Fundamentals and Basic Terminology
1.7 SOUND INTENSITY
Sound waves propagating through a fluid result in a transmission of energy. The time averaged rate at which the energy is transmitted is the acoustic intensity. This is a vector quantity, as it is associated with the direction in which the energy is being transmitted. This property makes sound intensity particularly useful in many acoustical applications.
The measurement of sound intensity is discussed in Section 3.13 and its use for the determination of sound power is discussed in Section 6.5. Other uses include identifying noise sources on items of equipment, measuring sound transmission loss of building partitions, measuring impedance and sound-absorbing properties of materials and evaluating flanking sound transmission in buildings. Here, discussion is restricted to general principles and definitions, and the introduction of the concepts of instantaneous intensity and time average intensity. The concept of time average intensity applies to all kinds of noise and for simplicity, where the context allows, will be referred to in this text as simply the intensity.
For the special case of sounds characterised by a single frequency or a very narrow-frequency band of noise, where either a unique or at least an approximate phase can be assigned to the particle velocity relative to the pressure fluctuations, the concept of instantaneous intensity allows extension and identification of an active component and a reactive component, which can be defined and given physical meaning. Reactive intensity is observed in the near field of sources (see Section 6.4), near reflecting surfaces and in standing wave fields. The time average of the reactive component of instantaneous intensity is zero, as the reactive component is a measure of the instantaneous stored energy in the field, which does not propagate. However, this extension is not possible for other than the cases stated. For the case of freely propagating sound; for example, in the far field of a source (see Section 6.4), the acoustic pressure and particle velocity are always in phase and the reactive intensity is identically zero in all cases.
1.7.1 Definitions
In the following analysis and throughout this book, vector quantities are represented as bold font. The subscript, 0, is used to represent an amplitude.
Sound intensity is a vector quantity determined as the product of sound pressure and the component of particle velocity in the direction of the intensity vector. It is a measure of the rate at which work is done on a conducting medium by an advancing sound wave and thus the rate of power transmission through a surface normal to the intensity vector. As the process of sound propagation is cyclic, so is the power transmission and consequently an instantaneous and a time-average intensity may be defined. However, in either case, the intensity is the product of pressure and particle velocity. For the case of single frequency sound, represented in complex notation, this has the meaning that intensity is computed as the product of like quantities; for example, both pressure and particle velocity must be real quantities.
Ii(r ,t) ' p(r ,t)u (r,t) (1.64)
I(r ) ' ¢p(r ,t)u (r ,t)¦ ' lim
T64 1 Tm
T
0
p(r ,t)u (r ,t) dt (1.65a,b)
u (r ,t) ' n j
ωρ Mp(r,t)
Mr ' n
ωρ &p0Mθp Mr %jMp0
Mr e j(ωt%θp(r)) (1.66a,b)
In (r ,t) ' & n
ωρ p02Mθp
Mrcos2(ωt%θp)%p0Mp0
Mr cos(ωt%θp) sin(ωt%θp) (1.67)
In (r ,t) ' & n
2ωρ p02Mθp
Mr 1%cos2(ωt%θp) %p0Mp0
Mr sin2 (ωt%θp) (1.68) The instantaneous sound intensity, Ii(r , t), in an acoustic field at a location given by the field vector, r , is a vector quantity describing the instantaneous acoustic power transmission per unit area in the direction of the vector particle velocity, u (r , t). The general expression for the instantaneous sound intensity is:
A general expression for the sound intensity I(r ), is the time average of the instantaneous intensity given by Equation (1.64). Referring to Equation (1.57), let f(t) be replaced with p(r , t) and g(t) be replaced with u(r , t), then the sound intensity may be written as follows:
Integration with respect to time of Equation (1.14), introducing the unit vector n = r/r, taking the gradient in the direction n and introduction of Equation (1.36c) gives the following result:
Substitution of the real parts of Equations (1.36b) and (1.37a) into Equation (1.64) gives the following result for the sound intensity in direction, n :
The first term in brackets on the right-hand side of Equation (1.67) is the product of the real part of the acoustic pressure and the in-phase component of the real part of the particle velocity and is defined as the active intensity. The second term on the right-hand side of the equation is the product of the real part of the acoustic pressure and the in-quadrature component of the real part of the particle velocity and is defined as the reactive intensity. The reactive intensity is a measure of the energy stored in the field during each cycle but is not transmitted.
Using well known trigonometric identities (Abramowitz and Stegun, 1965), Equation (1.67) may be rewritten as follows:
Equation (1.68) shows that both the active and the reactive components of the instantaneous intensity vary sinusoidally but the active component has a constant part.
