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BASIC EQUATIONS OF ACOUSTICS
CHAPTER 1 BASIC EQUATIONS OF ACOUSTICS
To understand the physics of noise control, it is first necessary to understand acoustics, the science of sound. To make the type of problems that we will address in this book tractable, we will only consider linear acoustics and further assume that the ambient quantities of pressure, density, velocity, etc. do not vary as a function of either time or space. As we shall see, these assumptions considerably simplify the associated mathematics. The scope of our required acoustic analysis is further narrowed by the type of problem which is typically encountered in the design of a quiet structure. In many applications, noise, the unwanted sound, is generated by the surface vibrations of a machine which in turn interact with the surrounding acoustic medium. The movement of the surface of the machine compresses the adjacent fluid and causes sound waves to radiate. Only this mechanism for generating noise will be considered, and thus mechanisms such as sound generation by turbulence, combustion, etc. are not considered. Even with these restrictions, a very broad range of design problems can be addressed, as will be seen subsequently.
1.1 Derivation of the Wave Equation The spatial and time dependence of the sound waves radiating away from a vibrating structure are governed by the physics of the acoustic medium. The change in pressure associated with a sound wave is usually very small in comparison to the ambient pressure, so that the pressure, density, and particle velocity of an acoustic disturbance can be written in terms of an ambient quantity, denoted by a subscript 0, plus a small fluctuating quantity, denoted by a superscript " Thus, the pressure, density, and particle velocity at a point in the fluid are given as p = P0 + P', P = P0 + P', and v = v', respectively, where v 0 = 0 because there is no ambient fluid flow. Defining the acoustic variables in this form allows products of the variables to be linearized; for example, p v - ( P0 + P') v' ~ P0 v' because the nonlinear term, p' v', is small in comparison to the linear term, P0 v'. The linearization process is revisited later in this section, after the partial differential equations governing sound waves have been derived. The propagation and radiation of sound waves is governed by three basic equations of fluid dynamics: (1) the equation of continuity or conservation of mass, (2) Euler's equation of motion of a fluid, and (3) the equation of state. Only a brief derivation of each of these equations is given in the subsequent text because similar derivations can be found in any book on acoustics, including Pierce ( 1989, pp. 6-20), Beranek ( 1986, pp. 16-23), Morse (1976, pp. 217-222), Morse and Ingard (1968, 227-305), Temkin (1981, pp. 1-58), and Kinsler et al. (1982, pp. 98-110). These three equations can be combined to give the wave equation, which is the basic partial differential equation governing the spatial and time dependence of an acoustic field.
2 DESIGNING QUIET STRUCTURES
C o n s e r v a t i o n of mass
The most physically intuitive of the basic equations of fluid dynamics is the equation of continuity or conservation of mass. To derive this equation, consider a fixed "control volume" V through which fluid is free to flow, as shown in Figure 1.1.
Figure 1.1 Surface S enclosing a volume of fluid V. The total mass of fluid within the control volume at a particular moment is given as I I I v P (x; t) dV(x) , so that the rate of change of mass within the control volume is (c3/c3t) IIIV p ( x ; t ) d V ( x ) . The rate of change of mass within the control volume can also be written in terms of the velocity of the particles flowing through the surface of the control volume as IIS p ( x ; t ) v ( x ; t ) ' n
dS(x),
where n is the outward normal direction at the point x of the surface S. Equating these quantities gives the integral form of the equation of continuity as (c3/c3t) I I I v p ( x ; t ) d V ( x ) = - I I s
p(x;t) v(x;t).n dS(x),
(1.1)
which simply implies that the mass of fluid within the control volume must be conserved. To convert from integral to differential form, take the time derivative within the integral on the left hand side of Equation (1.1) and convert the surface integral on the right hand side to a volume integral using Gauss' theorem. Rewriting the result as a single volume integral yields
CHAPTER 1: BASIC EQUATIONS OF ACOUSTICS 3
ff~v { ( c 3 / c 3 t ) p ( x ; t ) + V ' [ o ( x ; t ) v ( x ; t ) ] }
dV(x)=O
(1.2)
Because Equation (1.2) must be valid for any control volume within the fluid, the integrand must be zero such that Op ~+V.[p Ot
v]-O,
(1.3)
which is the desired result for the differential form of the equation of continuity.
Euler's equation of motion The second of the basic equations of fluid dynamics is Euler's equation of motion, which is simply Newton's second law for a fluid element in motion. To derive the equation of motion, consider a fixed volume of fluid, V t , enclosed within an imaginary, deformable surface, S t . The volume is referred to as a material volume, because the "material" within the volume does not change with time, and the subscript t indicates that the shape of the material volume changes with time. The momentum of the fluid within the material volume is affected by surface forces acting on S t, and by body forces acting on all the particles within V t . Mathematically, the rate of change of momentum of the fluid within the material volume can be written as (d/dt)
ffff~vt Pv dV(x't)=ffffSt fs d S ( x ' t ) + ~
fb d V ( x ' t )
(1.4)
where fs constitutes the surface forces and fb constitutes the body forces. The surface forces include the pressure and shear forces exerted at the boundary of the material volume and the body force is primarily due to gravity. In the applications considered in this text, the body forces due to gravity and the shear forces due to viscosity are negligible. The surface forces due to the pressure outside the material volume can be written as fs = - n p ,
(1.5)
such that the surface integral in Equation (1.4) can be rewritten using Gauss' theorem as j'j'st
fs
dS(x ; t ) - - - ~ V
t V p d V ( x ',t ) .
(1 6)
Equation (1.4) then reduces to
(d/dt) ~ g t ~ V dV(x; t)=-f~fv t
V p dV(x't) ,
.
(1 7)
4 DESIGNINGQUIET STRUCTURES To convert Equation (1.7) to a differential form, the total differential of the integral on the left-hand-side needs to be brought inside the integral, which is complicated by the fact that the material volume can change with time. Without going into the details, which can be found in the text by Temkin (1981, pp. 16-20), the term on the left hand side of Equation (1.7) can be transformed as (d / d t) I I I v t PV dV(x" ' t ) - I I I v
t
p(Dv/Dt)
dV(x;t),
(1.8)
where D/Dt=O/Ot+v.V
(1.9)
is the total derivative operator. Equation (1.7) then reduces to
.flJvt [ p ( D v / D t ) + V p ]
dV(x;t)-0
.
(1.10)
Because Equation (1.10) has to be valid for any choice of the material volume, the integrand must be identically zero so that p(Dv/Dt)+Vp=0.
(1..11)
Equation (1.11) is the differential form of Euler's equation of motion for a fluid. Physically, it simply states that the change in the momentum of a fluid particle is equal to the external forces applied to the particle. The equation of state
The equation of state is the last of the basic equations required to derive the wave equation. The derivation begins from an assumption that the pressure at a point in the fluid depends on the density and the specific entropy of the fluid, or p - p ( P, s ). If the fluid medium is uniform, the pressure is only a function of the ambient specific entropy So, so that the pressure is then written functionally as p - p ( P, s0). The fundamental question involved in deriving the equation of state is whether sound propagation is an isothermal or adiabatic process. During the 1700's, the basic theoretical models of sound propagation assumed it was an isothermal process. This assumption leads to a theoretical prediction for the speed of sound which differed from the experimentally measured values by about 16%. In hindsight, we know that sound propagation occurs with negligible heat flow because the changes of state occur so rapidly that there is no time for the temperature to equalize with the surrounding medium, and thus the process is adiabatic. If we consider only gaseous fluids, such as air, the assumption that the process is adiabatic leads to the relation P P - y = constant,
(1.12)
CHAPTER 1: BASIC EQUATIONS OF ACOUSTICS 5
where y is the specific heat ratio. This result was derived in 1816 by Laplace, and finally brought the theoretical predictions into close agreement with the measured data. The interested reader is referred to the text edited by Lindsay (1973) for a compendium of related literature. Linearization
The three basic equations of fluid dynamics derived in the previous analysis are nonlinear because they contain products of the dependent variables, such as the p v term in the conservation of mass equation. The first step in deriving the wave equation is to linearize each of the equations. Substituting for the pressure, density, and particle velocity in Equation (1.3) in terms of ambient plus fluctuating quantities, the conservation of mass equation becomes
0 (po + p , ) + V . [ ( p o + p,)v, ] = 0 ~3t
(1.13)
or, noting that c3P0 / c3t = 0, and that p' v' is a small quantity, the linearized version of the conservation of mass equation is 09' ~+90 ~t
V.v' = 0 .
(1.14)
In a similar way, the linearized version of the conservation of momentum equation can be determined as P0 ~c3v' =-V c3t
(1 15)
p',
the derivation of which is left as an exercise for the reader in problem 1-1. To linearize the equation of state we proceed by first evaluating Equation (1.12) in the ambient state yielding P0 90 ~ = constant,
(1.16)
so that, after substituting for the constant, Equation (1.12) becomes (po + P ' ) ( P o + P ' ) -~ = Po 9o ~ B
(1 17)
~
o
Equation (1.16) can then be rewritten as P0+P P0 and thus
= 1+ p = P0
90+9 90
~=
1 + ~9' P0
~l+y
9 90
(1.18)
6 DESIGNINGQUIET STRUCTURES
p'z(yPo/90)9'=c29
' .
