Pressure radiation from a duct terminated in an infinite baffle

An expression has been derived for the pressure field due to a rigid piston with a velocity amplitude of $v$ on the surface of a baffle. Consider a uniform duct terminated in an infinite baffle. The plane wave mode will have a velocity source identical to that of the piston. Recalling equation (2.27), the axial velocity due to the $m$th mode is given by the modal velocity amplitude $U_m/S$ multiplied by that mode's transverse velocity profile $\psi_m(x_0,y_0)$. Replacing the piston velocity source in (3.3) with this velocity distribution gives the total pressure field assuming only the $m$th mode is present to provide a velocity source on the surface $S$.

$$p(x,y,z) = \frac{i \omega \rho}{2\pi S} \int_S dS_0 U_m \psi_m(x_0,y_0) \frac{e^{-ikh}}{h}.$$ (3.4)

In general, the total pressure field will be the sum of the contribution by all the modes:

$$p(x,y,z) = \sum_{m=0}^\infty \frac{i \omega \rho}{2\pi S} \int_S dS_0 U_m \psi_m(x_0,y_0) \frac{e^{-ikh}}{h}.$$ (3.5)