1. Mode count, modal density and group velocity 2. Damping loss factor
3. Coupling loss factors to connected subsystems 4. Input power from the external sources of excitation
where the defi nition and evaluation method of the damping loss factor have been given in Chapter 3, and are not duplicated in this chapter.
7.6.3 Evaluating the response variables
After defi ning the system model and evaluating the subsystem parameters, the response variables can be evaluated by
[D]{ε} = {〈Pin〉} (7.85)
where [D] is a symmetric matrix of damping and coupling loss factors. The modal energy vector {ε} can be found from
{ε} = [D]−1{〈Pin〉} (7.86)
7.7 Evaluation of the statistical energy analysis subsystem parameters
There are three main SEA parameters for the SEA wave approach:
1. Input power from external excitation – input power as shown in Fig. 7.12(a)
2. Energy storage capacity of the reverberant fi eld – modal density/group velocity as shown in Fig. 7.12(b)
3. Energy transmission from the reverberant fi eld to direct fi elds of adjacent subsystems – coupling loss factor as shown in Fig. 7.12(c).
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In order to calculate the SEA subsystem parameters using a wave approach, we need to know the following:
• Types of waves that can propagate in a given subsystem, which depends on subsystem dimension, cross-section, etc.
• The dispersion properties of each wave type.
7.7.1 Modal density and group velocity for one-dimensional wave propagation
The exponential displacement fi eld along the one-dimensional waveguide is assumed to be
u = U(x, y)eikzeiωt (7.87)
From the equation of motion for the assumed displacement fi eld, the roots of the homogeneous equation can be solved for values of U and k that satisfy the homogeneous equation at a given frequency ω. Each root rep- resents a wave type of the waveguide; roots with real k represent propagat- ing waves, while roots with imaginary k represent evanescent waves.
U refl ects wave type and k represents wave number in rad/m, which is given by
k f
c c
= 2π =2π = λ
ω
p p
(7.88) where λ is wavelength in m, cp is phase velocity in m/s, and the group veloc- ity cg (m/s) is defi ned as
cg= ∂k
∂
ω (7.89)
which was mentioned in Equation 7.13. The group velocity is the speed at which energy propagates by a wave. The dispersive relation is given by
External excitation
(a) (b) (c)
7.12 Three main SEA parameters.
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curves or plots which show how the wave number varies with frequency for each wave type where the group and phase velocity are related to the local and global slopes of the dispersion curve at a given frequency. If group velocity varies with frequency then the wave is said to be dispersive as shown in Fig. 7.13. For a one-dimensional waveguide of length L, one mode is expected on average for every increment in wave number of π/L radians, which is shown in Fig. 7.14 which refl ects wave mode duality.
Modal density is defi ned as the expected number of modes per unit fre- quency, which is given by:
n N
kL L
Z Z c
Z Z
( ) = ∂ ( )
∂ = ∂
∂ ( π) = π g
(7.90) where the mode count N(ω) = kL/π is defi ned as the expected number of modes below frequency ω. Equation 7.90 gives
E n
E L c
=π
( )
g (7.91)This means that the modal energy E/n is equal to the energy density π(E/L) multiplied by the speed cg at which energy fl ows, which is the incident
w
∂w
∂k k
cg =
w cp = k
7.13 Dispersion curves.
w k
4p/L
L 0
3p/L 2p/L p/L
7.14 Wave mode duality.
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power. From a wave viewpoint, the modal density relates the incident power in a reverberant fi eld to the total energy contained within the rever- berant fi eld.
7.7.2 Modal density and group velocity for two-dimensional wave propagation
The exponential displacement fi eld for plates/shells which support wave propagation in two dimensions is assumed to be
u U z= ( )eik xx eik yy eiZt (7.92) Wave types are calculated in the same way as for a one-dimensional wave- guide, but now U(z) describes the displacement fi eld through the thickness;
the wave number and wave velocities become vector quantities.
It is helpful to view the response of a panel in wave number space where the wave number space description is obtained by taking the two- dimensional Fourier transform of the physical displacement fi eld. The wave number indicates the number of ‘wiggles’ per unit distance in a given direc- tion, which is extremely useful for understanding wave propagation and acoustic radiation. For example, infi nite isotropic panel resonant wave numbers lie on a circle in wave number space as shown in Fig. 7.3, where the number of modes in a given band is proportional to the area contained between two dispersion curves in the wave number space.
