# What is the physical origin of acoustic modes in a duct?

Lets deal with wave propagation in a cylindrical duct.

We ask the question: "what is the general form of a pressure wave which can propagate through the duct?" In answering this, we assume that the pressure wave in question has harmonic time dependence, i.e. $\exp(i\omega t)$ dependence.

The critical thing about solutions of the wave equation is that they must obey the boundary conditions of the duct. Without going into detail, for an infinite length duct of radius $a$ the solution is:

$p(r,\theta,x,t) = \exp(i(\omega t-n\theta) J_n(z_{mn}r/a) \left( A_{mn}\exp(-ik_{mn}x) + B_{mn}\exp(ik_{mn}x) \right)$

Where $J$ is the bessel function with corresponding zeros $z_{mn}$. Mathematically the significance of $m,n$ is simply that we can assign any positive integer value to these as a feasible solution. This determines the mode mathematically. As an aside it should be pointed out that for a particular value of $\omega = \omega_c$ only certain modes with low enough $\max(m.n)$ can actually propagate- any "higher order" modes will decay rapidly along $x$. $\omega_c$ is therefore known as the cut-off frequency.

My question now is how are modes created in the first place by an acoustic source? Does the source select a particular mode or are many present at the same time? Please explain.

To start with an answer I thought of a vibrating disc or membrane in the duct which vibrates with velocity $v = \hat{V}\exp(i\omega t)$ in the $x$ (axial) direction. Assuming linearity, the pressure wave form should have same harmonic time dependence. Now it is important for the sake of this argument that $v$ is exactly sinusoidal, i.e. I am not assuming there would be any higher order coefficients in the Fourier decomposition of $v$. So what mode or modes would initially (even if they are cut-off) be set up by this scenario?