(1.19)
where c is the speed of sound in the acoustic medium. In performing the derivation, we began from Equation (1.12), which is only valid for gases. The equation of state for a liquid medium is identical to that given in Equation (1.19), but is derived in a slightly different manner, as discussed by Pierce (1989, pp. 30-34). The wave equation
Equations (1.14), (1.15), and (1.19) represent a system of three linear partial differential equations in three dependent variables, p', 9', and v'. The physical quantity which is most easily measured for a sound wave is the pressure, and thus the equations will be reduced to a single partial differential equation for the acoustic pressure. We take the partial derivative with respect to time of the linearized equation of conservation of mass [Equation (1.14)] giving 029 ' _ c3t 2 ~ p ~
c3v' c3t = 0 ,
(1.20)
and take V. of the linearized equation of the conservation of momentum [Equation (1.15)] giving PO V . c 3 v ' _V 2 , c3--T = p .
(.1.21)
Subtracting Equation (1.21) from Equation (1.20) yields c32p , ~=V 2p 8t 2
' ,
(1.22)
or, after substituting for 9' from the linearized equation of state [Equation (1.19)], 1 02p ' c2 c3t2 = VZp .
(1.23)
To simplify the notation, we drop the superscript "so that p is the small fluctuating component of the pressure, and v is the acoustic velocity. Also, we take the ambient density of the fluid to be P, without the subscript 0. Equation (1.23) can then be written as V2 p
c
12 02P t 2 - O. 0
(1.24)
which is the desired result for the wave equation. As a simple example, consider an acoustic wave which does not depend on the y or z coordinate. The wave equation then reduces to
CHAPTER 1" BASIC EQUATIONS OF ACOUSTICS 7
c32p c3x 2
1 c32p c 2 c3t 2
= 0,
(1.25)
which has the general solution p= F (t-x / c)+G(t+x
/ c).
(1.26)
In Equation (1.26), F and G are arbitrary functions representing plane waves traveling in the positive and negative x-directions, respectively. 1.2 The Helmholtz Equation for Time-Harmonic Vibrations
In many of the design methods that we discuss in this book, the surface vibration and associated sound field occur at a constant frequency, a typical example of which is the hum of a transformer at 120 Hz. After an initial transient period, the pressure field generated by the vibration will reach steady state and fluctuate at the same frequency as the excitation. In this instance, the time dependence of the wave equation can be eliminated. When the sound field has reached steady state, we write the acoustic pressure at any point in the fluid as p ( x;t ) = Ppk (x) cos ( m t - ~)) = Re { Ppk (x) e - i ( o3 t-d0 )},
(1.27)
p ( x ; t ) = Re {15(x) e -imt} ,
(I.28)
or
where 15(x)= Ppk (x) e i *. In Equations (1.27) and (1.28), m is the angular frequency of the excitation, P pk is the peak value of the acoustic pressure, ~ is the phase angle of the acoustic pressure, and 15 is called the complex amplitude of the acoustic pressure. In most of the subsequent analysis, the complex amplitude of the acoustic pressure is the quantity of interest, and we will simply refer to it as the "acoustic pressure". Substituting for the acoustic pressure in the wave equation gives
Re{V215(x)e-i~~ ,
}
{15(x)~t2[e-i~~ ] -0,
(1.29)
or
Re { [ V219 (x)+ k2 19(x)] e - i m t } = 0 ,
(1.30)
where k - m / c is the acoustic wavenumber. The only way Equation (1.30) can be satisfied for all time is if g21~ ( x ) + k 2 1~ ( x ) - 0 ,
(1.31)
8 DESIGNING QUIET STRUCTURES which is known as the Helmholtz equation. Equation (1.31) is a homogeneous partial differential equation because the right hand side is identically zero at every point in the fluid: If the vibration excitation contains more than one frequency, each frequency can be analyzed separately and the results can be superposed to give the total sound field. It is convenient at this juncture to derive the time-harmonic version of the linearized equation of conservation of momentum. Substituting for the acoustic pressure and velocity in terms of their complex amplitudes, Equation (1.14) becomes - i m p'~ (x) = - V t3 (x).
(1.32)
Taking the dot product of both sides of Equation (1.32) with the unit vector n, and noting that co = k c, the result for the component of ;e (x) in the direction of n is ~r ( x ) . n
= ~
1
ikpc
V 15 ( x ) . n .
(1.33)
In the subsequent text, Equation (1.33) is simply referred to as Euler's equation. Returning to the example of one-dimensional waves, when the sound field is timeharmonic the acoustic pressure can be written in terms of waves traveling in the positive and negative x-direction as 15- A e ikx + B e - i k x .
(1.34)
The particle velocity in the x-direction can be determined from Euler's equation as ~. e x =
1 dl3 , ikpcdx
(1.35)
or, substituting for the pressure from Equation (1.34), ~ . e x = 1 [ A eikx - B e pc
ikx] .
(1.36)
Taking B = 0 in Equations (1.34) and.(1.36), the pressure and particle velocity of a plane wave traveling in the positive x-direction are related as s
A x =--e pc
ikx
t3 =-- . pc
(1.37)
The quantity p c is called the "characteristic impedance" of the fluid. In situations where the pressure and particle velocity are directly related through the characteristic impedance, the resulting wave motion can be thought of as nearly equivalent to idealized, one-dimensional wave motion.
CHAPTER 1: BASIC EQUATIONS OF ACOUSTICS 9 1.3 Boundary Conditions for Acoustic Boundary Value Problems As discussed in the chapter introduction, the only mechanism for generating acoustic energy considered in this book is the coupling between the vibrations of a surface and the motion of the adjacent fluid. The surface vibration can always be broken down into two components: one normal to the surface, and one tangential to the surface. The normal component of the surface vibration compresses the adjacent fluid causing sound waves to radiate away from the surface. The tangential component of the surface vibration shears the fluid and results in only a small motion of the fluid which is confined to the viscous boundary layer. In general, then, the interaction of the tangential motion of the vibrating surface with the adjacent fluid does not cause sound to radiate from the boundary surface. We thus ignore the tangential component of the surface vibration, and the boundary condition becomes 1 ;r ( x ) . n = ~ V 15 (x)- n , ikpc
(1.38)
i.e., the normal component of the surface velocity must equal the gradient of the pressure in the adjacent fluid. One could alternatively specify the surface pressure, but it is difficult to accurately measure the surface pressure because of the influence of nearby surfaces which scatter the sound waves radiating from the structure. In most cases, the normal surface velocity can be accurately measured with small, lightweight accelerometers or with laser vibrometers without significantly changing the vibration of the radiating structure. Also, the surface velocity of a vibrating structure can be computed numerically using the finite element method for a wide range of practical problems, whereas no such method exists for computing the surface pressure. An additional boundary condition is necessary if the vibrating structure is idealized as being surrounded by an infinite fluid medium. In this case, the acoustic waves radiating from the vibrating structure are distributed over an ever increasing area. Conservation of energy requires the amplitude of the acoustic waves to diminish as the area of the wavefront increases, which is expressed mathematically as the Sommerfeld radiation condition, lim [ r ( ~ - p c ~ . e r ) ] = 0 r~oo
.
(1.39)
In Equation (1.39), the coordinate r is centered on the vibrating structure, such that the quantity 15- p c ~,- e r must go to zero at least as 1 / r as r ~ oo. Simply setting 15- p c ~,. e r, Equation (1.39) is identically satisfied. Physically, this implies that as sound radiates away from a source the wavefronts become very nearly planar, and thus the pressure and radial velocity must be directly related through the characteristic impedance of the fluid medium.
10 DESIGNING QUIET STRUCTURES
1.4 The Time-Averaged Acoustic Power Output of a Vibrating Structure
In most of our design applications, we will find that the main quantity of interest is the time-averaged acoustic power output. To compute the power output of a vibrating structure, we calculate the rate at which energy flows through a closed surface surrounding the structure. The acoustic energy flux or acoustic intensity is defined as the average rate of flow of energy through a given area, which can be written in terms of the acoustic variables as
I = Force Velocity = p v . Area
(1.40)
For time-harmonic acoustic fields, we determine the time-averaged acoustic intensity by averaging Equation (1.40) over one period as
lj.T0 p v d t ,
lav=-T
(1.41)
where T is equal to one period. The component of the time-averaged intensity in the direction of the unit normal n is given by l a v . n = ~1sT0 p v . n d t ,
(1.42)
Substituting for the pressure and normal velocity in terms of their complex amplitudes, Equation (1.42) becomes |av . n = ~ _1
-TS0 =
T
o e iot}] at
[1151c~ (~ t - * P ) Iv" nl c~ (m t - * v ) ] at
I ll 'nl 27I:
1
--
lIT0
T
SO [ c o s ( , p - , v ) + C O S ( 2
51011,..I cos(,p -,v) 9
co t - , p - * v ) ]
at
(143
To illustrate Equation (1.43),take [ l ~ ] = 0 . 5 N / m 2 ] ~ . n [ = 2 m / s ,p=n/4 ~v = -3 x / 4, and o~ = 2 n and plot the pressure, particle velocity in the direction n, and intensity in the direction n as a function of time from 0 to 2 seconds, as shown in Figure 1.2. We note that the intensity in Figure 1.2 is at double the frequency of the pressure and particle velocity, and that it is displaced from zero by a constant equal to 0.5 cos (n). To convert Equation (1.43) from magnitude and phase to real and imaginary quantities, rewrite the equation as
CHAPTER 1: BASIC EQUATIONS OF ACOUSTICS 11
l
p
|~
V~
r
t
t
Figure 1.2 Plots of the pressure, particle velocity in the direction n and intensity in the direction n as a function of time from 0 to 2 seconds.
