Pressure Amplitude: Sound wave is a longitudinal wave in a gaseous medium. This wave generally can be represented by displacement wave equation as,

\(y=\sin\left(\omega t-kx\right)\) ------- (i)

where a is the amplitude, ω is the angular frequency, t is the time k is the wavenumber. Here x and y are parallel. Sound waves can be expressed in terms of pressure variations at various points in a medium. In the air, the pressure fluctuations sinusoidally above and below the atmospheric pressure 'P' during the formation of compression and rarefaction. The human air can sense such fluctuations.

Let ΔP be the instantaneous pressure fluctuation in a sound wave at a point X at time t. Consider an imaginary air cylinder of cross-section area A and length Δx, such that the change in volume V = A.Δx when there is no wave. When the wave is produced, the size of the cylinder disturbed. Let the left cross-section is displaced by y₁ and the right cross-section is displaced by y₂. if  y₂ > y₁, volume increases pressure decreases and vice versa.

Now,

\(\Delta V=A\left(y_2-y_1\right)=A\Delta y\)

In the limit \(\Delta x\rightarrow0\),

\(\frac{dV}V=\lim_{\Delta x\rightarrow0}\frac{A\Delta y}{A\Delta x}=\frac{\operatorname dy}{\operatorname dx}\)

∴ \(P=-B\frac{\triangle V}V\) , B is the Bulk modulus of air,
Or, \(P=-B\frac{\operatorname dy}{\operatorname dx}=Bak\cos\left(\omega t-kx\right)\)
Or, \(P=P_m\cos\left(\omega t-kx\right)\) ------ (ii)

Where, \(P_m=Bak\) is the pressure amplitude

∴ \(P_m=v^2\rho ak\)        \(\left[\because v=\sqrt{\frac B\rho},\;B=v^2\rho\right]\)

The equation (ii) is pressure amplitude which is directly proportional to the displacement amplitude. From equation (i) and (ii) we can differentiate that the displacement wave is out of phase by 90° withy pressure wave. it means that, when the displacement at a point is zero, the pressure change is a maximum, and vice-versa.