When boundary impedance is random, averaged modal densities for two- and three-dimensional wave propagation are given by [9]
n kA
ω c ( ) =
2π g
(7.93)
n kV
ω c ( ) =
2π2 g
(7.94) respectively, where A is the surface area of the two-dimensional structure and V is the volume of the three-dimensional acoustic cavity where bound- ary corrections are sometimes applicable.
For a three-dimensional room with rigid walls, the averaged modal density is given by [9]:
n n f L L H
c L L L H L H
c
L L
x y
x y x y x y
ω ω
ω
( ) = ( ) =
+ ( 2 + 2 + )+ +
4
1 2 2
2 3
1 2 1 2
2
1 2
π π
π
++H c
4π (7.95)
where L1x and L2y are the length and width of the room, H is the depth of the room, c is the speed of sound in air, f is the frequency in Hz and ω is the radial frequency in rad/s.
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In summary, energy storage in the SEA wave approach is represented by modal density, which is related to group velocity and dimensions of a subsystem/wave fi eld. The SEA parameters for the wave fi elds can be found from dispersion curves.
7.7.3 Coupling loss factor
The coupling loss factor describes the rate at which energy fl ows from the reverberant fi eld of the ith subsystem to the direct fi eld of the jth subsystem per unit energy in the reverberant fi eld of the ith subsystem, which is expressed as
ηij ω
ij i
P
= coup,E (7.96)
According to the SEA wave approach assumption shown in Fig. 7.11(a), that is neglecting direct fi eld transmission between two junctions connected to the same subsystem, or assuming that transmission is incoherent via the reverberant fi eld, the coupling loss factor for each junction can be calcu- lated in isolation. This implies that the SEA ensemble average response does not capture transmission in which coherent phase information is propagated across several subsystems; localization, wave number fi ltering effects and global modes are not revealed in the SEA ensemble average predictions.
In order to calculate the coupling loss factor, the semi-infi nite impedance to a junction has to be added to the direct fi elds of each subsystem, and the junction has to be loaded by diffuse reverberant loading with a particular type of forcing. The diffuse fi eld wave transmission coeffi cient from subsys- tem i to subsystem j is calculated when a diffuse reverberant fi eld is incident upon the connection. For line and area connections, wave transmission calculation is often simplifi ed by assuming that junctions are planar and/or large compared with a wavelength.
For pointed connected subsystems, direct fi eld impedances of all subsys- tems connected to a junction are assembled fi rst, that is the 6 × 6 dynamic stiffness matrix for connection degrees of freedom, and this process is repeated and looped over the excited wave fi eld/subsystem. The force on the junction due to the diffuse incident wave fi eld (with unit energy density) in the excited subsystem is then found. This force is then applied to the junction and the input power to the direct fi elds of each of the receiving subsystems is calculated. The columns of the coupling loss factor matrix are calculated and the process is repeated.
For line-connected subsystems, the maximum and minimum trace-wave numbers in the excited subsystem are fi rst sought, then a set of incident trace-wave numbers are chosen to describe the diffuse fi eld, and the process loops over all trace-wave numbers. For each trace-number, direct fi eld
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impedances of all subsystems connected to the junction are assembled into a 6 × 6 matrix for connection degrees of freedom. The force on the junction due to an incident trace-wave number (with unit energy density) is then sought in the excited subsystem. This force to the junction is then applied and the input power to the direct fi elds of each of the receiving subsystems is calculated, the results being averaged over all incident trace-wave numbers.
A formulation is available for predicting diffuse fi eld transmission through pointed and line-connected subsystems such as plates/shells, etc., which can include lumped masses/isolators, in-line beams, etc.; subsystem impedances can be computed from local wave impedances (for example, line wave impedance) which are related to dispersion properties of waves.
Coupling loss factors can be related to the matrix of subsystem wave trans- mission coeffi cients from which the coupling loss factor matrix is obtained for a given point or line junction. This matrix describes scattering/coupling/
transmission between different wave fi elds.