_l 2
1
v )=-~Re
ll?ll,~.n [ c o s ( , p - , = -1Re{t3~'*. n 2
{ 1151eiOOp ]~,.nle- iOOv}
t,
( 1.44)
where the * indicates complex conjugation. Thus, the time-average of the acoustic intensity can be calculated easily using the complex amplitude notation. The power output of the vibrating structure is defined as the rate at which energy flows through a surface completely surrounding the structure which includes no other acoustic sources. The power flowing through a surface of unit area is equal to the time-averaged acoustic intensity, and thus integrating the intensity over a surface surrounding the vibrating structure gives the time-averaged acoustic power output as l-lav = I I S l a v ' n d S
'
(1.45)
or, in terms of the pressure and velocity, 1-'Iav = 21I I S
Re { 15v* 9n } dS .
(1.46)
The most convenient form for computing the power output of a vibrating structure is given by Equation (1.46) with the surface S chosen as the boundary surface of the vibrating structure. The other convenient choice for the surface S is a very large spherical surface centered on the vibrating structure, which is only applicable if there is only one acoustic source. The boundary surface of the vibrating structure and the very large spherical surface are shown in Figure 1.3 as S and S', respectively. Equation (1.46) shows that if the pressure and normal velocity can be forced into quadrature at each point on the boundary surface such that q~p - q~v ~ 90 o_, the overall power output will be very small.
12 DESIGNING QUIET STRUCTURES
Figure 1.3 Boundary surface of a vibrating structure and a large spherical surface of radius r. In the design methods presented in the subsequent chapters, we will thus try to develop simple methods for forcing quadrature between the pressure and normal velocity over the boundary surface.
1.5 The Inhomogeneous Form of the Helmholtz Equation and Green's Functions The equation of mass conservation given in Equation (1.14) assumes that no mass sources are present in the fluid. If a very small mass source is present in the fluid at the point x s, then the equation of mass conservation becomes c~O+v.(pv) c3t
c~ms 6 ( x Ot
Xs)
(1.47)
where c3m s / a t is the rate at which mass is added to the fluid outside a small fixed volume enclosing the source. A small vibrating body whose volume is undergoing periodic oscillations qualifies as a mass source, because as the volume of the source increases, the air surrounding the source is forced out of the fixed control volume surrounding the source, as shown in Figure 1.4. In Equation (1.47), we denote 6 ( x - x s) as the Dirac delta function which is defined as III
v
6(x-x
s
)dV(x)={1
0
ifx s eV
ifx s~V"
(1.48)
The Dirac delta function must have units of 1 / volume if the right hand side of Equation (1.48) is to be dimensionless. Following the same steps as in the derivation of the homogeneous form of the Helmholtz equation, we determine the inhomogeneous form of the Helmholtz equation to be
CHAPTER 1: BASICEQUATIONSOF ACOUSTICS 13
Figure 1.4 Small source whose volume is undergoing periodic oscillations.
V2t3 (x)+ k2 t3 (x)= o32 l'hs a ( X - X s ) .
(1.49)
Equation (1.49) can now be related to a Green's function for an acoustic boundary value problem. The Green's function represents the basic solution of an inhomogeneous differential equation, from which other solutions can be determined by superposition. To conform with the text by Morse and Feshbach (1953), we rewrite the inhomogeneous Helmholtz equation
as V2~(x)+k 2t3(x):-4rtg
5(X-Xs),
(1.50)
where - 4 n ~ = m 2 rhs. A Green's function of the Helmholtz equation is defined as any solution of Equation (1.50) with ~ = 1. Denoting the Green's function for time--harmonic problems as 6 ( x / x s), the inhomogeneous partial differential equation for CJ is
V2G(X/Xs)+k2G(x/Xs):-4n8 ( X - X s )
9
(1.51)
In the next section, the solution of Equation (1.51) for a small spherical surface undergoing radial vibrations is examined in detail.
1.6 The Free-Space Green's Function To avoid a lengthy discussion of the solution of partial differential equations, we will simply list the free-space Green's function, and prove by direct substitution that it is the correct solution of the Helmholtz equation for a small spherical source in an infinite
14 DESIGNINGQUIETSTRUCTURES fluid medium. We use a lower case ~ to distinguish between the free-space Green's function and the more general form for the Green's function denoted by an upper case (]. The solution for the three-dimensional free-space Green's function is given by
~(X/Xs)=eikR/R
(1.52)
where R = l x - X s ] = ~ ( X - X s ) 2 + ( y - y s ) 2 +(Z-Zs) 2
(1.53)
is the distance from the source point to the field point. Before proceeding with the proof, it is instructive to discuss the nature of the singularity in the Green's function. Expanding the exponential in Equation (1.52) gives ~ ( x / X s ) = cos(k R) + i k sin(k R) . kR R
(1.54)
In the limit as R--+ 0, the real component of the Green's function becomes infinite, but using L'Hopital's rule we can show that the imaginary component is finite. This fact is generally true of all Green's functions and will become important in the computation of the power output. Proceeding with the proof, we integrate Equation (1.51) over the volume enclosed within a small sphere of radius e centered on the point x s, and substitute for the free-space Green's function giving
~V ( V2 +k2){ e i k R / R } d V ( x ) = - 4 ~ ~v ~ (X-Xs) dV(x)
(1.55)
We evaluate the right hand side of Equation (1.55) using the sifting property of the Dirac delta function yielding - 4 n. The left hand side is rewritten as ~V
( V2+k2){eikR/R} dV(x)=~V + k2 ~ V
[ e i k R / R ] dV(x),
V.{V[e i k R / R ] } dV(x) (1.56)
or, using Gauss' theorem ~v(V2+k2){ + k2 ~ v
eikR/R} d V ( x ) = ~ s { ( d / d R ) [ e i k R / R ] } R = e dS(x) [e i k R / R ] dV(x) .
Evaluating the integrals gives
(1.57)
CHAPTER 1: BASICEQUATIONSOF ACOUSTICS 15 IIIv(V2+k2){eikR/R}dV(x):{4xR2
d [ ikRe / R ] } dR R=~
2rt rt e[ ikR ]R 2 + k2 I0 I0 I0 e /R sin 0 dR dO d~).
(1.58)
As e goes to zero, the volume integral on the right hand side of Equation (1.58) goes to zero, as can be easily verified by direct evaluation. The remaining term can be evaluated by performing the indicated differentiation, substituting for R = e, and taking the limit as goes to zero, giving- 4 n. Thus, Equation (1.58) reduces to lim I I I v ( v 2 + k 2 ) { e i k R / R } d V ( x ) = - 4 x . e--~0
(1.59)
We have shown by direct substitution that the free-space Green's function given in Equation (1.52) is the correct solution of Equation (1.52) for a small spherical source in an infinite fluid medium.
1.7 The Kirchhoff-Helmholtz Equation In an acoustic radiation problem, we need to calculate the acoustic field of a vibrating structure submerged in an infinite fluid medium. In general, analytical solutions for the acoustic field cannot be found, and it is necessary to determine the acoustic field numerically. Because the fluid is infinite, methods of solving partial differential equations which seek to discretize the entire solution domain are difficult to apply to this problem. It is possible, though, to derive the solution of the problem in the form of an integral equation (i.e., an equation with unknown function appearing under an integral sign), where the integration extends only over the boundary surface of the radiator. The resulting integral equation, called the Kirchhoff-Helmholtz equation, gives the solution for the pressure at a field point in terms of the specified normal surface velocity and the unknown surface pressure. The Kirchhoff-Helmholtz equation for the acoustic radiation problem is derived in this section and the numerical solution of the equation is examined in detail in Chapters 2 and 3. The derivation of the Kirchhoff-Helmholtz equation proceeds from the vector identity:
Vs. {6(x/Xs) Vs ( Xst- (Xstv, 6(x/Xs)} = G (x / Xs) V2s 1~( X s ) - P (Xs) Vs2(~(X/Xs) = ( ~ ( X / X s ) ( V 2s + k 2 ) l ~ ( X s ) - P ( X s ) ( V ' 2s +k 2 ) G ( X / X s ) ,
(1.60)
16 DESIGNING QUIET STRUCTURES
where the V s notation indicates that the derivatives are taken with respect to the variable x s . The pressure field must satisfy Equation (1.31), and the Green's function must satisfy Equation (1.51) so that Equation (1.60) reduces to
~s {6(,,/,,s) ~s f,(Xs)-~ (,,s)Vs 6(:,,/,,s) } = 4 = 15(Xs) 5 ( X - X s )
(1.61)
.
