Fundamentals and Basic Terminology
1.4 WAVE EQUATION
1.4.2 Plane Wave Propagation
For the case of plane wave propagation, only one spatial dimension, x, the direction of propagation is required to describe the acoustic field. An example of plane wave propagation is sound propagating along the centre line of a rigid wall tube. In this case, Equation (1.15) written in terms of the potential function, φ, reduces to:
A solution for Equation (1.16), which may be verified by direct substitution, is:
u ' Kf)(ct ± x) (1.18)
p ' ρcf)(ct ± x) (1.19)
p/u ' ±ρc (1.20)
φ ' Acos(k(ct±x) % β) (1.21)
kct % β ' 0 (1.22)
φ ' Acoskx ' Acos(2πx/λ) (1.23a,b)
The function, f, in Equation (1.17) describes a distribution along the x axis at any fixed time, t, as well as the variation with time at any fixed place, x, along the direction of propagation. If the argument (ct ± x) is fixed and the positive sign is chosen then with increasing time, t, x, must decrease with speed, c. Alternatively, if the argument (ct ± x) is fixed and the negative sign is chosen then with increasing time, t, x, must increase with speed, c. Consequently, a wave travelling in the positive x direction is represented by taking the negative sign and a wave travelling in the negative x direction is represented by taking the positive sign in the argument of Equation (1.17).
A very important relationship between acoustic pressure and particle velocity will now be determined. A prime sign, N, will indicate differentiation of a function by its argument as for example, df(w)/dw = fN(w). Substitution of Equation (1.17) in Equation (1.10) gives Equation (1.18) and substitution in Equation (1.11) gives Equation (1.19) as follows:
Division of Equation (1.19) by Equation (1.18) gives a very important result, the characteristic impedance, ρc, of a plane wave:
In Equation (1.20), the positive sign is taken for positive travelling waves, while the negative sign is taken for negative travelling waves. The characteristic impedance is one of three kinds of impedance used in acoustics. It provides a very useful relationship between acoustic pressure and particle velocity in a plane wave. It also has the property that a duct terminated in its characteristic impedance will respond as an infinite duct as no wave will be reflected at its termination.
Fourier analysis enables the representation of any function, f (ct ± x), as a sum or integral of harmonic functions. Thus it will be useful for consideration of the wave equation to investigate the special properties of harmonic solutions. Consideration will begin with the following harmonic solution for the acoustic potential function, where k is a constant, which will be investigated, and β is an arbitrary constant representing an arbitrary relative phase:
In Equation (1.21) as β is arbitrary, for fixed time β may be chosen so that:
In this case, Equation (1.17) reduces to the following representation of the spatial distribution:
2π/λ ' k (1.24)
β ± kx ' 0 (1.25)
φ ' Acoskct ' Acos 2π
T t (1.26a,b)
2π/kc ' T (1.27)
(a)
(b)
+ +
wavelength acoustic
pressure
patm pmax
Figure 1.2 Representation of a sound wave: (a) compressions and rarefactions of a sound wave in space at a fixed instance in time;
(b) graphical representation of sound pressure variation.
From Equations (1.23) it may be concluded that the unit of length, λ, defined as the wavelength of the propagating wave and the constant, k, defined as the wave number are related as follows:
An example of harmonic (single frequency) plane wave propagation in a tube is illustrated in Figure 1.2. The type of wave generated is longitudinal, as shown in Figure 1.2(a) and the corresponding pressure fluctuations as a function of time are shown in Figure 1.2(b).
The distribution in space has been considered and now the distribution in time for a fixed point in space will be considered. The arbitrary phase constant, β, of Equation (1.17) will be chosen so that, for fixed position x:
Equation (1.21) then reduces to the following representation for the temporal distribution:
The unit of time, T is defined as the period of the propagating wave:
2π/T ' 2πf ' ω (1.28a,b)
k ' ω/c (1.29)
fλ ' c (1.30)
u ' ± B
ρcsin (ωtKkx%β) (1.31)
p ' ±Bsin(ωtKkx%β) (1.32)
cg ' dω/dk (1.33)
10 20 20
50 100 200 500 1000 2000 5000 10000 20000
aud ble frequency (Hz) wavelength (m)
5 2 1 0.5 0.2 0.1 0.05 0.02
Figure 1.3 Wavelength in air versus frequency under normal conditions.