For area-connected subsystems, resonant energy transmission is gov- erned by the radiation effi ciency of resonant modes, while non-resonant energy transmission is governed by the mass law and can be described by additional coupling loss factors. For example, the coupling loss factor between a cavity and a plate is given by [9]
η ρ σ
12 ωρ
= 0c ht
(7.97) where ρ0 is the mass density of air, ρ is the mass density of the plate, ht is the thickness of the plate, and σ is the radiation effi ciency of the plate, which is given by [9]
σ λ= c ∗ ( − ) ( < B)
B B
2 1 2
S 10 f f
f f f .
(7.98)
σ λ= c ( B < < A)
2
S f f f, f (7.99)
σ λ= c ( − ) ( A< < c)
A A 2 0 18
10 0 25
S f f f f
f f f .
, . (7.100)
σ λ
= ⎛ λ
⎝⎜
⎞
⎠⎟ ⎛
⎝⎜ ⎞
⎠⎟
− ( − ) − −
c
c c
c c A
A c
c
2 4 4
3 0 18 4
3
4
10 2
S
P
f f
f f f
f f f
f
f f
. cc
c
c c
6f (0 25. f < f, f < f ) (7.101)
σ = ⎛ λ
⎝⎜ ⎞
⎠⎟− ( < < )
P f f f f
f f
2 2
1 2
c
c c
c , (7.102)
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where the critical frequency is given by [10]
f c
c c h
Lt t
= 2
1 8. (7.103)
where
c E
Lt =
(
−)
ρ 1 ν2 (7.104)
and the critical wavelength is given by λc
c
= c
f (7.105)
The perimeter of the plate is given by
P = 2(L1x+ L2y) (7.106)
The radiation area is given by
S = L1xL2y (7.107)
Frequency limits for the radiation ratio are given by
f c
Sf P
B S
c
= ⎛ −
⎝⎜ ⎞
⎠⎟
2 2
2 8 1 (7.108)
f c
A P
=100
( )
λc2 (7.109)The radiation effi ciency can also be calculated from the Leppington–Heron radiation effi ciency [2].
7.7.4 Transfer matrices and insert loss of a trim lay-up
It is assumed that there are three characteristic wave types in poroelastic materials: the fl uid standing wave, the solid standing wave, and a shear wave of both solid and fl uid. Based on inertial/stiffness coupling, Biot equations provide a simple model (semi-empirical) way to describe the three wave types in terms of various measurable properties of a poroelastic material for which the response of a poroelastic material can be solved from known properties, domain and boundary conditions.
Noise control treatment is often encountered by planar lay-ups of poro- elastic material where the transfer matrix method provides an effi cient numerical method for computing wave propagation through such lay-ups.
The method assumes the trim lay-ups are infi nite in the lateral direction and homogeneous, and the response across each layer is sought analytically based on transfer properties/matrices for a given layer.
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7.7.5 Leaks
Acoustic leakage paths are signifi cant for acoustic energy transmission.
Typical leaks arise because of access holes/pass-throughs, grillages, gaps, imperfect coverage, etc. It is important to account for leakage in the SEA model, which can be represented by additional coupling loss factors at area junctions.
7.7.6 Energy inputs
There are fi ve types of energy inputs for the SEA wave approach: direct fi eld impedance and input power, distributed random loading, turbulent boundary layer loading, diffuse acoustic fi eld loading and propagating wave fi eld loading.
Input power from point force excitation applied to point/line/area-type connections can be computed from the direct fi eld impedance, which is identical to calculations performed when calculating coupling loss factors.
Input power from distributed pressure loads is calculated from the auto- spectrum and cross-spectrum. The auto-spectrum function of turbulent boundary layer loading depends on the boundary layer thickness and fl ow condition, and is typically obtained empirically from wind tunnel tests. The cross-spectrum of turbulent boundary layer loading contains information about spatial correlation structures within a fl ow in a Corcos model. For convection fl ows, different characteristic decay lengths in the fl ow and cross-fl ow directions are observed. The decay coeffi cients and convection wavelengths are important parameters.
Loading from diffuse acoustic fi eld excitation can be calculated by the use of the diffuse fi eld reciprocity principle. Spatial correlation in a diffuse fi eld is given by a sine function. Input power from diffuse acoustic fi eld excitation is proportional to radiation effi ciency, in principle.
Input power from propagating wave fi eld excitation can be calculated from modal joint acceptance or wave number space integrals in which trace velocities are derived from components of acoustic wave numbers and cor- relation decays are generally low.