Integrating Equation (1.61) over the volume of fluid outside of the surface of the vibrating structure and inside a large sphere of radius r centered on the vibrating structure, as shown previously in Figure 1.3, gives
IIIv Vs'{G(X/Xs)VsP(Xs)-p(Xs)VsG(x/Xs)} d V ( x s ) = 4 ~: I I I V 15( X s ) ~ 5 ( X - X s ) d V ( x s ) = 4
(1.62)
rc 15(x) ,
where we use the sifting property of the Dirac delta function to simplify the integral on the right hand side. We next convert the integral on the left hand side of Equation (1.62) to surface integrals using Gauss' theorem as
IIIv ~s {6(X/Xs)~s~(Xs)-~(Xs)~s6(X/Xs)} dV(xs) --Ils {6(~/~s)~s~(Xst-~(~s)~s6(~/~s)}
+IIS'
.s .S(xst
{6(X/Xs)Vsf~(Xs)-f)(Xs)VsG(x/Xs)} "er d S ( x s )
'
(1.63)
where n s is the outward normal to the surface S at the point x s , e r is the unit vector in the radial direction, and the negative sign for the surface integral over S is necessary because the outward normal points into the volume of fluid. The integral over S' goes to zero as the radial distance r goes to infinity because of the Sommerfeld radiation condition, and thus Equation (1.63) reduces to
-ffs {~(~/Xs)~s~/~s)-~t~s)~s6(~/~s)} = 4 r~ 15(x)
.s dSi~st (1.64)
which is referred to as the Kirchhoff-Helmholtz equation. In Equation (1.64), the point x is restricted to lie in the volume V because of the use of the sifting property of the Dirac delta function in Equation (1.62). If the field point x is taken to be inside S, then the sifting property of the Dirac delta function yields
CHAPTER 1: BASIC EQUATIONS OF ACOUSTICS 17
(1.65)
4 ~ I I ~ v 1~(Xs) 6 ( x - Xs) dV ( X s ) - 0 , and Equation (1.64) becomes ~IS { G ( X / X s ) V s t 3 ( X s ) - t 3 ( X s ) V s G ( x / X s ) }
"nsdS(xs)-0"
(1.66)
The pressure field thus has a "jump" as one proceeds from the interior to the exterior of the boundary surface, which is a mathematical artifact of the use of Gauss' theorem, and does not have any physical significance. In the subsequent analysis, we will always assume that the field point lies within the volume V, and thus the pressure field is determined from Equation (1.64). The Green's function in the Kirchhoff-Helmholtz equation is required to satisfy two conditions: (1) it must be a solution of the inhomogeneous form of the Helmholtz equation, and (2) it must satisfy the Sommerfeld radiation condition, such that the integral over S' in Equation (1.63) is negligible. An infinite variety of Green's functions exist, but only two forms for the Green's function are commonly used. The simplest form for the Green's function is the free-space Green's function, which was discussed previously in section 1.6. The other common form for the Green's function is called the Green's function of the second kind, following the nomenclature of Kellogg (1953). The Green's function of the second kind represents the solution of Equation (1.51) when a rigid object is submerged in the fluid along with the mass source. Because the acoustic field scattered by the rigid object is generally unknown, this form of the Green's function is not known analytically. Given these two choices, the free-space Green's function seems much more attractive because it is known analytically, and, indeed, computational analyses of the acoustic radiation problem conventionally begin by substituting the free-space Green's function into the Kirchhoff-Helmholtz equation. Still, many analytical results can be derived using the Green's function of the second kind that cannot be derived using the free-space Green's function. A thorough examination of the Green's function of the second kind is given in the subsequent chapters, but first, the free-space Green's function is used to derive some basic results for the acoustic field of a small source. 1.8 Sound Radiation from a Very Small Source
The size of an acoustic source is only relevant in comparison to the acoustic wavelength, and a very small source is one that is much smaller than the acoustic wavelength. Thus, we determine the solution for a very small source by keeping the size of the boundary surface fixed and taking the limit as the wavelength goes to infinity or the frequency goes to zero. In this section, the solution for the sound radiation from a very small source of any shape is derived by substituting the free-space Green's function into the Kirchhoff-Helmholtz equation, and taking the limit as the frequency goes to zero.
18 DESIGNING QUIET STRUCTURES Consider a small vibrating structure with boundary surface S as shown in Figure 1.5, where we assume the origin of coordinates lies within the boundary surface.
V __X s
Xs
P~[
Figure 1.5 Vibrating structure with boundary surface S. Substituting the flee-space Green's function into the Kirchhoff-Helmholtz equation, we determine the acoustic radiation of the source as 4 x l ~ ( X ) = - I I S [ e i k R / R ] Vst3(Xs).ns d S ( x s ) +IIS t3(Xs)Vs[eikR/R] "nsdS(xs)"
(1.67)
The distance from the source point x s to the field point x is written as R = ~/ r 2 - 2 x . x
s + r 2s = r ~/ 1 - 2 x . x s / r 2 + r 2sj/ r 2 ,
(1.68)
where r is the distance from the origin to the field point, and r s is the distance from the origin to the source point. At intermediate distances from the boundary surface (somewhere between the nearfield and farfield), r >> r s and Equation (1.68) can be approximated as R~r(1-x.x
s/r2)=r-er.Xs
.
(1.69)
Multiplying Equation (1.69) by the acoustic wavenumber, the nondimensional quantity k R is approximately equal to kR~kr-er.(kxs).
(1.70)
CHAPTER 1: BASICEQUATIONSOF ACOUSTICS 19 For a very small structure, the quantity k x s is negligible, and thus k R ~ k r. Substituting this result into Equation (1.67) gives 4 r t 1 5 ( x ) = - II S [ e i k r / r ] V s t3(x s ) . n s d S ( x s ) +IIst3(xs) Vs[eikr/r] "nsdS(xs)'
(1.71)
but V s [ ( 1 / r ) exp (i k r)] = 0, so that 1 e i k r l I V 15(x )-n s d S ( x s ) 4rtr S s s 9
(1.72)
We next write the integral in Equation (1.72) in terms of the boundary condition for the normal surface velocity using Equation (1.33). The final result for the pressure field of a small vibrating structure is t3(x),~
ikpc 4rcr
e ikr JJ II
pt I
S
v ( x ).n s d S ( x s ) = s
-
ikpcfi 4rtr
e
ikr
,
(1.73)
where the quantity fi is called the volume velocity of the source. Equation (1.73) shows that the acoustic field of any small source with nonzero volume velocity is functionally the same as that of a small spherical source, except in the nearfield of the source, where the approximation in Equation (1.69) is not valid. The power output of the small source can be computed using Equation (1.46). Taking the surface for the integration of the acoustic intensity to be a large sphere of radius R centered on the source, the power output becomes
1[ R2Re {.15 {' "eR }1,
1-Iav = ~ 4rc
(1.74)
where the radial symmetry of the acoustic field is used to simplify the integrations. The radial component of the velocity is determined from Equation (1.33) as 1
s
c315 Or
-fi- ( i k R - 1 ) 4rcR 2
e
ikR
(1.75)
r=R Substituting this result into Equation (1.74), and taking the limit as k R + m, gives
1 - I a v = 2 7 t R 2 R e { [ i k iRk4lpr[tciRkf if i *e-
k 2 pc 8rt
lfil2
4r~R e _ i k R l }
(1.76)
20 DESIGNING QUIET STRUCTURES
This result is useful for checking the accuracy of numerical and experimental results, especially when discrepancies between the two exist. References
Beranek, L. L. (1986), Acoustics, Acoustical Society of America, Woodbury, N.Y. Kellogg, O. D. (1953), Foundations of Potential Theory, Dover, N.Y. Kinsler, L. E., Frey, A. R., Coppens, A. B., and Sanders, J. V. (1982), Fundamentals of Acoustics, 3rd ed., Wiley, N.Y. Lindsay, R. B., ed., (1973)Acoustics, Dowden, Hutchinson and Ross, Stroudsburg, PA. Morse, P. M. (1976), Vibration and Sound, Acoustical Society of America, Woodbury, N.Y. Morse, P. M., and Feshbach, H. (1953), Methods of Theoretical Physics, McGraw-Hill, N.Y. Morse, P. M., and Ingard, K. U. (1968), Theoretical Acoustics, Princeton University Press, Princeton, N.J. Pierce, A. D. (1989), Acoustics: An Introduction to its Physical Principles and Applications, Acoustical Society of America, Woodbury, N.Y. Temkin, S. (1981), Elements of Acoustics, Wiley, N.Y. Problems
1-1 Show the steps involved in linearizing Euler's equation of motion, as given in Equation (1.11). 1-2 A spherical surface of radius a is radially oscillating with normal velocity, -~. (a)
Solve for the acoustic field of the sphere by assuming a solution in the form 19(r) = A
eikr
and determining the constant A by matchingthe radial velocity on the surface of the sphere.
(b)
(c)
(d)
Determine the relationship between t3 and -;,.e r as r goes to infinity. Does the result agree with Equation (1.37) ? What is the surface impedance on the sphere, i.e., plot the real and imaginary part of the ratio of pressure/velocity on the surface of the sphere as a function ofka. Compare the power output from the source by integrating the time-averaged
CHAPTER 1: BASIC EQUATIONS OF ACOUSTICS 21
intensity over the surface of the sphere and over a spherical surface in the farfield. Do the results agree? 1--3'~ Compute the pressure field of the sphere using the Kirchhoff-Helmholtz equation by substituting the free-space Green's function for (] ( x / x s ). Perform a similar analysis for field points located within the sphere. 1-4 Show that the function R in Equation (1.68) can be approximated as given in Equation (1.69) when r >> r s. 1-5 Determine the low frequency limit for the acoustic field radiated by a cube with one face of the cube vibrating as a piston with amplitude ~, and the other faces of the cube quiescent.
