the intensity of the sound at distance$\mathit{d}\mathbf{=}\mathbf{}\mathbf{3}\mathbf{.0}\mathbf{}\mathit{m}$ from the center of the source? (b) What is the ratio of the wave amplitude at$\mathit{d}\mathbf{=}\mathbf{4}\mathbf{.0}\mathbf{}\mathit{m}$ to that at$\mathit{d}\mathbf{=}\mathbf{3}\mathbf{.0}\mathbf{}\mathit{m}$ ?

Short answer

1. The intensity of the sound at distance $d_2=3.0\text{m}$ from the center of the source is$0.088\frac{\text{W}}{{\text{m}}^{\text{2}}}$
2. The ratio of the wave amplitude $d_1=4.0\text{m}$ to that at $d_2=3.0\text{m}$ is$0.75$

Step-by-step solution

The given data

1. The power of the spherical loudspeaker is $P_s=10\text{\hspace{0.17em}W}$
2. The point of intensity from source of sound is$d_1=4.0\text{\hspace{0.17em}m}$
3. The point of intensity from source of sound is $d_2=3.0\text{\hspace{0.17em}m}$

Understanding the concept of the variation of intensity

Use the concept of variation of intensity with distance. The intensity of the sound from an isotropic point source decreases with the square of the distance from the source. The intensity is proportional to the square of the amplitude.

Formulae:

The intensity of the wave, $I=\frac{P_s}{4\pi r^2}$ (i)

The intensity of a wave is directly proportional to the square of the amplitude,

$I\propto A^2$ (ii)

Calculation of the intensity of the sound

Using equation (i), the intensity of the sound wave at 3.0 m is given as:

$\begin{array}{c}I=\frac{10W}{4\times 3.14\times {(3.0\text{m})}^2}\\ =0.088\frac{\text{W}}{{\text{m}}^{\text{2}}}\end{array}$

Hence, the value of the intensity of the sound is$0.088\frac{\text{W}}{{\text{m}}^{\text{2}}}$

b) Calculation of the ratio of the wave amplitudes

The intensity of sound is proportional to the square of the amplitude. The intensity of sound at distance $d_1=4.0m$ is $I_1$ and wave amplitude is $A_1$. Hence, using equation (ii), the intensity relation is given as:

$I_1\propto A_1^2$, $I_2\propto A_2^2$ ….. (1)

The intensity of sound at distance $d_2=3.0m$is $I_2$ and wave amplitude is $A_2$. Hence, using equation (ii), the intensity relation is given as

$I_2\propto A_2^2$ ….. (2)

Divide equation (1) by equation (2), we get

$\begin{array}{c}\frac{I_1}{I_2}=\frac{A_1^2}{A_2^2}\\ \frac{A_1}{A_2}=\sqrt{\frac{I_1}{I_2}}\end{array}$ ….. (3)

Using equation (i), the equation (3) becomes:

$\begin{array}{c}\frac{A_1}{A_2}=\sqrt{\frac{\left(\frac{P_s}{4\pi r_1^2}\right)}{\left(\frac{P_s}{4\pi r_2^2}\right)}}\\ =\frac{r_2}{r_1}\\ =\frac{3.0m}{4.0m}\\ =0.75\end{array}$

Hence, the value of the ratio of amplitudes is $0.75$.