Insertion loss: radiation into the ambient air

How the insertion loss treats the air the outlets radiate into: the ambient air of the Material step (Ambient air temperature); there is no separate setting. A hot outlet gas radiating into cold air has a higher radiation resistance than the same pipe radiating into its own gas, so the outlet reflects less and the insertion loss changes.

Last updated 2026-11-11

Theory

The termination of an outlet (unflanged, flanged, or the baffled piston below the radiation domain's switch frequency) is a normalised mouth impedance $z = Z/(\rho c)$ seen by the plane wave in the outlet pipe. For a duct exhausting a hot gas, (Hirschberg and Hoeijmakers, 2014) show that at low frequency the transition from the hot jet to the cold surroundings is compact: the acoustic volume flow leaving the pipe radiates as a monopole into the OUTSIDE air.

The radiation resistance is therefore that of the outside air, equation (1).
Expressed in the wavenumber of the pipe gas, this becomes equation (2): $(T_p/T_o)^{3/2}$ times the same-gas value for one ideal gas (3.87 for 450 °C into 20 °C).
According to the same authors, citing earlier measurements, temperature has little effect on the end correction (the mass of the near field, which lies in the hot jet).

The program applies equation (3) at every frequency: the resistance of the same-gas termination $z_1$ evaluated at the outside wavenumber $k_o a$ and scaled by $\rho_o c_o / (\rho c)$ , and the reactance of $z_1$ at the pipe's own $ka$ . At low $ka$ this reduces to equation (1); at high $ka$ the resistance tends to $\rho_o c_o/(\rho c)$ , the value of a plane interface. The reflection coefficient is $R = (z-1)/(z+1)$ . The reference pipe's outlet radiates into the same air, so both powers of the insertion loss use it. With a radiation-domain outlet, the matched layer of the half-space is filled with the outside air, while the stub and the hemisphere in front of the mouth keep the outlet gas.

The outside air is dry air (the case's gas model with the fluid set to air) at this temperature and the ambient pressure of the Material step. Its density and sound speed follow the ideal gas law, $\rho_o = p M/(R T_o)$ and $c_o = \sqrt{\gamma R_s T_o}$ . When the outlet gas is air at the outside temperature, the termination is exactly the same-gas one.

Equations

\mathrm{Re}\,z = \frac{\rho_o c_o}{\rho_p c_p}\,\frac{k_o^2 S}{4\pi}

(1)

\mathrm{Re}\,z = \frac{\rho_o c_p}{\rho_p c_o}\,\frac{k_p^2 S}{4\pi}

(2)

where	meaning	unit
$\rho_p, c_p$	Density and sound speed of the outlet gas	kg/m³, m/s
$\rho_o, c_o$	Density and sound speed of the outside air	kg/m³, m/s
$k_p, k_o$	Wavenumbers $\omega/c_p$ , $\omega/c_o$	1/m
$S$	Outlet area ( $\pi a^2$ )	m²

z = \frac{\rho_o c_o}{\rho_p c_p}\,\mathrm{Re}\,z_1(k_o a) + i\,\mathrm{Im}\,z_1(k_p a)

(3)

where	meaning	unit
$z_1$	Same-gas mouth impedance: Levine & Schwinger (unflanged) or baffled piston (flanged)	-
$a$	Radius of the circle of the outlet's area	m

Source

Equations (1) and (2): (Hirschberg and Hoeijmakers, 2014), eqs. (5) and (10). The paper compares equation (2) with four measured data sets for $k_p a \le 0.8$ and $T_p/T_o$ = 1 to 3.5 and finds agreement within the experimental scatter. Above $k_p a \approx 0.8$ , equation (3) is an extension made in this program and has not been compared with measurements. Same-gas terminations: (Levine and Schwinger, 1948) (unflanged).

Valid range

Cold outlet

When the outlet gas is air at the outside temperature, nothing changes.

Above k a ≈ 0.8

The temperature correction is validated against measurements only up to $k_p a$ = 0.8; above that, the form of equation (3) is an extension.

No mean flow

The model has no mean flow at the mouth (vortex shedding at the edge). A hot exhaust always flows; at low Mach numbers this causes a further loss that the model does not include.

Theory

Equations

Source

Valid range

Related 4

References 2