Problems with an * are intended for advanced students.
To understand the physics of noise control, it is first necessary to understand acoustics, the science of sound. To make the type of problems that we will address in this book tractable, we will only consider linear acoustics and further assume that the ambient quantities of pressure, density, velocity, etc. do not vary as a function of either time or space. As we shall see, these assumptions considerably simplify the associated mathematics. The scope of our required acoustic analysis is further narrowed by the type of problem which is typically encountered in the design of a quiet structure. In many applications, noise, the unwanted sound, is generated by the surface vibrations of a machine which in turn interact with the surrounding acoustic medium. The movement of the surface of the machine compresses the adjacent fluid and causes sound waves to radiate. Only this mechanism for generating noise will be considered, and thus mechanisms such as sound generation by turbulence, combustion, etc. are not considered. Even with these restrictions, a very broad range of design problems can be addressed, as will be seen subsequently.
1.1 Derivation of the Wave Equation The spatial and time dependence of the sound waves radiating away from a vibrating structure are governed by the physics of the acoustic medium. The change in pressure associated with a sound wave is usually very small in comparison to the ambient pressure, so that the pressure, density, and particle velocity of an acoustic disturbance can be written in terms of an ambient quantity, denoted by a subscript 0, plus a small fluctuating quantity, denoted by a superscript " Thus, the pressure, density, and particle velocity at a point in the fluid are given as p = P0 + P', P = P0 + P', and v = v', respectively, where v 0 = 0 because there is no ambient fluid flow. Defining the acoustic variables in this form allows products of the variables to be linearized; for example, p v - ( P0 + P') v' ~ P0 v' because the nonlinear term, p' v', is small in comparison to the linear term, P0 v'. The linearization process is revisited later in this section, after the partial differential equations governing sound waves have been derived. The propagation and radiation of sound waves is governed by three basic equations of fluid dynamics: (1) the equation of continuity or conservation of mass, (2) Euler's equation of motion of a fluid, and (3) the equation of state. Only a brief derivation of each of these equations is given in the subsequent text because similar derivations can be found in any book on acoustics, including Pierce ( 1989, pp. 6-20), Beranek ( 1986, pp. 16-23), Morse (1976, pp. 217-222), Morse and Ingard (1968, 227-305), Temkin (1981, pp. 1-58), and Kinsler et al. (1982, pp. 98-110). These three equations can be combined to give the wave equation, which is the basic partial differential equation governing the spatial and time dependence of an acoustic field.
2 DESIGNING QUIET STRUCTURES
C o n s e r v a t i o n of mass
The most physically intuitive of the basic equations of fluid dynamics is the equation of continuity or conservation of mass. To derive this equation, consider a fixed "control volume" V through which fluid is free to flow, as shown in Figure 1.1.
Figure 1.1 Surface S enclosing a volume of fluid V. The total mass of fluid within the control volume at a particular moment is given as I I I v P (x; t) dV(x) , so that the rate of change of mass within the control volume is (c3/c3t) IIIV p ( x ; t ) d V ( x ) . The rate of change of mass within the control volume can also be written in terms of the velocity of the particles flowing through the surface of the control volume as IIS p ( x ; t ) v ( x ; t ) ' n
dS(x),
where n is the outward normal direction at the point x of the surface S. Equating these quantities gives the integral form of the equation of continuity as (c3/c3t) I I I v p ( x ; t ) d V ( x ) = - I I s
p(x;t) v(x;t).n dS(x),
(1.1)
which simply implies that the mass of fluid within the control volume must be conserved. To convert from integral to differential form, take the time derivative within the integral on the left hand side of Equation (1.1) and convert the surface integral on the right hand side to a volume integral using Gauss' theorem. Rewriting the result as a single volume integral yields
CHAPTER 1: BASIC EQUATIONS OF ACOUSTICS 3
ff~v { ( c 3 / c 3 t ) p ( x ; t ) + V ' [ o ( x ; t ) v ( x ; t ) ] }
dV(x)=O
(1.2)
Because Equation (1.2) must be valid for any control volume within the fluid, the integrand must be zero such that Op ~+V.[p Ot
v]-O,
(1.3)
which is the desired result for the differential form of the equation of continuity.
Euler's equation of motion The second of the basic equations of fluid dynamics is Euler's equation of motion, which is simply Newton's second law for a fluid element in motion. To derive the equation of motion, consider a fixed volume of fluid, V t , enclosed within an imaginary, deformable surface, S t . The volume is referred to as a material volume, because the "material" within the volume does not change with time, and the subscript t indicates that the shape of the material volume changes with time. The momentum of the fluid within the material volume is affected by surface forces acting on S t, and by body forces acting on all the particles within V t . Mathematically, the rate of change of momentum of the fluid within the material volume can be written as (d/dt)
ffff~vt Pv dV(x't)=ffffSt fs d S ( x ' t ) + ~
fb d V ( x ' t )
(1.4)
where fs constitutes the surface forces and fb constitutes the body forces. The surface forces include the pressure and shear forces exerted at the boundary of the material volume and the body force is primarily due to gravity. In the applications considered in this text, the body forces due to gravity and the shear forces due to viscosity are negligible. The surface forces due to the pressure outside the material volume can be written as fs = - n p ,
(1.5)
such that the surface integral in Equation (1.4) can be rewritten using Gauss' theorem as j'j'st
fs
dS(x ; t ) - - - ~ V
t V p d V ( x ',t ) .
(1 6)
Equation (1.4) then reduces to
(d/dt) ~ g t ~ V dV(x; t)=-f~fv t
V p dV(x't) ,
.
(1 7)
4 DESIGNINGQUIET STRUCTURES To convert Equation (1.7) to a differential form, the total differential of the integral on the left-hand-side needs to be brought inside the integral, which is complicated by the fact that the material volume can change with time. Without going into the details, which can be found in the text by Temkin (1981, pp. 16-20), the term on the left hand side of Equation (1.7) can be transformed as (d / d t) I I I v t PV dV(x" ' t ) - I I I v
t
p(Dv/Dt)
dV(x;t),
(1.8)
where D/Dt=O/Ot+v.V
(1.9)
is the total derivative operator. Equation (1.7) then reduces to
.flJvt [ p ( D v / D t ) + V p ]
dV(x;t)-0
.
(1.10)
Because Equation (1.10) has to be valid for any choice of the material volume, the integrand must be identically zero so that p(Dv/Dt)+Vp=0.
(1..11)
Equation (1.11) is the differential form of Euler's equation of motion for a fluid. Physically, it simply states that the change in the momentum of a fluid particle is equal to the external forces applied to the particle. The equation of state
The equation of state is the last of the basic equations required to derive the wave equation. The derivation begins from an assumption that the pressure at a point in the fluid depends on the density and the specific entropy of the fluid, or p - p ( P, s ). If the fluid medium is uniform, the pressure is only a function of the ambient specific entropy So, so that the pressure is then written functionally as p - p ( P, s0). The fundamental question involved in deriving the equation of state is whether sound propagation is an isothermal or adiabatic process. During the 1700's, the basic theoretical models of sound propagation assumed it was an isothermal process. This assumption leads to a theoretical prediction for the speed of sound which differed from the experimentally measured values by about 16%. In hindsight, we know that sound propagation occurs with negligible heat flow because the changes of state occur so rapidly that there is no time for the temperature to equalize with the surrounding medium, and thus the process is adiabatic. If we consider only gaseous fluids, such as air, the assumption that the process is adiabatic leads to the relation P P - y = constant,
(1.12)
CHAPTER 1: BASIC EQUATIONS OF ACOUSTICS 5
where y is the specific heat ratio. This result was derived in 1816 by Laplace, and finally brought the theoretical predictions into close agreement with the measured data. The interested reader is referred to the text edited by Lindsay (1973) for a compendium of related literature. Linearization
The three basic equations of fluid dynamics derived in the previous analysis are nonlinear because they contain products of the dependent variables, such as the p v term in the conservation of mass equation. The first step in deriving the wave equation is to linearize each of the equations. Substituting for the pressure, density, and particle velocity in Equation (1.3) in terms of ambient plus fluctuating quantities, the conservation of mass equation becomes
0 (po + p , ) + V . [ ( p o + p,)v, ] = 0 ~3t
(1.13)
or, noting that c3P0 / c3t = 0, and that p' v' is a small quantity, the linearized version of the conservation of mass equation is 09' ~+90 ~t
V.v' = 0 .
(1.14)
In a similar way, the linearized version of the conservation of momentum equation can be determined as P0 ~c3v' =-V c3t
(1 15)
p',
the derivation of which is left as an exercise for the reader in problem 1-1. To linearize the equation of state we proceed by first evaluating Equation (1.12) in the ambient state yielding P0 90 ~ = constant,
(1.16)
so that, after substituting for the constant, Equation (1.12) becomes (po + P ' ) ( P o + P ' ) -~ = Po 9o ~ B
(1 17)
~
o
Equation (1.16) can then be rewritten as P0+P P0 and thus
= 1+ p = P0
90+9 90
~=
1 + ~9' P0
~l+y
9 90
(1.18)
6 DESIGNINGQUIET STRUCTURES
p'z(yPo/90)9'=c29
' .
