Question

Find the ratios (greater to smaller) of the (a) intensities, (b) pressure amplitudes, and (c) particle displacement amplitudes for two sounds whose sound levels differ by $\mathbf{37}\mathbf{\text{\hspace{0.17em}dB}}$.

Short answer

1. Ratio of intensities is,$5000$ .
2. Ratio of pressure amplitude is, $71$.
3. Ratio of particle displacement amplitudes is, $71$.

Step-by-step solution

The given data

Sound levels differ by $\mathrm{\beta }=37\text{\hspace{0.17em}db}$.

Understanding the concept of sound intensity and pressure

We have to use the formula for sound level in terms of intensity to calculate the intensity ratio. To calculate the ratio of pressure amplitudes, we will use the formula for pressure amplitude in terms of displacement amplitude.

Formula:

The scale of sound intensity level,

$\mathbf{\Delta \beta }\mathbf{=}\mathbf{\left(}\mathbf{10}\mathbf{\text{\hspace{0.17em}dB}}\mathbf{\right)}\mathbf{log}\left(\frac{I_1}{I_2}\right)$ …(i)

The ratio of pressure amplitudes,

$\frac{{\mathbf{\Delta p}}_{\mathbf{m}\mathbf{1}}}{{\mathbf{\Delta p}}_{\mathbf{m}\mathbf{2}}}\mathbf{=}\sqrt{\frac{{\mathbf{I}}_{\mathbf{1}}}{{\mathbf{I}}_{\mathbf{2}}}}$ …(ii)

The ratio of displacement amplitude,

$\frac{{\mathbf{s}}_{\mathbf{m}\mathbf{1}}}{{\mathbf{s}}_{\mathbf{m}\mathbf{2}}}\mathbf{=}\sqrt{\frac{{\mathbf{I}}_{\mathbf{1}}}{{\mathbf{I}}_{\mathbf{2}}}}$ …(iii)

a) Calculation of ratio of intensities 

We are given the difference in sound levels. We can find intensity ratio by taking antilog on both sides of equation (i) as:

$\begin{array}{c}\frac{{\mathrm{I}}_1}{{\mathrm{I}}_2}={10}^{\mathrm{\Delta \beta }/10\text{\hspace{0.17em}dB}}\\ ={10}^{\frac{37\text{\hspace{0.17em}dB}}{10\text{\hspace{0.17em}dB}}}\\ ={10}^{3.7}\\ =5011.872\\ \frac{{\mathrm{I}}_1}{{\mathrm{I}}_2}\simeq 5012\end{array}$

Hence, the value of ratio of intensities is $5012$.

b) Calculation of ratio of pressure amplitude

Using equation (ii), and the value of intensity ratio, we get the ratio of pressure amplitude as:

$\begin{array}{c}\frac{{\mathrm{\Delta p}}_{\mathrm{m}1}}{{\mathrm{\Delta p}}_{\mathrm{m}2}}=\sqrt{\frac{{\mathrm{I}}_1}{{\mathrm{I}}_2}}\\ =\sqrt{5011.87}\\ =70.79\\ \frac{{\mathrm{\Delta p}}_{\mathrm{m}1}}{{\mathrm{\Delta p}}_{\mathrm{m}2}}=71\end{array}$

Hence, the value of ratio of pressure amplitude is$71$ .

c) Calculation of ratio of displacement amplitude

Using equation (iii) and the value of intensity ratio, we can get the displacement amplitude as:

$\begin{array}{c}\frac{{\mathrm{s}}_{\mathrm{m}1}}{{\mathrm{s}}_{\mathrm{m}2}}=\sqrt{\frac{{\mathrm{I}}_1}{{\mathrm{I}}_2}}\\ =\sqrt{5011.87}\\ =70.79\\ \frac{{\mathrm{s}}_{\mathrm{m}1}}{{\mathrm{s}}_{\mathrm{m}2}}\simeq 71\end{array}$

Hence, the value of ratio of displacement amplitude is$71$ .