2.2.1 Conservation of Mass

For simplicity we consider first the flow in the $x$-direction as in Figure 2.1. The mass flow balance contains the following quantities:

1. The elemental mass $m=\rho dV=\rho A$ with $A=dydz$.
2. Mass flow into the volume ${(\rho v_{\mathrm{x}}A)}_x$.
3. Mass flow out of the volume ${(\rho v_{\mathrm{x}}A)}_{x+dx}$.
4. Mass input from external sources $\dot{m}$.

leading to equation

  (2.1)

for mass conservation. Expanding the second term on the right hand side in a Taylor series gives

and finally

  (2.2)

This one dimensional equation of mass conservation in $x$-direction can be extended to three dimensions:

  (2.3)

The second term of (2.3) may be confusing, but it says that the change of density is not only determined by a gradient in the velocity field but also by a gradient of the density.

2.2.2 Newton’s law – Conservation of Momentum

The same procedure is applied to the momentum of the fluid. As shown in Figure 2.2 we get for flow in $x$-direction:

1. The momentum of the control volume is $\rho v_{\mathrm{x}}dV=\rho v_{\mathrm{x}}Adx$.
2. The momentum flow into the volume ${(\rho v_{\mathrm{x}}^{2}A)}_x$.
3. mass flow out of the volume ${(\rho v_{\mathrm{x}}^{2}A)}_{x+dx}$.
4. The force at position $x$ is ${(PA)}_x$.
5. The force at position $x+dx$ is ${(PA)}_{x+dx}$.
6. External volume force density $f_x$.

Thus, the conservation of momentum in $x$ reads

  (2.4)

Using Taylor expansions for ${(\rho v_{\mathrm{x}}^{2}A)}_{x+dx}$ and ${(PA)}_{x+dx}$ gives

  (2.5)

Here, $f_x=F_x/(Adx)$ is the volume force density (force per volume). Using the chain law the partial derivatives of the first and second term lead to

  (2.6)

The term in brackets is the homogeneous continuity Equation (2.1), and Equation (2.6) simplifies to

  (2.7)

As with the conservation of mass, this can be extended to three dimensions:

  (2.8)

This equation is the non-linear, inviscid momentum equation called the Euler equation.

2.2.3 Equation of State

The above equations relate pressure, velocity and density. For further reducing this set we need a third equation. The easiest way would be to introduce the . Here we start with the first law of thermodynamics in order to show the difference between isotropic (or adiabatic) equation of state and other relationships.

  (2.9)

With the following specific quantities per unit mass

  (2.10)

With the specific entropy $ds=\frac{dq+dr}{T}$ we get:

  (2.10)

The relation $dv=d(1/\rho )$ comes from the fact that $v$ is a mass specific value and therefore the reciprocal of the density $\rho =1/v$. For an ideal gas we have

  (2.11)

$c_{\mathrm{p}}$ and $c_{\mathrm{v}}$ are the specific thermal heat capacities for constant pressure and volume, respectively. That is the ratio of temperature change $\mathrm{\partial }T$ per increase of heat $\mathrm{\partial }q$. From the total differential

  (2.12)

we can derive

  (2.13)

Using all above relations the change in density $d\rho$ is:

  (2.14)

with $\kappa =c_{\mathrm{v}}/c_{\mathrm{p}}$. In most acoustic cases the process is isotropic: i.e. time scales are too short for heat exchange in a free gas; thus $ds=0$, and the change of pressure per density is

  (2.15)

In case of constant temperature (isothermal) $dT=0$ we get with (2.12) and the ideal gas law (2.11):

  (2.16)

As we will later see, $c_0$ is the . Newton calculated the wrong speed of sound based on the assumption of constant temperature that was later corrected by Laplace by the conclusion that the process is adiabatic. For fluids and liquids like water a different quantity is used because there is no such expression as the ideal gas law. The bulk modulus is defined as:

  (2.17)

Due to (2.15) and (2.16) the relationship between the bulk modulus $K$ and $c_0$ is:

  (2.18)

The bulk modulus can be defined for gases too, but we must distinguish between isothermal or adiabatic processes.