(1.19)
where c is the speed of sound in the acoustic medium. In performing the derivation, we began from Equation (1.12), which is only valid for gases. The equation of state for a liquid medium is identical to that given in Equation (1.19), but is derived in a slightly different manner, as discussed by Pierce (1989, pp. 30-34). The wave equation
Equations (1.14), (1.15), and (1.19) represent a system of three linear partial differential equations in three dependent variables, p', 9', and v'. The physical quantity which is most easily measured for a sound wave is the pressure, and thus the equations will be reduced to a single partial differential equation for the acoustic pressure. We take the partial derivative with respect to time of the linearized equation of conservation of mass [Equation (1.14)] giving 029 ' _ c3t 2 ~ p ~
c3v' c3t = 0 ,
(1.20)
and take V. of the linearized equation of the conservation of momentum [Equation (1.15)] giving PO V . c 3 v ' _V 2 , c3--T = p .
(.1.21)
Subtracting Equation (1.21) from Equation (1.20) yields c32p , ~=V 2p 8t 2
' ,
(1.22)
or, after substituting for 9' from the linearized equation of state [Equation (1.19)], 1 02p ' c2 c3t2 = VZp .
(1.23)
To simplify the notation, we drop the superscript "so that p is the small fluctuating component of the pressure, and v is the acoustic velocity. Also, we take the ambient density of the fluid to be P, without the subscript 0. Equation (1.23) can then be written as V2 p
c
12 02P t 2 - O. 0
(1.24)
which is the desired result for the wave equation. As a simple example, consider an acoustic wave which does not depend on the y or z coordinate. The wave equation then reduces to
CHAPTER 1" BASIC EQUATIONS OF ACOUSTICS 7
c32p c3x 2
1 c32p c 2 c3t 2
= 0,
(1.25)
which has the general solution p= F (t-x / c)+G(t+x
/ c).
(1.26)
In Equation (1.26), F and G are arbitrary functions representing plane waves traveling in the positive and negative x-directions, respectively. 1.2 The Helmholtz Equation for Time-Harmonic Vibrations
In many of the design methods that we discuss in this book, the surface vibration and associated sound field occur at a constant frequency, a typical example of which is the hum of a transformer at 120 Hz. After an initial transient period, the pressure field generated by the vibration will reach steady state and fluctuate at the same frequency as the excitation. In this instance, the time dependence of the wave equation can be eliminated. When the sound field has reached steady state, we write the acoustic pressure at any point in the fluid as p ( x;t ) = Ppk (x) cos ( m t - ~)) = Re { Ppk (x) e - i ( o3 t-d0 )},
(1.27)
p ( x ; t ) = Re {15(x) e -imt} ,
(I.28)
or
where 15(x)= Ppk (x) e i *. In Equations (1.27) and (1.28), m is the angular frequency of the excitation, P pk is the peak value of the acoustic pressure, ~ is the phase angle of the acoustic pressure, and 15 is called the complex amplitude of the acoustic pressure. In most of the subsequent analysis, the complex amplitude of the acoustic pressure is the quantity of interest, and we will simply refer to it as the "acoustic pressure". Substituting for the acoustic pressure in the wave equation gives
Re{V215(x)e-i~~ ,
}
{15(x)~t2[e-i~~ ] -0,
(1.29)
or
Re { [ V219 (x)+ k2 19(x)] e - i m t } = 0 ,
(1.30)
where k - m / c is the acoustic wavenumber. The only way Equation (1.30) can be satisfied for all time is if g21~ ( x ) + k 2 1~ ( x ) - 0 ,
(1.31)
8 DESIGNING QUIET STRUCTURES which is known as the Helmholtz equation. Equation (1.31) is a homogeneous partial differential equation because the right hand side is identically zero at every point in the fluid: If the vibration excitation contains more than one frequency, each frequency can be analyzed separately and the results can be superposed to give the total sound field. It is convenient at this juncture to derive the time-harmonic version of the linearized equation of conservation of momentum. Substituting for the acoustic pressure and velocity in terms of their complex amplitudes, Equation (1.14) becomes - i m p'~ (x) = - V t3 (x).
(1.32)
Taking the dot product of both sides of Equation (1.32) with the unit vector n, and noting that co = k c, the result for the component of ;e (x) in the direction of n is ~r ( x ) . n
= ~
1
ikpc
V 15 ( x ) . n .
(1.33)
In the subsequent text, Equation (1.33) is simply referred to as Euler's equation. Returning to the example of one-dimensional waves, when the sound field is timeharmonic the acoustic pressure can be written in terms of waves traveling in the positive and negative x-direction as 15- A e ikx + B e - i k x .
(1.34)
The particle velocity in the x-direction can be determined from Euler's equation as ~. e x =
1 dl3 , ikpcdx
(1.35)
or, substituting for the pressure from Equation (1.34), ~ . e x = 1 [ A eikx - B e pc
ikx] .
(1.36)
Taking B = 0 in Equations (1.34) and.(1.36), the pressure and particle velocity of a plane wave traveling in the positive x-direction are related as s
A x =--e pc
ikx
t3 =-- . pc
(1.37)
The quantity p c is called the "characteristic impedance" of the fluid. In situations where the pressure and particle velocity are directly related through the characteristic impedance, the resulting wave motion can be thought of as nearly equivalent to idealized, one-dimensional wave motion.
CHAPTER 1: BASIC EQUATIONS OF ACOUSTICS 9 1.3 Boundary Conditions for Acoustic Boundary Value Problems As discussed in the chapter introduction, the only mechanism for generating acoustic energy considered in this book is the coupling between the vibrations of a surface and the motion of the adjacent fluid. The surface vibration can always be broken down into two components: one normal to the surface, and one tangential to the surface. The normal component of the surface vibration compresses the adjacent fluid causing sound waves to radiate away from the surface. The tangential component of the surface vibration shears the fluid and results in only a small motion of the fluid which is confined to the viscous boundary layer. In general, then, the interaction of the tangential motion of the vibrating surface with the adjacent fluid does not cause sound to radiate from the boundary surface. We thus ignore the tangential component of the surface vibration, and the boundary condition becomes 1 ;r ( x ) . n = ~ V 15 (x)- n , ikpc
(1.38)
i.e., the normal component of the surface velocity must equal the gradient of the pressure in the adjacent fluid. One could alternatively specify the surface pressure, but it is difficult to accurately measure the surface pressure because of the influence of nearby surfaces which scatter the sound waves radiating from the structure. In most cases, the normal surface velocity can be accurately measured with small, lightweight accelerometers or with laser vibrometers without significantly changing the vibration of the radiating structure. Also, the surface velocity of a vibrating structure can be computed numerically using the finite element method for a wide range of practical problems, whereas no such method exists for computing the surface pressure. An additional boundary condition is necessary if the vibrating structure is idealized as being surrounded by an infinite fluid medium. In this case, the acoustic waves radiating from the vibrating structure are distributed over an ever increasing area. Conservation of energy requires the amplitude of the acoustic waves to diminish as the area of the wavefront increases, which is expressed mathematically as the Sommerfeld radiation condition, lim [ r ( ~ - p c ~ . e r ) ] = 0 r~oo
.
(1.39)
In Equation (1.39), the coordinate r is centered on the vibrating structure, such that the quantity 15- p c ~,- e r must go to zero at least as 1 / r as r ~ oo. Simply setting 15- p c ~,. e r, Equation (1.39) is identically satisfied. Physically, this implies that as sound radiates away from a source the wavefronts become very nearly planar, and thus the pressure and radial velocity must be directly related through the characteristic impedance of the fluid medium.
10 DESIGNING QUIET STRUCTURES
1.4 The Time-Averaged Acoustic Power Output of a Vibrating Structure
In most of our design applications, we will find that the main quantity of interest is the time-averaged acoustic power output. To compute the power output of a vibrating structure, we calculate the rate at which energy flows through a closed surface surrounding the structure. The acoustic energy flux or acoustic intensity is defined as the average rate of flow of energy through a given area, which can be written in terms of the acoustic variables as
I = Force Velocity = p v . Area
(1.40)
For time-harmonic acoustic fields, we determine the time-averaged acoustic intensity by averaging Equation (1.40) over one period as
lj.T0 p v d t ,
lav=-T
(1.41)
where T is equal to one period. The component of the time-averaged intensity in the direction of the unit normal n is given by l a v . n = ~1sT0 p v . n d t ,
(1.42)
Substituting for the pressure and normal velocity in terms of their complex amplitudes, Equation (1.42) becomes |av . n = ~ _1
-TS0 =
T
o e iot}] at
[1151c~ (~ t - * P ) Iv" nl c~ (m t - * v ) ] at
I ll 'nl 27I:
1
--
lIT0
T
SO [ c o s ( , p - , v ) + C O S ( 2
51011,..I cos(,p -,v) 9
co t - , p - * v ) ]
at
(143
To illustrate Equation (1.43),take [ l ~ ] = 0 . 5 N / m 2 ] ~ . n [ = 2 m / s ,p=n/4 ~v = -3 x / 4, and o~ = 2 n and plot the pressure, particle velocity in the direction n, and intensity in the direction n as a function of time from 0 to 2 seconds, as shown in Figure 1.2. We note that the intensity in Figure 1.2 is at double the frequency of the pressure and particle velocity, and that it is displaced from zero by a constant equal to 0.5 cos (n). To convert Equation (1.43) from magnitude and phase to real and imaginary quantities, rewrite the equation as
CHAPTER 1: BASIC EQUATIONS OF ACOUSTICS 11
l
p
|~
V~
r
t
t
Figure 1.2 Plots of the pressure, particle velocity in the direction n and intensity in the direction n as a function of time from 0 to 2 seconds.
