Equivalent source surface
2.10 Other Measures of Acoustic Energy
tative of a point sound source. For 3-D analyses the sound source is spherical and for 2-D analyses the source is cylindrical. The effective surface area associated with the sound source is dependent on the local fluid element size.
TheFLOW load can be written mathematically for a harmonic source as
FLOW =
where ω is the circular frequency in radians / s, x is the particle displacement, A is the effective surface area associated with the node, and ρ0 is the density of the acoustic fluid. The volume velocity of a source is equal to the particle velocity times the effective nodal area Q = (jωx)A. (Note that the ANSYS theory manual [20, Eq. (8.1)] unfortunately uses Q as a mass source, whereas many acoustic textbooks define Q as a volume velocity source.) Hence the ANSYS FLOW load can be written in terms of an equivalent acoustic volume velocity as
FLOW = jωρ0Q . (2.36)
A mass source excitation applied to a vertex has units of kg/s (see Ta-ble 2.18) and is defined as
Mass Source = ρ0Av = ρ0Q , (2.37) where v is the particle velocity of the node. Hence the relation between a flow load and a mass source is
FLOW = jω × [ Mass Source ] . (2.38)
2.10 Other Measures of Acoustic Energy
The main acoustic results from an ANSYS analysis is acoustic pressure, acous-tic pressure gradient, or acousacous-tic paracous-ticle velocity. By further post-processing, these results can be transformed into other measures of acoustic energy such as
• sound intensity
• sound power
• acoustic potential energy
• acoustic energy density
and these are further discussed in the following sub-sections.
80 2. Background 2.10.1 Sound Intensity
The sound intensity of a wave is the average rate of flow of energy per unit area that is perpendicular to the direction of the propagation of the wave, as shown in Figure 2.20.
FIGURE 2.20
Sketch showing the area that is normal to the direction of wave propagation.
The instantaneous sound intensity Ii at time t describes the sound power per unit area at a given location and is calculated as the product of the pressure p(r, t) and acoustic particle velocity vector ~v(r, t) at a point r as [47, Eq. (1.64)]
Ii(r, t) = p(r, t)~v(r, t) . (2.39) As sound pressure and velocity vary with time and location, it is more useful to describe the sound energy at a point by the time-averaged sound intensity as [47, Eq. (1.65)] [102, p. 125, Eq. (5.9.1)] For a monofrequency wave, T is the period. It can be shown [64, p. 48] that the time-averaged active sound intensity is given by [91, p. 53, Eq. (6.13)]
I = 1
2Re{pv∗} , (2.41)
where the superscript ∗ indicates the complex conjugate. The active intensity corresponds with the local net transport of sound energy. The time-averaged reactive intensity is calculated as
Ireactive= 1
2Im{pv∗} , (2.42)
The reactive intensity is a measure of the energy stored in the sound field during each cycle but is not transmitted.
A harmonic sound wave with pressure p and acoustic particle velocity v can be defined as
p = Pmaxcos(ωt + θp) (2.43)
v = Vmaxcos(ωt + θv) , (2.44)
2.10. Other Measures of Acoustic Energy 81 where Pmaxand Vmaxare the peak amplitude of the sound pressure and acous-tic paracous-ticle velocity, ω is the circular frequency, and θp and θv are the phase angles of the pressure and particle velocity, respectively. The corresponding active and reactive intensities are given by [47, Eqs. (1.72) and (1.73)]
I =1 These equations indicate that the difference in phase angles between the pres-sure and acoustic particle velocity (θp− θv) are crucial in determining the sound intensity. There are two cases of interest that will be discussed further:
(1) a progressive traveling wave, where it is assumed that the difference in phase angles is zero, and (2) for a standing wave configuration, where the difference in phase angles is 90◦.
For a traveling progressive harmonic sound wave, such as from a plane, cylindrical, or spherical spreading wave, the pressure is defined as
p(x, t) = Pmaxcos(k(x − ct)) , (2.47) and the corresponding particle velocity in the far field is assumed to be in phase with the pressure and defined as
v(x, t) = p(x, t)
ρ0c0 . (2.48)
The maximum intensity is given by [46, p. 33, Eq. (2.24)] [91, p. 53, Eq. (6.15)]
Imax= Pmax2 2ρ0c0
=p2RMS ρ0c0
, (2.49)
where pRMS is the square root of the mean (time) square value of p(x, t).
However for the general case, where the sound intensity is not related to only the sound pressure, both sound pressure and particle velocity must be evaluated at the same instant of time and location.
The second case of interest is a standing wave configuration, such as an un-damped duct with rigid ends, which is examined in more detail in Section 3.3.
For this case, the pressure and acoustic particle velocity are in quadrature, which means |θp − θv| = 90◦, and therefore when this is substituted into Equation (2.45), the time-averaged active sound intensity is zero [65, p. 80].
The sound intensity field is characterized by the reactive sound intensity from Equation (2.46), where sound energy oscillates locally during each cycle but is not transmitted along the duct.
When a duct has some acoustic damping installed, the difference in the phase angle between pressure and velocity is not 90◦, which results in a non-zero value of active sound intensity indicating the net transport of sound energy along the duct. This situation is further described in Section 5.5.
82 2. Background The sound intensity level is calculated as [46, Eq. (1.17)]
LI = 10 log10
where I is the sound intensity in units of W/m2, and Iref. is the reference sound intensity that has a value of 10−12 W/m2.
For further information about sound intensity, see [47, p. 33] [64] [65, p. 76]
[91, p. 51].
It has been shown that sound intensity is a function of the pressure and acoustic particle velocity, and these results are available from ANSYS simula-tions. For pressure-formulated acoustic elements (FLUID29, FLUID30, FLUID220, FLUID221) the pressure at each node is one of the degrees of freedom of the element, and this result is always available. The estimate of the acoustic par-ticle velocity can be obtained from the pressure gradient results, or if the displacement degrees of freedom of the acoustic elements are activated, by multiplying the displacement at the nodes of the acoustic elements by jω (for harmonic waves).
2.10.2 Sound Power
The sound power W radiated by a source can be evaluated by integrating the sound intensity over a surface that encloses the sound source. Figure 2.21 illustrates the concept where an oscillating piston installed in an infinite baffle generates sound that radiates outward. A hypothetical hemispherical surface is shown that encloses the sound source. The sound power can be calculated
Infinite plane baffle
Example of the hypothetical surface that encloses a sound source for evalua-tion of sound power. An oscillating piston in an infinite plane baffle radiates sound, and the sound power is evaluated by integrating the sound intensity over a hemispherical surface.