Theory for the Pipe Acoustics Boundary Conditions

The simplest boundary conditions to specify are to prescribe the pressure or the velocity at the pipe ends. These result in the Pressure condition

and the Velocity condition

and can be set independently of each other leaving the other dependent variable free. A special subclass of the velocity condition is the Closed condition where

this corresponds to the sound-hard wall condition in pressure acoustics. It is also assumed here that u0 = 0 at a closed boundary.

End Impedance Condition

At the end of pipes the relation between the pressure and the velocity can be defined in terms of an end impedance Zend. The End Impedance condition is in the Pipe Acoustics interface given by

(3-21)

where Zend = p/u (SI unit: (Pa⋅s)/m). Different models for the end impedance exist in the Pipe Acoustics interfaces. The variety depend on if the transient or the frequency domain equations are solved.

Transient End-Impedance Models

In the transient version of the physics interface the end impedance can be user-defined or set to mimic an infinite long pipe for low Mach number background flow conditions. In this case it is assumed that the pipe continues with constant cross section A and that there is no external body force F and drag τw. Because the acoustic waves are, by design, always normal to the pipe ends. In order to define the relation between the pressure and the velocity (the impedance) the dispersion relation for a plane wave needs to be determined.

In order to do so insert the assumed plane wave form

into the governing Equation 3-18 and solve for the desired relations. After some manipulation this results in

with the dispersion relation

(3-22)

This dispersion relation is nonlinear in k. In the limit where βA tends to zero and for small Mach numbers M (= u0/c) the expression is expanded to

Hence, the infinite pipe (low Mach number limit) end impedance relation reads

(3-23)

where the sign in front of c depends on the direction of propagation of the wave.

Frequency Domain End-Impedance Models

In the frequency domain many engineering relations exist for the end impedance or radiation impedance of a pipe or waveguide. Most of the relations apply only to a specific geometry or frequency range. The relations available in the Pipe Acoustics, Frequency Domain interface are:

•

Infinite pipe (low Mach number limit): This is the same relation as for the transient study and the end impedance is given by Equation 3-23. This can be thought of as the characteristic impedance of the tube.

•

Infinite pipe: This relation uses the full dispersion relation given in Equation 3-22 and yields the expression

(3-24)

where the wave number k at the right hand side is a user input. In the frequency domain a good estimate for this quantity is simply ω/c.

•

Flanged pipe, circular: In the case of a circular pipe terminated in an infinite baffle (a flanged pipe) an analytical expression exists for the radiation impedance (see Ref. 1),

(3-25)

where J1 is the Bessel function of order 1, H1 is the Struve function of order 1, a is the pipe radius, and k is the wave number. The Struve function is approximated according to Ref. 3 by

(3-26)

In the low frequency limit (small ka) Equation 3-25 reduces to the classical expression for the radiation impedance

(3-27)

•

Flanged pipe, rectangular: In the case of a pipe of rectangular cross-section (with sides wi and hi) terminated in an infinite baffle (a flanged pipe) the radiation impedance can be approximated by

(3-28)

see Ref. 4 and Ref. 5.

•

Unflanged pipe, circular (low ka limit): In the case of a circular pipe of radius a ending in free air the classical low ka limit for the radiation impedance is given by

(3-29)

see Ref. 1 and Ref. 5.

•

Unflanged pipe, circular: A solution for the unflanged pipe exists for the case when

, it is presented in Ref. 6 and is based on solving the Wiener-Hopf integral, it reads

(3-30)

where δ is an interpolation function found by numerical integration for ka = 0, δ = 0.6133.

Common for the last four radiation impedance relations is that they do only apply when there is no background flow present u0 = 0 (or at least when it is very small).