Evanescent modes per port

The number of transverse modes above the cut-off kept at each inlet and outlet, in addition to all modes that propagate up to $f_{max}$ . They allow the port to absorb the decaying near field of an area change close to the port.

Last updated 2026-11-03

Theory

Each port face carries its own transverse modes $\psi_m$ , obtained from a two-dimensional eigenproblem on the face mesh, equation (1); the plane wave is $k_t = 0$ . A mode propagates above its cut-off frequency, equation (2), and decays exponentially along the duct below it (evanescent), equation (3).

The port boundary condition is non-reflecting for every mode it includes: each included mode leaves the model without reflection.
A mode that is NOT included but is excited near the port (by an area step or a tube end close to it) is reflected by the port as if the port were rigid for that mode.
Keeping the first few evanescent modes removes most of this spurious reflection.

The program includes every mode with $f_c \le f_{max}$ plus this number of further modes, in order of increasing $k_t$ . The evanescent wavenumber is taken on the decaying branch, so the additional modes are stable.

Equations

-\nabla_s^2\psi_m = k_{t,m}^2\,\psi_m \ \text{on the port face}, \qquad \frac{\partial\psi_m}{\partial n} = 0 \ \text{on its rim}

(1)

f_{c,m} = \frac{c\,k_{t,m}}{2\pi}

(2)

k_{z,m} = \sqrt{k^2 - k_{t,m}^2}\ \ (k \ge k_{t,m}), \qquad k_{z,m} = -\mathrm{i}\sqrt{k_{t,m}^2 - k^2}\ \ (k < k_{t,m}), \qquad k = \frac{\omega}{c}

(3)

where	meaning	unit
$\psi_m$	Transverse shape of port mode $m$ (unit mean square over the face)	-
$k_{t,m}$	Transverse wavenumber of mode $m$	1/m
$f_{c,m}$	Cut-off frequency of mode $m$	Hz
$k_{z,m}$	Axial wavenumber (time dependence $e^{\mathrm{i}\omega t}$ )	1/m
$c$	Speed of sound at the port	m/s

Default and rule

Default

Rule

1 further mode per port.

Source

Validation of the modal port:

The modes of circular ports are checked against the zeros of $J_m'$ (tolerance 0.5 %; measured 0.017 % on 30 mm pipes and 0.01-0.17 % on the ports of case V2, a two-inlet single-chamber muffler with internal inlet and outlet tubes).
A straight duct transmits with $\lvert T\rvert = 1.0000$ and no mode coupling.
The power-normalised scattering matrix of six propagating modes at 5 kHz is unitary and reciprocal to $10^{-9}$ .

Valid range 0 to 3

Values

0 to 3. The mode table on this step lists the modes each port includes.

Theory

Equations

Default and rule

Valid range 0 to 3

Related 3