Its reciprocal is the more familiar frequency, f. Since the angular frequency, ω, is quite often used as well, the following relations should be noted:
and from Equations (1.27) and (1.28b):
and from Equations (1.24), (1.28b), and (1.29):
The relationship between wavelength and frequency is illustrated in Figure 1.3.
Note that the wavelength of audible sound varies by a factor of about one thousand. The shortest audible wavelength is 17 mm (corresponding to 2000 Hz) and the longest is 17 m (corresponding to 20 Hz). Letting A = B/ρω in Equation (1.21) and use of Equation (1.29) and either (1.10) or (1.11) gives the following useful expressions for the particle velocity and the acoustic pressure respectively for a plane wave:
The wavenumber, k, may be thought of as a spatial frequency, where k is the analog of frequency, f, and wavelength, λ, is the analog of the period, T. It may be mentioned in passing that the group speed, briefly introduced in Section 1.3.5, has the following form:
By differentiating Equation (1.29) with respect to wavenumber k, it may be concluded that for non-dispersive wave propagation where the wave speed is
φ ' Aej(ωt±kx%β) ' Acos(ωt±kx%β) % jAsin(ωt±kx%β) (1.34a,b)
p(r,t) ' p0(r ) e jk(ct%*r*%θ/k) ' p0(r ) e j(ωt%θp(r)) ' Ae jωt (1.36a-c)
u (r ,t) ' u 0(r ) e j(ωt%θu(r)) ' Be jωt (1.37a,b) Ae jωt ' A(cosωt%jsinωt) (1.35) independent of frequency, as for longitudinal compressional waves in unbounded media, the phase and group speeds are equal. Thus, in the case of longitudinal waves propagating in unbounded media, the rate of acoustic energy transport is the same as the speed of sound, as earlier stated.
A convenient form of harmonic solution for the wave equation is the complex solution written in either one or the other of following equivalent forms:
In either form the negative sign represents a wave travelling in the positive x-direction, while the positive sign represents a wave travelling in the negative x-direction.
The real parts of Equations (1.34) are just the solutions given by Equation (1.21).
The imaginary parts of Equations (1.34) are also solutions, but in quadrature (90Eout of phase) with the former solutions. By convention, the complex notation is defined so that what is measured with an instrument corresponds to the real part; the imaginary part is then inferred from the real part. The complex exponential form of the harmonic solution to the wave equation is used as a mathematical convenience, as it greatly simplifies mathematical manipulations, allows waves with different phases to be added together easily and allows graphical representation of the solution as a rotating vector in the complex plane. Setting β = 0 and x = 0, allows Equation (1.34a,b) to be rewritten in the following simplified useful form:
Equation (1.35) represents harmonic motion that may be represented at any time, t, as a rotating vector of constant magnitude A, and constant angular velocity, ω, as illustrated in Figure1.4. Referring to the figure, the projection of the rotating vector on the abscissa, x-axis, is given by the real term on the RHS of Equation (1.35) and the projection of the rotating vector on the ordinate, y-axis, is given by the imaginary term.
For the special case of single frequency sound, complex notation may be introduced. For example, the acoustic pressure of amplitude, p0, and the particle velocity of amplitude, u 0, may then be written in the following general form where the wavenumber, k, is given by Equation (1.25):
and
where A and B are complex numbers.
y t( ) = sinωX t
xtXt() = cosω
X
X
0
0
-X
-X
π
π
2π
2π
3π 4π
3π4π
ωt
ωt
xt()
y t( )
X x ωIm y
Re ωt 0
Figure 1.4 Harmonic motions represented as a rotating vector.
In writing Equations (1.36b) and (1.37a) it has been assumed that the origin of vector r is at the source centre and θp and θu are, respectively, the phases of the pressure and particle velocity and are functions of location r .
Use of the complex form of the solution makes integration and differentiation particularly simple. Also, impedances are conveniently described using this notation.
For these reasons, the complex notation will be used throughout this book. However, care must be taken in the use of the complex notation when multiplying one function by another. In the calculation of products of quantities expressed in complex notation it is important to remember that the product implies that only like quantities are multiplied. In general, the real parts of the quantities are multiplied. This is important, for example, in the calculation of intensity associated with single frequency sound fields expressed in complex notation.