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Appendix C
Symbols

Table C.1 Greek Symbols I

$α_{s}$	Absorption coefficient
$β_{x}, β_{y}, β_{z}$	Positive rotation around $x$ , $y$ , and $z$ -axis
$Δ ω$	Angular frequency bandwidth
$Δ T$	Sampling interval
$α$	Decay rate
$α_{x}$	Strain damping coefficient
$delta left-parenthesis dot right-parenthesis$	Dirac delta function
$δ_{B C}$	Correction factor for certain boundary condition
$ζ$	Critical damping ratio
$η$	Damping loss factor
$η_{m}, η_{m m}$	Damping loss factor of $m$ th system
$η_{m n}$	Coupling loss factor of system $m$ to $n$
$ϑ$	Azimuthal angle
$κ = c_{v} / c_{p}$	Specific heat ratio
$λ$	Wavelength, first Lamé coefficient
$μ$	Frequency ratio, ratio factor
$ν$	Poisson ratio
$ρ$	Density
$ρ_{f g}$	Correlation between $f$ and $g$
$ϕ$	Phase angle
$φ$	Polar angle
$Φ$	Velocity potential (chapter 2)
$Φ_{n}$	Normalized mode shape
$Ψ_{n}$	Mode shape
$Ψ_{L / S}$	Amplitude of plate longitudinal and shear wave
$bold upper Psi Subscript upper B Baseline 1 slash 2$	Amplitude of evanescent and propagating plate bending waves
$Π$	Power
$Π_{i n}$	Input power
$normal upper Pi Subscript i right-arrow j$	Transmitted power from subsystem $i$ to $j$

Table C.2 Greek Symbols II

$σ$	Stress
$σ_{rad}$	Radiation efficiency
$σ^{'}$	Surface porosity
$τ$	Decay rate
$τ_{m n}$	Transmission coefficient
$ϕ$	Wave incident angle
$ζ = c_{v} / c_{v c}$	Ratio of viscous to critical viscous damping
$ω$	Angular frequency
$omega 0$	Angular resonance frequency of undamped harmonic oscillator
$ω_{d}$	Angular resonance frequency of damped harmonic oscillator
$ω_{d}$	Angular frequency of maximum amplitude of damped harmonic oscillator
$ω_{n}$	Modal angular frequency of mode $n$
$omega Subscript s Baseline equals 2 pi slash normal upper Delta t$	Angular sampling frequency

Table C.3 Capital latin Symbols I

$A$	Surface, area
$A_{c}$	Cross section of beam or tube
$bold-italic upper A$	Complex amplitude
$B$	Plate bending stiffness
$B_{y}, B_{z}$	Beam bending stiffness for $y$ and $x$ -axis
$[\begin{matrix} C \end{matrix}]$	Viscous damping matrix
$[\begin{matrix} D \end{matrix}]$	Dynamic stiffness matrix
$E$	Energy, Young’s modulus
$E_{kin}$	Kinetic energy
$E_{pot}$	Potential energy
$F$	Force
$bold-italic upper F left-parenthesis omega right-parenthesis comma bold-italic upper G left-parenthesis omega right-parenthesis$	Functions in frequency domain
$bold-italic upper F$	Complex force amplitude
$G$	Shear modulus
$bold-italic upper H left-parenthesis omega right-parenthesis$	Frequency response function
He $= k L$	Helmholtz number
$[\begin{matrix} I \end{matrix}]$	Unit matrix
$J$	Torsional moment of rigidity
$I_{i i}$	Area inertial moment
$K$	Bulk modulus
$[\begin{matrix} K \end{matrix}]$	Stiffness matrix
$L$	Length
$L_{j}$	Length of junction
$L_{p}$	Pressure level
$L_{v}$	Velocity level
$L_{j}$	Junction length
$[\begin{matrix} L \end{matrix}]$	Nonsymmetric power SEA matrix
$[\begin{matrix} L^{'} \end{matrix}]$	Symmetric SEA matrix
$M$	Mass (total)
$bold-italic upper M$	Moment vector
$M_{x}$	Moment in $x$ -direction
$[\begin{matrix} M \end{matrix}]$	Mass matrix
$N$	Mode count
$N_{x / y}^{'}$	Force per length in plates
$P$	Perimeter
$Q$	Quality factor
$bold-italic upper Q$	Volume source strength
$Q_{x / y}^{'}$	Shear stress per length in plates
$R$	Radius
$bold-italic upper R$	Reflection factor
$upper R Subscript f g Baseline left-parenthesis t 1 comma t 2 right-parenthesis$	Cross correlation function
$S_{f f}$	Power spectral density
$S$	Surface, section area
$S_{j}$	Surface of junction
$bold-italic upper S Subscript f g$	Cross spectral density
$T$	Time interval
$bold-italic upper T$	Transmission factor
$T_{mem}$	Membrane tension
$T L$	Transmission loss
$V$	Volume
$bold-italic upper W left-parenthesis omega right-parenthesis$	Spectrum of window function
$bold-italic upper Y$	Mobility
$bold-italic upper Z equals bold-italic upper F slash bold-italic upper V$	Mechanical impedance
$bold-italic upper Z Subscript a Baseline equals bold-italic p Subscript surf Baseline slash bold-italic upper Q Subscript surf$	Radiation impedance