  (2.19)

2.2.4 Linearized Equations

Equations (2.3) and (2.8) can be linearized if small changes around a certain equilibrium are considered:

  (2.20)

  (2.21)

  (2.22)

Inserting (2.22) into the equation of continuity (2.3), neglecting all second order terms as far as source terms, and setting¹ ${\mathbf{v}}_0=0$ the linear equation of continuity is:

  (2.23)

Doing the same for the equation of motion (2.8) leads to:

  (2.24)

Using the curl$(\mathrm{\nabla }\times )$ of this equation it can be shown that the acoustic velocity $\mathbf{v}{}^{\prime }$ can be expressed using a so-called velocity potential which will be useful for the calculation of some wave propagation phenomena.

  (2.25)

2.2.5 Acoustic Wave Equation

From the following operation

follows

  (2.26)

With the equation of state (2.15) for the density we get the linear wave equation for the acoustic pressure $p$

  (2.27)

Inserting the velocity $\mathbf{v}{}^{\prime }=\mathrm{\nabla }\mathrm{\Phi }$ derived from the potential $\mathrm{\Phi }$ into the linear equation of motion (2.24) provides the required relation between pressure and the velocity potential

  (2.28)

Thus, the relationship between pressure $p$ and the velocity potential $\mathrm{\Phi }$ is

  (2.29)

Entering this into the wave equation (2.27) and eliminating one time derivative gives:

  (2.30)

The definition of the velocity potential (2.25) and equation (2.29) can be applied for the derivation of a relationship between acoustic velocity and pressure:

  (2.31)

2.3.1 Harmonic Waves

According to D’Alambert every function of the form $p(x,t)=Af(x-c_{0}t)+Bg(x+c_{0}t)$ is a solution of the one-dimensional wave equation. In the following we will consider harmonic motion or waves so we replace the functions $f$ and $g$ by the exponential function with

  (2.33)

and get

  (2.34)

The first term of the right hand side of this equation is travelling in positive directions, the second in negative directions². Harmonic waves are characterized by two quantities, the angular frequency $\omega$ and the wavenumber $k$. The first is the frequency (in time) as for the harmonic oscillator, and the second is a frequency in space. A similar relationship can be found between the time period $T$ and the wavelength $\lambda$. Space and time domains are coupled by the sound velocity $c_0$ as shown in Table 2.1.

The time integration in Equation (2.31) corresponds to the factor $1/(j\omega )$ and reads in the frequency domain:

  (2.35)

For one-dimensional waves in the $x$-direction this leads to:

  (2.36)

Depending on the wave orientation the ratio between pressure and velocity is given by:

  (2.37)

In accordance with the impedance concept from section 1.2.3 we define the ratio of complex pressure and velocity as specific acoustic impedance $\mathit{z}$

  (2.38)

also called acoustic impedance. For plane waves this leads to:

  (2.39)

$z_0={\rho }_0{c}_0$ is called the characteristic acoustic impedance of the fluid. The specific acoustic impedance $\mathit{z}$ is complex, because for waves that are not plane the velocity may be out of phase with the pressure. However, for plane waves the specific acoustic impedance is real and an important fluid property.

The above description of plane waves can be extended to three-dimensional space by introducing a wavenumber vector $\mathbf{k}$.

  (2.40)

2.3.2 Helmholtz equation

Entering (2.40) into the wave Equation (2.27) provides

  (2.41)

The $e^{j\omega t}$ term is often omitted and with $k=\omega /c_0$ we get the homogeneous.

  (2.42)

2.3.3 Field Quantities: Sound Intensity, Energy Density and Sound Power

A sound wave carries a certain amount of energy that is moving with the speed of sound. We start with the instantaneous acoustic power $\mathrm{\Pi }$:

  (2.43)

$\mathbf{F}$ is the force acting on a fluid particle and $\mathbf{v}$ the associated velocity. The acoustic intensity $\mathbf{I}$ is defined as the power per unit area $\mathbf{A}=A\mathbf{n}$ in the direction of the unit vector $\mathbf{n}$ and with $\mathbf{F}=pA\mathbf{n}$ we get:

  (2.44)

As in Equation (1.48) the time average is given by:

  (2.45)

Using the harmonic plane wave solutions for pressure (2.34) and velocity (2.36)

the time averaged mean intensity yields:

  (2.46)

and finally:

  (2.47)

We see that the specific impedance $z_0={\rho }_0{c}_0$ relates the intensity to the squared pressure.

