Calculating Sound Intensity Levels: Sound Waves

We are given Δp, so we can calculate I using the equation $I=\frac{{\left(\Delta p\right)}^2}{2\rho v_{}}$ . Using I , we can calculate β straight from its definition in $\beta \text{ (}d\text{B)}=10{\mathrm{ log}}_{10}\left(\frac{I}{I_0}\right)$ $\beta \text{ (dB)}=10{\text{ log}}_{10}\left(\frac{I}{I_0}\right)$ .

(1) Identify knowns:

Sound travels at 331 m/s in air at 0 °C.

Air has a density of 1.29 kg/m³ at atmospheric pressure and 0ºC.

(2) Enter these values and the pressure amplitude into $I=\frac{{\left(\Delta p\right)}^2}{2\rho v_w}$ .

$$I=\frac{{\left(\Delta p\right)}^2}{2\rho v_w}=\frac{{\left(0.656\text{ Pa}\right)}^2}{2\left(1.29{\text{ kg/m}}^3\right)\left(331\text{ m/s}\right)}=5.04\times {10}^{-4}{\text{ W/m}}^2\text{.}$$

(3) Enter the value for I and the known value for I₀ into $\beta \text{ (dB)}=10{\text{ log}}_{10}\left(\frac{I}{I_0}\right)$ . Calculate to find the sound intensity level in decibels.

$$10{\text{ log}}_{10}\left(5.04\times {10}^8\right)=10\left(8.70\right)\text{ dB}=87.0\text{ dB.}$$