91Ó°ÊÓ

A source sound level is increased by $30\text{ }dB$.

Intensity of the sound level can be characterized as the original intensity sound level and final intensity sound level. It is denoted by${\mathbf{β}}_1\mathbf{}$and${\mathbf{β}}_2$respectively. Also, pressure amplitude of sound waves can be characterized by taking the square root of the total intensity.

The total intensity is given by,

Final level intensity – Original level intensity = Total Intensity

The expression for the sound level is given by,

$\mathit{β}\mathbf{=}\mathbf{(}\mathbf{10}\mathbf{}\mathbf{\text{dB}}\mathbf{)}\mathit{l}\mathit{o}\mathit{g}\mathbf{}\frac{\mathbf{I}}{{\mathbf{I}}_{\mathbf{0}}}\mathbf{}$

Here, $\mathrm{β}$is sound level and l is the final intensity and$I_0$ is the initial intensity.

To find the increased intensity, consider the original intensity$I_1$and final intensity$I_2$.

Also, the original and final sound level can be given as,

$$\begin{gathered} {\mathrm{β}}_1=10\left(\text{dB}\right)\text{log}\left(\frac{I_1}{I_o}\right) \\ {\mathrm{β}}_2=10\left(\text{dB}\right)\text{log}\left(\frac{I_2}{I_o}\right) \end{gathered}$$

${\mathrm{β}}_2={\mathrm{β}}_1+30.\text{0dB}$

Therefore, the above equation yields,

$10\left(\text{dB}\right)\log \left(\frac{I_2}{I_0}\right)=10\left(\text{dB}\right)\log \left(\frac{I_1}{I_0}\right)+30.0\text{dB}$

i.e.,$10\left(\text{dB}\right)\log \left(\frac{I_2}{I_0}\right)−10\left(\text{dB}\right)\log \left(\frac{I_1}{I_0}\right)=30.0\text{dB}$

Dividing the above equation by 10 dB and using identity,

$log\left(\frac{I_2}{I_0}\right)-log\left(\frac{I_1}{I_0}\right)=log\left(\frac{I_2}{I_1}\right)$

Therefore,

$log\left(\frac{I_2}{I_1}\right)=3\text{dB}$

Take each side as a exponent of 10.

Therefore,

$$\begin{gathered} {10}^{log\left(\frac{I_2}{I_1}\right)}=\left(\frac{I_2}{I_1}\right) \\ \left(\frac{I_2}{I_1}\right)={10}^3 \end{gathered}$$

Therefore, the intensity is increased by a multiple factor of $1×{10}^3$.

To find the pressure amplitude, it is the square root of the intensity. Therefore, it is increased by the factor is,

$\sqrt{1000}=31.62≈32$

Hence, the pressure amplitude is increased by a multiple of 32.