The kinetic energy density $e_{\mathrm{k}\mathrm{i}\mathrm{n}}$ in a control volume $V_0$ is written as

  (2.48)

The potential energy density $e_{\mathrm{p}\mathrm{o}\mathrm{t}}$ follows from the adiabatic work integral as in equation (2.9)

  (2.49)

If we use Equation (2.15) we get the change in density as a start for the change in volume

With unit mass $M$ in the control volume $V_0$ it follows from $\rho =M/V_0$ that

Finally we get:

  (2.50)

Pressure and velocity are in phase for plane waves; the same is true for the potential and kinetic energy density, so the total energy density is given by:

  (2.51)

Using Equation (2.34) the time average over one period leads to:

  (2.52)

Finally, we can see that the speed of sound relates energy density to the sound intensity.

  (2.53)

All those above expressions are useful for the description and evaluation of sound fields. Especially in case of statistical methods that are based on the energy density of acoustic subsystems they link the wave fields to the energy in the systems and the power irradiating at the system boundaries.

If the intensity can be determined over a certain surface the source power is calculated by integrating the intensity component perpendicular to the surface

  (2.54)

2.3.4 Damping in Waves

There is no motion without damping, and a sound wave propagating over a long distance will vanish. This is considered by adding a damping component to the one-dimensional solution of the wave equation similar to the decay rate in (1.22)

  (2.55)

Here $\alpha$ is the damping constant. There are several reasons for the attenuation of acoustic waves:

• Viscous damping due to inner viscosity.
• Thermal damping due to irreversible heat flow during wave propagation.
• Molecular damping due to excitation of degrees of freedom of molecules (for additional content of the gas, e.g. humidity in air).

The damping loss $\eta$ as defined in (1.68) is based on the amount of energy dissipated during one cycle of wave motion. The harmonic pressure wave performs one cycle of oscillation in one period in time $T$ or space $\lambda$. So we get for $\eta$:

  (2.56)

For small damping the exponential function can be approximated by $e^x\approx 1+x-\dots$ providing the relationship between damping loss and fluid wave attenuation.

  (2.57)

Hence, the attenuation can be given by:

  (2.58)

An appropriate way to consider this relationship in the solution of the wave equation is to include this into a complex wavenumber $\mathit{k}$:

  (2.59)

This complex wavenumber naturally impacts the speed of sound

  (2.60)

and the acoustic impedance

  (2.61)

The shown quantities of the plane wave field can also be applied in three-dimensional space and they are summarized in Table 2.2.

2.4.1 Monopoles – Spherical Sources

The most simple geometry we might think of is a point in space. For simple derivation of the sound field of a point source the spherical coordinate system is introduced as shown in Figure 2.4 and defined by the following coordinate transformation

  (2.62a)

  (2.62b)

  (2.62c)

Using this coordinate system and neglecting the angular components the Laplace operator $\mathrm{\Delta }$ reads as

  (2.63)

The wave equation for the velocity potential (2.30) becomes

  (2.64)

The two right terms can be written in a different form using $r\mathrm{\Phi }$ as argument

  (2.65)

Equation (2.30) is the one-dimensional wave equation for the argument $r\mathrm{\Phi }$, so we can use the D’Alambert solution

  (2.66)

The first term represents an outgoing wave travelling away from the source, the second an incoming wave travelling to the source. As we are interested in sound being emitted from the source we consider the outgoing harmonic solution with complex amplitude $\mathit{A}$

  (2.67)

Consider a pulsating sphere of radius $R$ in the centre with normal surface velocity ${\mathit{v}}_R$. With the velocity potential the radial velocity can be easily derived from the solution (2.67):

  (2.68)

Substituting Equation (2.67) into (2.68) and solving for $\mathit{A}$ gives

  (2.69)

Hence,

  (2.70)

The strength $\mathit{Q}(t)$ of the source is defined by the volume flow rate. This is the surface of the sphere times normal velocity $v_R$

  (2.71)

With the harmonic source strength

  (2.72)

the spherical wave solution is

  (2.73)

Using equations (2.25) and (2.35) pressure and velocity are given by

  (2.74)

and

  (2.75)

2.4.1.1 Field Properties of Spherical Waves

The acoustic impedance $\mathit{z}$ is according to Equation (2.38)

  (2.76)

In contrast to the plane wave, the specific acoustic impedance is not real. It contains a resistive and a reactive part. When the resistive part is dominant the pressure is in phase with the velocity. When the reactive part dominates, the velocity is out of phase to the pressure. The out of phase component does not generate any power in the sound field as it was the case for moving a mass or driving a spring. The motion is partly introduced into the local kinetic energy, and this part can be recovered as it is the case for an oscillating mass. For the acoustic field of a spherical source the reactive field represents the near-field fluid volume that is carried by the sphere motion but not emitting a wave.

