Sound pressure Level

Recall that intensity is proportional to sound pressure amplitude squared:

$$I = p^2/(\rho c).$$

Though $\rho$ and $c$ are dependent on temperature, their product is often approximated by 400. When $\rho c = 400$, sound pressure level $L_p$ (SPL) is equivalent to sound intensity level, and is expressed in dB by:

$$L_p = 10\log I/I_0$$

$$= 10\log p^2/(\rho cI_0)$$

$$= 10\log p^2/(4 \times 10^{-10})$$

$$= 10\log \left(p/(2\times 10^{-5})\right)^2$$

$$= 20\log p/(2\times 10^{-5})$$

$$= 20\log p/p_0.$$

where $p_0 = 2\times 10^{-5}$ is the threshold of hearing for amplitude of pressure variations.