Table C.4 Small latin Symbols I

$a_{n}, b_{n}, c_{n}$	Fourier coefficients
$c$	Wave speed
$c 0$	Wave speed in fluids
$c_{gr}$	Wave group speed
$c_{ϕ}$	Wave phase speed
$c_{v}$	Viscous damping
$c_{L}$	Longitudinal wave speed in solids
$c_{L B}$	Longitudinal wave speed in beams
$c_{L P}$	Longitudinal wave speed in plates
$c_{S}$	Shear wave speed
$c Subscript v c Baseline equals StartRoot 4 m k Subscript s Baseline EndRoot$	Critical viscous damping
$c_{v eq}$	Equivalent viscous damping
$d = {\begin{matrix} u, v, w \end{matrix}}^{T}$	Displacement vector
$e$	Energy density
$e$	Euler constant
$f$	Frequency
$f_{c}$	Critical frequency
$f_{q}$	General source term of acoustic wave equation
$f left-parenthesis t right-parenthesis comma g left-parenthesis t right-parenthesis$	Functions in time domain
$j$	Imaginary number
$h$	Thickness of a plate
$k$	Wavenumber
$bold-italic k$	Complex wavenumber
$k_{s}$	Spring stiffness
$n left-parenthesis omega right-parenthesis$	Modal density
$m$	Mass
$m^{'}$	Mass per unit length
$m double-prime$	Mass per unit area
$m_{n}$	Modal mass
$p$	Acoustic pressure
$q$	Generalised displacement
$q_{s}$	Volume source strength density
$r_{s}$	Reflection coefficient
$t$	Time
$u, v, w$	Displacement in $x$ , $y$ , and $z$ -direction
$v_{x}, v_{y}, v_{z}$	Velocity in $x$ , $y$ , and $z$ -direction
$u_{r}$	Displacement at resonance
$v Subscript x Baseline 0$	Initial velocity in $x$ -direction
$bold-italic v Subscript x$	Complex velocity amplitude in $x$ -direction
$w left-parenthesis t right-parenthesis$	Window function in signal analysis
$z 0$	Characteristic acoustic impedance
$bold-italic z equals bold-italic p slash bold-italic v$	Acoustical impedance
$bold-italic z Subscript a Baseline equals bold-italic p Subscript surf Baseline slash bold-italic v Subscript surf$	Acoustic radiation impedance

Table C.5 Other Symbols

$upper I m bold-italic upper Z$	Imaginary part of $bold-italic upper Z$
$upper R e bold-italic upper Z$	Real part of $bold-italic upper Z$
$\infty$	Infinity
$mathematical left-angle dot mathematical right-angle$	Average operator
$mathematical left-angle dot mathematical right-angle Subscript upper T$	Average over time operator
$mathematical left-angle dot mathematical right-angle Subscript upper E$	Ensemble average operator
$mathematical left-angle dot mathematical right-angle Subscript upper V$	Average over volume operator
$mathematical left-angle dot mathematical right-angle Subscript upper S$	Average over area operator
$mathematical left-angle dot mathematical right-angle Subscript upper L$	Average over length operator
$upper E left-bracket dot right-bracket$	Expected value
$script upper F left-brace dot right-brace$	Fourier transform
$script upper F Superscript negative 1 Baseline left-brace dot right-brace$	Inverse Fourier transform
$\hat{\cdot}$	Real valued amplitude
${\begin{matrix} \cdot \end{matrix}}$	Column vector (of coefficients)
$r$	Vector, geometrical
$[\begin{matrix} \cdot \end{matrix}]$	Matrix
$left-parenthesis dot right-parenthesis prime$	Denoted quantity per length
$left-parenthesis dot right-parenthesis double-prime$	Denoted quantity per area
$\dot{x}$	Time derivative
$ModifyingAbove x With two-dots$	Double time derivative

Table C.6 Indexes

i,j	Index for degrees of freedom
i	Index for inner degrees of freedom
s	Index for surface degrees of freedom
jXY	Index for junction degrees of freedom connected to system X and Y
m,n	Index for systems
(m)	Index for system when subindex used by, e.g. other degrees of freedom
−1	Inverse of a matrix
T	Transpose of a vector
H	Hermitian of a matrix or vector
−H	Hermitian of the inverse of a matrix
ff	Cross correlation of force degrees of freedom
uu	Cross correlation of general displacement degrees of freedom
in	Denoting input into a system
out	Denoting output leaving a system
ref	Denoting reference values
refl	Denoting reflection
trans	Cross correlation of force degrees of freedom
∞	Index denoting quantities of infinite or semi infinite systems

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Table of Contents for
C Symbols