Question

Show that the sound level, $\beta$, can be written in terms of the pressure amplitude, $\Delta P_M$, as
$\beta \left(dB\right)=20\log \frac{\Delta P_M}{\Delta P_{M_0}}$
where $\Delta P_{M_0}$ is the pressure amplitude at some reference level.

Answer 1

As we know,

$\beta \left(dB\right)=10log\frac{I}{I_0}$

And

$I=\frac{\Delta P_M^2}{2\rho v_w}$

ie,

$I\propto (\Delta P{)}^2$

so,

$\beta \left(dB\right)=10log\frac{\Delta P_M^2}{\Delta P_{M_0}^2}$

hence taking 2 from the power we get,

$\beta \left(dB\right)=20log\frac{\Delta P_M}{\Delta P_{M_0}}$