_l 2
1
v )=-~Re
ll?ll,~.n [ c o s ( , p - , = -1Re{t3~'*. n 2
{ 1151eiOOp ]~,.nle- iOOv}
t,
( 1.44)
where the * indicates complex conjugation. Thus, the time-average of the acoustic intensity can be calculated easily using the complex amplitude notation. The power output of the vibrating structure is defined as the rate at which energy flows through a surface completely surrounding the structure which includes no other acoustic sources. The power flowing through a surface of unit area is equal to the time-averaged acoustic intensity, and thus integrating the intensity over a surface surrounding the vibrating structure gives the time-averaged acoustic power output as l-lav = I I S l a v ' n d S
'
(1.45)
or, in terms of the pressure and velocity, 1-'Iav = 21I I S
Re { 15v* 9n } dS .
(1.46)
The most convenient form for computing the power output of a vibrating structure is given by Equation (1.46) with the surface S chosen as the boundary surface of the vibrating structure. The other convenient choice for the surface S is a very large spherical surface centered on the vibrating structure, which is only applicable if there is only one acoustic source. The boundary surface of the vibrating structure and the very large spherical surface are shown in Figure 1.3 as S and S', respectively. Equation (1.46) shows that if the pressure and normal velocity can be forced into quadrature at each point on the boundary surface such that q~p - q~v ~ 90 o_, the overall power output will be very small.
12 DESIGNING QUIET STRUCTURES
Figure 1.3 Boundary surface of a vibrating structure and a large spherical surface of radius r. In the design methods presented in the subsequent chapters, we will thus try to develop simple methods for forcing quadrature between the pressure and normal velocity over the boundary surface.
1.5 The Inhomogeneous Form of the Helmholtz Equation and Green's Functions The equation of mass conservation given in Equation (1.14) assumes that no mass sources are present in the fluid. If a very small mass source is present in the fluid at the point x s, then the equation of mass conservation becomes c~O+v.(pv) c3t
c~ms 6 ( x Ot
Xs)
(1.47)
where c3m s / a t is the rate at which mass is added to the fluid outside a small fixed volume enclosing the source. A small vibrating body whose volume is undergoing periodic oscillations qualifies as a mass source, because as the volume of the source increases, the air surrounding the source is forced out of the fixed control volume surrounding the source, as shown in Figure 1.4. In Equation (1.47), we denote 6 ( x - x s) as the Dirac delta function which is defined as III
v
6(x-x
s
)dV(x)={1
0
ifx s eV
ifx s~V"
(1.48)
The Dirac delta function must have units of 1 / volume if the right hand side of Equation (1.48) is to be dimensionless. Following the same steps as in the derivation of the homogeneous form of the Helmholtz equation, we determine the inhomogeneous form of the Helmholtz equation to be
CHAPTER 1: BASICEQUATIONSOF ACOUSTICS 13
Figure 1.4 Small source whose volume is undergoing periodic oscillations.
V2t3 (x)+ k2 t3 (x)= o32 l'hs a ( X - X s ) .
(1.49)
Equation (1.49) can now be related to a Green's function for an acoustic boundary value problem. The Green's function represents the basic solution of an inhomogeneous differential equation, from which other solutions can be determined by superposition. To conform with the text by Morse and Feshbach (1953), we rewrite the inhomogeneous Helmholtz equation
as V2~(x)+k 2t3(x):-4rtg
5(X-Xs),
(1.50)
where - 4 n ~ = m 2 rhs. A Green's function of the Helmholtz equation is defined as any solution of Equation (1.50) with ~ = 1. Denoting the Green's function for time--harmonic problems as 6 ( x / x s), the inhomogeneous partial differential equation for CJ is
V2G(X/Xs)+k2G(x/Xs):-4n8 ( X - X s )
9
(1.51)
In the next section, the solution of Equation (1.51) for a small spherical surface undergoing radial vibrations is examined in detail.
1.6 The Free-Space Green's Function To avoid a lengthy discussion of the solution of partial differential equations, we will simply list the free-space Green's function, and prove by direct substitution that it is the correct solution of the Helmholtz equation for a small spherical source in an infinite
14 DESIGNINGQUIETSTRUCTURES fluid medium. We use a lower case ~ to distinguish between the free-space Green's function and the more general form for the Green's function denoted by an upper case (]. The solution for the three-dimensional free-space Green's function is given by
~(X/Xs)=eikR/R
(1.52)
where R = l x - X s ] = ~ ( X - X s ) 2 + ( y - y s ) 2 +(Z-Zs) 2
(1.53)
is the distance from the source point to the field point. Before proceeding with the proof, it is instructive to discuss the nature of the singularity in the Green's function. Expanding the exponential in Equation (1.52) gives ~ ( x / X s ) = cos(k R) + i k sin(k R) . kR R
(1.54)
In the limit as R--+ 0, the real component of the Green's function becomes infinite, but using L'Hopital's rule we can show that the imaginary component is finite. This fact is generally true of all Green's functions and will become important in the computation of the power output. Proceeding with the proof, we integrate Equation (1.51) over the volume enclosed within a small sphere of radius e centered on the point x s, and substitute for the free-space Green's function giving
~V ( V2 +k2){ e i k R / R } d V ( x ) = - 4 ~ ~v ~ (X-Xs) dV(x)
(1.55)
We evaluate the right hand side of Equation (1.55) using the sifting property of the Dirac delta function yielding - 4 n. The left hand side is rewritten as ~V
( V2+k2){eikR/R} dV(x)=~V + k2 ~ V
[ e i k R / R ] dV(x),
V.{V[e i k R / R ] } dV(x) (1.56)
or, using Gauss' theorem ~v(V2+k2){ + k2 ~ v
eikR/R} d V ( x ) = ~ s { ( d / d R ) [ e i k R / R ] } R = e dS(x) [e i k R / R ] dV(x) .
Evaluating the integrals gives
(1.57)
CHAPTER 1: BASICEQUATIONSOF ACOUSTICS 15 IIIv(V2+k2){eikR/R}dV(x):{4xR2
d [ ikRe / R ] } dR R=~
2rt rt e[ ikR ]R 2 + k2 I0 I0 I0 e /R sin 0 dR dO d~).
(1.58)
As e goes to zero, the volume integral on the right hand side of Equation (1.58) goes to zero, as can be easily verified by direct evaluation. The remaining term can be evaluated by performing the indicated differentiation, substituting for R = e, and taking the limit as goes to zero, giving- 4 n. Thus, Equation (1.58) reduces to lim I I I v ( v 2 + k 2 ) { e i k R / R } d V ( x ) = - 4 x . e--~0
(1.59)
We have shown by direct substitution that the free-space Green's function given in Equation (1.52) is the correct solution of Equation (1.52) for a small spherical source in an infinite fluid medium.
1.7 The Kirchhoff-Helmholtz Equation In an acoustic radiation problem, we need to calculate the acoustic field of a vibrating structure submerged in an infinite fluid medium. In general, analytical solutions for the acoustic field cannot be found, and it is necessary to determine the acoustic field numerically. Because the fluid is infinite, methods of solving partial differential equations which seek to discretize the entire solution domain are difficult to apply to this problem. It is possible, though, to derive the solution of the problem in the form of an integral equation (i.e., an equation with unknown function appearing under an integral sign), where the integration extends only over the boundary surface of the radiator. The resulting integral equation, called the Kirchhoff-Helmholtz equation, gives the solution for the pressure at a field point in terms of the specified normal surface velocity and the unknown surface pressure. The Kirchhoff-Helmholtz equation for the acoustic radiation problem is derived in this section and the numerical solution of the equation is examined in detail in Chapters 2 and 3. The derivation of the Kirchhoff-Helmholtz equation proceeds from the vector identity:
Vs. {6(x/Xs) Vs ( Xst- (Xstv, 6(x/Xs)} = G (x / Xs) V2s 1~( X s ) - P (Xs) Vs2(~(X/Xs) = ( ~ ( X / X s ) ( V 2s + k 2 ) l ~ ( X s ) - P ( X s ) ( V ' 2s +k 2 ) G ( X / X s ) ,
(1.60)
16 DESIGNING QUIET STRUCTURES
where the V s notation indicates that the derivatives are taken with respect to the variable x s . The pressure field must satisfy Equation (1.31), and the Green's function must satisfy Equation (1.51) so that Equation (1.60) reduces to
~s {6(,,/,,s) ~s f,(Xs)-~ (,,s)Vs 6(:,,/,,s) } = 4 = 15(Xs) 5 ( X - X s )
(1.61)
.