Taking the time average of Equation (1.68) gives the following expression for the intensity:
I(r ) ' & n
2ωρp02Mθp
Mr (1.69)
In(r ,t) ' n p0u 0cos(ωt%θp) cos(ωt%θu) (1.70)
Ii(r ,t)' p0u 0
2 1%cos2 (ωt%θp) cos(θp&θu)%sin2 (ωt%θp) sin(θp&θu) (1.71)
I(r ) ' p0u 0
2 cos(θp&θu) ' 1
2Re{AB(} (1.72a,b)
Ir(r) ' p0u 0
2 sin(θp&θu) ' 1
2Im{AB(} (1.73a,b)
Equation (1.69) is a measure of the acoustic power transmission in the direction of the intensity vector.
Alternatively substitution of the real parts of Equations (1.36) and (1.37) into Equation (1.64) gives the instantaneous intensity:
Using well-known trigonometric identities (Abramowitz and Stegun, 1965), Equation (1.70) may be rewritten as follows:
Equation (1.71) is an alternative form to Equation (1.68). The first term on the right-hand side of the equation is the active intensity, which has a mean value given by the following equation:
The second term in Equation (1.71) is the reactive intensity, which has an amplitude given by the following equation (Fahy, 1995):
where the * indicates the complex conjugate (see Equations (1.36) and (1.37)).
1.7.2 Plane Wave and Far Field Intensity
Waves radiating outward, away from any source, tend to become planar.
Consequently, the equations derived in this section also apply in the far field of any source. For this purpose, the radius of curvature of an acoustic wave should be greater than about ten times the radiated wavelength.
For a propagating plane wave, the characteristic impedance ρc is a real quantity and thus, according to Equation (1.20), the acoustic pressure and particle velocity are in phase and consequently acoustic power is transmitted. The intensity is a vector quantity but where direction is understood the magnitude is of greater interest and will frequently find use throughout the rest of this book. Consequently, the intensity will be written in scalar form as a magnitude. If Equation (1.20) is used to replace u in
I ' ¢p2(r ,t)¦/ρc (1.74)
I ' ρc¢u 2(r ,t)¦ (1.75)
p
u ' ρce jβcosβ (1.76)
β ' tan&1[1 /(kr)] (1.77)
Equation (1.65a) the expression for the plane wave acoustic intensity at location r becomes:
In Equation (1.74) the intensity has been written in terms of the mean square pressure.
If Equation (1.20) is used to replace p in the expression for intensity, the following alternative form of the expression for the plane wave acoustic intensity is obtained:
where again the vector intensity has been written in scalar form as a magnitude. The mean square particle velocity is defined in a similar way as the mean square sound pressure.
1.7.3 Spherical Wave Intensity
If Equations (1.42) and (1.43) are substituted into Equation (1.65a) and use is made of Equation (5.2) (see Section 5.2.1) then Equation (1.74) is obtained, showing that the latter equation also holds for a spherical wave at any distance r from the source.
Alternatively, similar reasoning shows that Equation (1.75) is only true of a spherical wave at distances r from the source, which are large (see Section 1.4.3).
To simplify the notation to follow, the r dependence (dependence on location) and time dependence t of the quantities p and u will be assumed, and specific reference to these discrepancies will be omitted.
It is convenient to rewrite Equation (1.49) in terms of its magnitude and phase.
Carrying out the indicated algebra gives the following result:
where β ' (θp&θu) is the phase angle by which the acoustic pressure leads the particle velocity and is defined as:
Equation (1.71) gives the instantaneous intensity for the case considered here in terms of the pressure amplitude, p0, and particle velocity amplitude, u 0. Solving Equation (1.76) for the particle velocity in terms of the pressure shows that u 0 = p0/(ρc cos b). Substitution of this expression and Equation (1.77) into Equation (1.71) gives the following expression for the instantaneous intensity of a spherical wave, Isi(r, t):
Isi(r ,t) ' p02
2ρc 1%cos2 (ωt%θp) % 1
kr sin2 (ωt%θp) (1.78)
W ' m
S
I@n dS (1.79)
W ' 4πr2I (1.80)
Consideration of Equation (1.78) shows that the time average of the first term on the right-hand side is non-zero and is the same as that of a plane wave given by Equation (1.74), while the time average of the second term is zero and thus the second term is associated with the non-propagating reactive intensity. The second term tends to zero as the distance r from the source to observation point becomes large; that is, the second term is negligible in the far field of the source. On the other hand, the reactive intensity becomes quite large close to the source; this is a near field effect.
Integration over time of Equation (1.78), taking note that the integral of the second term is zero, gives the same expression for the intensity of a spherical wave as was obtained previously for a plane wave (see Equation (1.74)).