There are two limit cases in Equation (2.76):

1. $kr\ll 1$; the wave length $\lambda$ is much larger than distance $r$.
2. $kr\gg 1$; the wave length $\lambda$ is much smaller than distance $r$.

Introducing the above approximations into Equation (2.76) gives a fully reactive impedance for (i)

  (2.77)

and a resistive part equal to plane waves for (ii)

  (2.78)

2.4.1.2 Field Intensity, Power and Source Strength

The time averaged radiated intensity is

  (2.79)

The total radiated power can now be evaluated from Equation (2.54) and the integration surface $4\pi r^2$

  (2.80)

The mean square pressure can be derived from (2.74) and expressed by the intensity using (2.45)

  (2.81)

Replacing the intensity in (2.81) gives the rms pressure in the spherical sound field due to power

  (2.82)

2.4.1.3 Power and Radiation Impedance at the Surface Sphere

The characteristic impedance of the sphere exactly at the surface at radius $R$ can be translated into the radiation impedance of the sphere as a volume source. The radiation impedance is defined as the ratio of pressure to source strength at the vibrating surface

  (2.83)

If we assume a constant harmonic surface velocity ${\mathit{v}}_R$ we get for the radiation impedance of the breathing sphere and according to the acoustic impedance (2.76)

  (2.84)

The acoustic radiation impedance is the ratio pressure and normal velocity at the sphere’s surface

  (2.85)

We can now use this impedance to eliminate either $\mathit{p}$ or ${\mathit{v}}_r$. The power transmitted by a vibrating sphere using Equation (2.54) over the surface of the sphere

  (2.86)

  (2.87)

or for constant source pressure

  (2.88)

It is instructive to see the mechanical properties considering the limit cases from above and extract the mass that is moved by the surface. From Newtons’s law a force given by $F=4\pi R^{2}p$ leads to an in-phase acceleration of $j\omega v_r$ of a mass $m$

hence

  (2.89)

For $kR\ll 1$ we get: $m=4\pi R^3{\rho }_0=3V_{\text{sph}}{\rho }_0$. Thus, at low frequencies the source surface motion carries three times the fluid volume of the sphere. This motion near the source is called an evanescent wave, because it is oscillatory motion of fluid that does not radiate.

2.4.1.4 Point Sources

A point source is a spherical source with an infinitely small radius. Performing the limit $kR\to 0$ for Equation (2.73) leads to the velocity potential for point sources of strength $\mathit{Q}$

  (2.90)

The pressure and velocity field of such a source is given by

  (2.91)

and

  (2.92)

All other relations regarding power and intensity expressions remain. We see that the limit is expressed for $kR$ and not for the wavelength. The reason is that it is the ratio of a characteristic length (in this case the sphere radius) to the wavelength that determines if the geometrical details must be considered or not. In other words, a wave of a certain wavelength doesn’t care about details that are much smaller.

With the D’Alambert solution for spherical waves (2.66) we can also derive a point source in time domain

  (2.93)

The point source is of great importance for the solution of the inhomogeneous wave equation in combination with complex boundary conditions. Any source can be reconstructed by a superposition of point sources as shown in Section 2.7.

Performing the limit process with $kR\to 0$ and taking the power from equation 2.86 we get the intensity of the point source:

  (2.94)

and the total radiated power

  (2.95)

with radiation impedance following from this

  (2.96)

2.7.1 Acoustic Green’s Functions

This section presents the concept of the Green’s function that uses a formalism to calculate the sound field for arbitrary source and boundary configurations as shown for example by Morse and Ingard (1968).

The Green’s function is defined as the solution of the following inhomogeneous wave equation.

  (2.124)

The inhomogeneous part is the delta function which allows for this elegant derivation of the Kirchhoff integral. The delta function is introduced in the appendix A.1.3 in the time domain. However, it can also be applied in space. The multidimensional delta function is simply the product of three Dirac delta functions in space

  (2.125)

The sifting properties and the value of the integration is defined by volume integral

  (2.126)

The solution of Equation (2.124) is the point source (2.91)³.

