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?


You need to consider an impedance discontinuities at the ends of the tube. In simplified model the duct has non-zero impedance, ending of the stopped pipe has infinite impedance and open end zero impedance. Therefore the reflection occurs and a standing wave can be created.

Number, frequencies and amplitudes of the modes depend on the parameters of the source. Mathematically, you need to solve the inhomogenous wave equation with the source-term you have described. I would suggest you to study the 1D case of the string (e.g. here) and then:

  • generalise it for your 3D case or (and maybe better)
  • use the planar wave approximation and therefore the duct will be the 1D case as well.

Generally, modes are feature of the duct and you can think of them as masses on the springs to be brought in oscillation (effect of the source).


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