A certain sound source is increased in sound level by$\mathbf{\text{30.0dB}}$. By what multiple is (a) its intensity increased and (b) its pressure amplitude increased?

A source sound level is increased by $30dB$.

Intensity of the sound level can be characterized as the original intensity sound level and final intensity sound level. It is denoted by${\mathbf{\beta }}_1\mathbf{}$and${\mathbf{\beta }}_2$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,

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

Here, $\beta$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,

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

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

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 $1\times {10}^3$.

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

$\sqrt{1000}=31.62\approx 32$

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