  (2.127)

In order to achieve a common formulation we add an arbitrary solution $\chi$ of the homogeneous wave equation

  (2.128)

to the Green’s function to get the generalized Green’s function

  (2.129)

The purpose of the additional homogeneous solution is to create freedom to fulfill boundary conditions that do not occur in the free sound field. The task is to find the solution for the inhomogeneous wave equation

  (2.130)

The generalized Green’s function must be a solution of the following equation for the special case with $\mathbf{r},{\mathbf{r}}_0\in V$ and the boundary $\mathrm{\partial }V$ as shown in Figure 2.9.

  (2.131)

In order to receive a global solution we perform the operation

  (2.132)

This leads to

  (2.133)

Exchanging $\mathbf{r}$ and ${\mathbf{r}}_0$ and integrating ${\mathbf{r}}_0$ over the volume $V$ gives

  (2.134)

The last term on the RHS follows from the sifting property of the delta function

  (2.135)

With Green’s law of vector analysis

  (2.136)

some volume integrals can be transferred into surface integrals and we get finally

  (2.137)

The first term on the right-hand side is the volume integral over all sources ${\mathit{f}}_q(\mathbf{r})$ in the volume. So given a known source distribution we can calculate the according sound field. The two terms in the surface integral take care of the boundary condition. The pressure gradient in the first can be converted into the normal velocity using (2.35). The second surface integral allows establishing the correct surface impedance. Equation (2.137) is called the constant frequency version of the .

2.7.2 Rayleigh integral

The Rayleigh integral is a special solution of the Kirchhoff-Helmholtz integral applied to flat and infinite surfaces. We assume a configuration as shown in Figure 2.10. The integration volume is the right half space for $z>0$ closed by a half sphere of infinite radius. The Green’s function of any source at ${\mathbf{r}}_0=(x_0,y_0,z_0)$ with $z_0>0$ is as defined in equation (2.127). The rigid surface acts as a reflector. Thus, there is a mirror source located at ${\mathbf{r}}_0^{\prime }=(x_0,y_0,-z_0)$. This source is not in volume $V$, and the added wave field is therefore considered as a homogeneous solution in the volume. Hence, we get for the generalized Green’s function

  (2.138)

We enter this version of the Green’s function in Equation (2.137) and we get

  (2.139)

We assume a source-free half space so $f_q(\mathbf{r})=0$, and due to the mirror source symmetry $\frac{\mathrm{\partial }G({\mathbf{r}}_0,\mathbf{r})}{\mathrm{\partial }z}=0$ is also true. By clever selection of the Green’s function we fulfilled the boundary condition automatically. For the surface integral the contributions from the half sphere with infinite radius are supposed to be zero. From Equation (2.35) the first expression can be converted into an expression for the surface velocity ${\mathit{v}}_z$. Performing the limit process $z_0\to 0$ we get

  (2.140)

and with this Green’s function we can derive the Rayleigh integral that allows the calculation of infinite half space sound fields excited by a rigid vibrating plane with arbitrary velocity distribution ${\mathit{v}}_z(x_0,y_0)$.

  (2.141)

2.7.3 Piston in a Wall

A cylindrical loudspeaker in a wall can be modelled by a piston of radius $R$ vibrating with velocity ${\mathit{v}}_z$ located in a rigid wall. For convenience the surface integral will be expressed in cylindrical coordinates $r_0$ and ${\phi }_0$. The receiver coordinates are given as spherical coordinates $r$ and $\vartheta$(Figure 2.11). Without loss of generality the azimuthal angle $\phi$ is set to zero.

  (2.142)

In the far field approximation we assume $l\approx r$ and get

  (2.143)

So, the approximate result is

  (2.144)

The integral is the Bessel function of first order

  (2.145)

The results for some values of $kR$ are shown in Figure 2.12 over the angular range of $\pm \pi /2$. For a piston size small compared to the wavelength ($kR\le 1$) the radiation pattern is similar to a point source. The smaller the wavelength gets in relation to the piston radius $R$ the more a specific radiation pattern develops.