Integrating Equation (1.61) over the volume of fluid outside of the surface of the vibrating structure and inside a large sphere of radius r centered on the vibrating structure, as shown previously in Figure 1.3, gives
IIIv Vs'{G(X/Xs)VsP(Xs)-p(Xs)VsG(x/Xs)} d V ( x s ) = 4 ~: I I I V 15( X s ) ~ 5 ( X - X s ) d V ( x s ) = 4
(1.62)
rc 15(x) ,
where we use the sifting property of the Dirac delta function to simplify the integral on the right hand side. We next convert the integral on the left hand side of Equation (1.62) to surface integrals using Gauss' theorem as
IIIv ~s {6(X/Xs)~s~(Xs)-~(Xs)~s6(X/Xs)} dV(xs) --Ils {6(~/~s)~s~(Xst-~(~s)~s6(~/~s)}
+IIS'
.s .S(xst
{6(X/Xs)Vsf~(Xs)-f)(Xs)VsG(x/Xs)} "er d S ( x s )
'
(1.63)
where n s is the outward normal to the surface S at the point x s , e r is the unit vector in the radial direction, and the negative sign for the surface integral over S is necessary because the outward normal points into the volume of fluid. The integral over S' goes to zero as the radial distance r goes to infinity because of the Sommerfeld radiation condition, and thus Equation (1.63) reduces to
-ffs {~(~/Xs)~s~/~s)-~t~s)~s6(~/~s)} = 4 r~ 15(x)
.s dSi~st (1.64)
which is referred to as the Kirchhoff-Helmholtz equation. In Equation (1.64), the point x is restricted to lie in the volume V because of the use of the sifting property of the Dirac delta function in Equation (1.62). If the field point x is taken to be inside S, then the sifting property of the Dirac delta function yields
CHAPTER 1: BASIC EQUATIONS OF ACOUSTICS 17
(1.65)
4 ~ I I ~ v 1~(Xs) 6 ( x - Xs) dV ( X s ) - 0 , and Equation (1.64) becomes ~IS { G ( X / X s ) V s t 3 ( X s ) - t 3 ( X s ) V s G ( x / X s ) }
"nsdS(xs)-0"
(1.66)
The pressure field thus has a "jump" as one proceeds from the interior to the exterior of the boundary surface, which is a mathematical artifact of the use of Gauss' theorem, and does not have any physical significance. In the subsequent analysis, we will always assume that the field point lies within the volume V, and thus the pressure field is determined from Equation (1.64). The Green's function in the Kirchhoff-Helmholtz equation is required to satisfy two conditions: (1) it must be a solution of the inhomogeneous form of the Helmholtz equation, and (2) it must satisfy the Sommerfeld radiation condition, such that the integral over S' in Equation (1.63) is negligible. An infinite variety of Green's functions exist, but only two forms for the Green's function are commonly used. The simplest form for the Green's function is the free-space Green's function, which was discussed previously in section 1.6. The other common form for the Green's function is called the Green's function of the second kind, following the nomenclature of Kellogg (1953). The Green's function of the second kind represents the solution of Equation (1.51) when a rigid object is submerged in the fluid along with the mass source. Because the acoustic field scattered by the rigid object is generally unknown, this form of the Green's function is not known analytically. Given these two choices, the free-space Green's function seems much more attractive because it is known analytically, and, indeed, computational analyses of the acoustic radiation problem conventionally begin by substituting the free-space Green's function into the Kirchhoff-Helmholtz equation. Still, many analytical results can be derived using the Green's function of the second kind that cannot be derived using the free-space Green's function. A thorough examination of the Green's function of the second kind is given in the subsequent chapters, but first, the free-space Green's function is used to derive some basic results for the acoustic field of a small source. 1.8 Sound Radiation from a Very Small Source
The size of an acoustic source is only relevant in comparison to the acoustic wavelength, and a very small source is one that is much smaller than the acoustic wavelength. Thus, we determine the solution for a very small source by keeping the size of the boundary surface fixed and taking the limit as the wavelength goes to infinity or the frequency goes to zero. In this section, the solution for the sound radiation from a very small source of any shape is derived by substituting the free-space Green's function into the Kirchhoff-Helmholtz equation, and taking the limit as the frequency goes to zero.
18 DESIGNING QUIET STRUCTURES Consider a small vibrating structure with boundary surface S as shown in Figure 1.5, where we assume the origin of coordinates lies within the boundary surface.
V __X s
Xs
P~[
Figure 1.5 Vibrating structure with boundary surface S. Substituting the flee-space Green's function into the Kirchhoff-Helmholtz equation, we determine the acoustic radiation of the source as 4 x l ~ ( X ) = - I I S [ e i k R / R ] Vst3(Xs).ns d S ( x s ) +IIS t3(Xs)Vs[eikR/R] "nsdS(xs)"
(1.67)
The distance from the source point x s to the field point x is written as R = ~/ r 2 - 2 x . x
s + r 2s = r ~/ 1 - 2 x . x s / r 2 + r 2sj/ r 2 ,
(1.68)
where r is the distance from the origin to the field point, and r s is the distance from the origin to the source point. At intermediate distances from the boundary surface (somewhere between the nearfield and farfield), r >> r s and Equation (1.68) can be approximated as R~r(1-x.x
s/r2)=r-er.Xs
.
(1.69)
Multiplying Equation (1.69) by the acoustic wavenumber, the nondimensional quantity k R is approximately equal to kR~kr-er.(kxs).
(1.70)
CHAPTER 1: BASICEQUATIONSOF ACOUSTICS 19 For a very small structure, the quantity k x s is negligible, and thus k R ~ k r. Substituting this result into Equation (1.67) gives 4 r t 1 5 ( x ) = - II S [ e i k r / r ] V s t3(x s ) . n s d S ( x s ) +IIst3(xs) Vs[eikr/r] "nsdS(xs)'
(1.71)
but V s [ ( 1 / r ) exp (i k r)] = 0, so that 1 e i k r l I V 15(x )-n s d S ( x s ) 4rtr S s s 9
(1.72)
We next write the integral in Equation (1.72) in terms of the boundary condition for the normal surface velocity using Equation (1.33). The final result for the pressure field of a small vibrating structure is t3(x),~
ikpc 4rcr
e ikr JJ II
pt I
S
v ( x ).n s d S ( x s ) = s
-
ikpcfi 4rtr
e
ikr
,
(1.73)
where the quantity fi is called the volume velocity of the source. Equation (1.73) shows that the acoustic field of any small source with nonzero volume velocity is functionally the same as that of a small spherical source, except in the nearfield of the source, where the approximation in Equation (1.69) is not valid. The power output of the small source can be computed using Equation (1.46). Taking the surface for the integration of the acoustic intensity to be a large sphere of radius R centered on the source, the power output becomes
1[ R2Re {.15 {' "eR }1,
1-Iav = ~ 4rc
(1.74)
where the radial symmetry of the acoustic field is used to simplify the integrations. The radial component of the velocity is determined from Equation (1.33) as 1
s
c315 Or
-fi- ( i k R - 1 ) 4rcR 2
e
ikR
(1.75)
r=R Substituting this result into Equation (1.74), and taking the limit as k R + m, gives
1 - I a v = 2 7 t R 2 R e { [ i k iRk4lpr[tciRkf if i *e-
k 2 pc 8rt
lfil2
4r~R e _ i k R l }
(1.76)
20 DESIGNING QUIET STRUCTURES
This result is useful for checking the accuracy of numerical and experimental results, especially when discrepancies between the two exist. References
Beranek, L. L. (1986), Acoustics, Acoustical Society of America, Woodbury, N.Y. Kellogg, O. D. (1953), Foundations of Potential Theory, Dover, N.Y. Kinsler, L. E., Frey, A. R., Coppens, A. B., and Sanders, J. V. (1982), Fundamentals of Acoustics, 3rd ed., Wiley, N.Y. Lindsay, R. B., ed., (1973)Acoustics, Dowden, Hutchinson and Ross, Stroudsburg, PA. Morse, P. M. (1976), Vibration and Sound, Acoustical Society of America, Woodbury, N.Y. Morse, P. M., and Feshbach, H. (1953), Methods of Theoretical Physics, McGraw-Hill, N.Y. Morse, P. M., and Ingard, K. U. (1968), Theoretical Acoustics, Princeton University Press, Princeton, N.J. Pierce, A. D. (1989), Acoustics: An Introduction to its Physical Principles and Applications, Acoustical Society of America, Woodbury, N.Y. Temkin, S. (1981), Elements of Acoustics, Wiley, N.Y. Problems
1-1 Show the steps involved in linearizing Euler's equation of motion, as given in Equation (1.11). 1-2 A spherical surface of radius a is radially oscillating with normal velocity, -~. (a)
Solve for the acoustic field of the sphere by assuming a solution in the form 19(r) = A
eikr
and determining the constant A by matchingthe radial velocity on the surface of the sphere.
(b)
(c)
(d)
Determine the relationship between t3 and -;,.e r as r goes to infinity. Does the result agree with Equation (1.37) ? What is the surface impedance on the sphere, i.e., plot the real and imaginary part of the ratio of pressure/velocity on the surface of the sphere as a function ofka. Compare the power output from the source by integrating the time-averaged
CHAPTER 1: BASIC EQUATIONS OF ACOUSTICS 21
intensity over the surface of the sphere and over a spherical surface in the farfield. Do the results agree? 1--3'~ Compute the pressure field of the sphere using the Kirchhoff-Helmholtz equation by substituting the free-space Green's function for (] ( x / x s ). Perform a similar analysis for field points located within the sphere. 1-4 Show that the function R in Equation (1.68) can be approximated as given in Equation (1.69) when r >> r s. 1-5 Determine the low frequency limit for the acoustic field radiated by a cube with one face of the cube vibrating as a piston with amplitude ~, and the other faces of the cube quiescent.
Problems with an * are intended for advanced students.