2.7.3.1 Impedance Concept

The radiation impedance of the piston is calculated from the pressure averaged over the surface related to the piston velocity ${\mathit{v}}_z$. As shown by Lerch and Landes (2012) the mechanical impedance of the piston due to radiation is given by

  (2.146)

According to equation (2.141) assuming a constant velocity ${\mathit{v}}_z$ over the surface $A$ the pressure is

  (2.147)

Thus, we get the pressure at $r$ from integrating the contribution from the rest of the piston in circles of radius $s$. The angle integration over ${\phi }_0$ runs from $0$ to $2\pi$. From every angle ${\phi }_0$ follows the integration limits $s_{max}$ of the second integral.

  (2.148)

Using those limits gives

  (2.149)

Inserting equation (2.149) into (2.146) leads to the expression

  (2.150)

Running through quite a lot of algebraic modifications we get the expression for the impedance of a piston

  (2.151)

or

  (2.152)

$H_1(z)$ is the Hankel function of first order. In Figure 2.14 the real and imaginary parts of the acoustic radiation impedance are compared to those of the pulsating sphere. Both sources have a similar shape except some waviness for the piston resulting from interference effects from the integration over the piston surface. For large $kR$ the impedance is real for both radiators and approaches the acoustic impedance of a plane wave $z_0={\rho }_0{c}_0$.

With Equation (2.87) the radiated power of a piston of source strength $\mathit{Q}=\pi R^2{\mathit{v}}_z$ is

  (2.153)

The main use of Equation (2.153) is that the required velocity to achieve (or prevent) a certain sound power can be calculated from it, for example if one must define the boundary condition for a radiating piston in simulation software and only the radiated power is known.

2.7.3.2 Inertia Effects

The Bessel functions can be approximated by a series in $2kR$ taking the first series term of both functions (Jacobsen, 2011)

  (2.154)

This expression is valid for $ka<0.5$. From the imaginary part we get for the mass

  (2.155)

Assuming a cylindrical volume $V=\pi R^2{l}_c$ of the fluid above the piston we can calculate the length of the moving mass cylinder to be

  (2.156)

meaning that at low frequencies the piston is moving a fluid layer of $0.85$ times the radius acting as an inertia without radiation.

2.7.4 Power Radiation

For the radiated power calculation of the piston we took the pressure at the piston surface and integrated the pressure–velocity product over the surface. Due to the fact that the velocity is constant the surface integral involves mainly the pressure as a space-dependent property. In case of vibrating structures with complex shapes of vibration the velocity distribution over the surface is not homogeneous, and we need a more detailed approach.

  (2.157)

In the above equation a function with argument $(\mathbf{r}-{\mathbf{r}}_0)$ is multiplied by the velocity function for ${\mathbf{r}}_0$ and integrated over the two-dimensional space. Mathematically, this can be interpreted as a two-dimensional convolution in space

  (2.158)

Thus, when we apply the two-dimensional Fourier transform to the Rayleigh integral the result is the product of the Fourier transform of the vibration shape ${\mathit{v}}_z({\mathbf{r}}_0)$ and the Green’s function in wavenumber space leading to

  (2.159)

So, we have replaced the expensive convolution operation by a multiplication. This simplification is at the cost of two-dimensional Fourier transforms that are required to get the expressions in wavenumber domain.

The time averaged intensity of a sound field is given by the product of pressure and velocity (2.45). As the velocity is not uniform over the surface we perform a surface integration over the vibrating area to get the total radiated power

  (2.160)

Thus, for the determination of radiated power a double area integral is required that may become computationally expensive.

In the above expression we can also switch to the wavenumber domain. In this case the area integration is replaced by an integration over the two-dimensional wavenumber space.

  (2.161)

The double integral is replaced by a single two-dimensional wavenumber integration. Thus, once the shape function is available the power calculation in wavenumber space is much faster than in real space (Graham, 1996).

2.7.4.1 Radiation Efficiency

The radiation efficiency is a quantity that relates the power of a plane wave to the radiated power of a vibrating surface with same surface averaged velocity. The definition of the radiation efficiency was motivated by experimental procedures because it allows the estimation of the radiated power from the measurements of the vibration velocity. The squared average velocity of a vibrating surface is

  (2.162)

and the power radiated by a plane wave through the same area $S$ is given by (2.47)

  (2.163)

The radiation efficiency is defined as the ratio between the radiated power of a velocity profile ${\mathit{v}}_z(\mathbf{r})$ of a surface $S$ and the standardized power of the plane wave:

  (2.164)

The radiation efficiency is used to determine the radiated power of vibrating structures from calculated, estimated, or measured radiation efficiency of specific surfaces

  (2.165)