p(x, t) =|p(x)|cos (ωt+φ(x)) =Ren
|p(x)|ej(ωt+φ(x))o
. (3.18)
It is useful to introduce the complex amplitudep(x) =ˆ |p(x)|ejφ(x) that inherently contains the amplitude and the phase of the time-harmonic perturbations.
For a non time-harmonic signal, frequency dependent complex amplitudes p(x, ω)ˆ can be introduced by using the Fourier transformed variablesˆ·that satisfy the relation
inverse
The Fourier transformed variables can then be attained as
Fourier
if the corresponding integral exsists.1
Making use of the interchangeability of the second derivative with respect to time ∂2/∂t2 and the factor(jω)2for the Fourier transformed variables in equation (3.17) and introducing the acoustic wave numberk=ω/c, thehomogeneous Helmholtz equationis obtained as
Helmholtz equation
∇2p(x, ω) +ˆ k2p(x, ω) = 0.ˆ (3.21) The operatorH=∇2+k2is often referred to as theHelmholtz operator.
3.2 Wave propagation in cylindrical ducts
In this section the Helmholtz equation (3.21) is applied for solving acoustic wave propagation problems in cylindrical ducts. Since a cylindrical duct with perfectly rigid walls is the simplest model of a straight cylindrical organ pipe, cylindrical ducts are the essential elements of the acoustic model of the resonator of an organ pipe. First, an infinite cylindrical duct is examined, then the properties of finite pipes are discussed.
3.2.1 The infinite cylindrical duct
Let us assume an infinite cylindrical duct of inner radiusawith its axis of symmetry located at thez-axis of the cartesian coordinate system. It is useful to describe the problem in cylindrical coordinatesr= (r, θ, z), withr=p
x2+y2andθ= tan−1(y/x). The problem domain is defined asΩ = {r|r≤a, θ∈[0,2π), z∈R}, whereas the boundary is given asΓ = {r|r=a}. The Helmholtz equation in cylindrical coordinates—omitting the notation of the dependence of the pressure variable on space and angular frequency for simplicity—reads as
∂2 It is assumed for the time being that the walls of the duct are perfectly rigid, which means that the normal component of the particle velocityvnvanishes atr=a. Making use of the linearized Euler equation (3.12) this boundary condition is stated as
vn(a, θ, z) =− 1
1The existence of the integral in (3.20) is not required for the existence ofy(x, ω), see [34].ˆ
The general solution of the boundary value problem posed by the PDE (3.22) and the Neu-mann type boundary condition (3.23) reads as
ˆ withP standing for an arbitrary pressure amplitude,Jmdenoting themth order Bessel function of the first kind, φmnbeing an arbitrary angle andqmn defined to satisfy the boundary condi-tion (3.23), such that the derivativeJm0 (πqmn)is zero. The propagation wave numberkmnfor mode(m, n)is obtained by substituting equation (3.24) into (3.22) as
kmn2 =ω
The plane wave mode withm =n = 0can always propagate with the wave numberk00 = ω/c, whereas higher modes can only propagate above a certain angular frequency, often referred to as a cutoff angular frequency
ωcut,mn= πqmnc
a . (3.26)
Under its cutoff frequency the wave numberkmn is imaginary, and from equation (3.24) it is seen that the solution decays exponentially withz. The cutoff frequency of the first transversal mode(1,0)is of special interest both theoretically and practically. The corresponding coefficient is found as the first positive root of the Bessel functionJ10(πq10)givingπq10 ≈1.8412. This first cutoff frequency is often simply referred to as thecutoff frequency, and is given as
cutoff
frequency ωcut= 1.8412·c
a ⇐⇒ (ka)cut= 1.8412, (3.27)
with(ka)cutdenoting thenondimensional cutoff frequency.
Under the cutoff frequency of the first transversal modeωcutonly the plane wave mode can propagate inside the duct, and if the evanescent components can be neglected—which is the case in most applications—the problem can be considered one-dimensional. Then the model is significantly simplified and the one-dimensional Helmholtz equation can be utilized for its description, that is
1D Helm-holtz eq.
∂2p(z, ω)ˆ
∂z2 +k2p(z, ω) = 0ˆ r∈Ω. (3.28) The solution of (3.28) is given in the so-called d’Alembert form as
ˆ
p(z, ω) =p+e−jkz+p−ejkz, (3.29) with p+ andp− denoting arbitrarily chosen complex amplitudes. Similarly, utilizing the one-dimensional linearized Euler equation, i.e.jωρ0v=−∂p/∂z, the particle velocity is given as
ˆ denoted byZ0(spec) =ρ0cis known as thespecific plane wave impedanceand is measured in units kg/m2s. By using thevolume velocityU = vS (S denoting the cross sectional area of the duct), theacoustic plane wave impedancerelates the pressure and volume velocity amplitudes in a similar manner asZ0(acou)=ρ0c/Sand is measured in unitskg/m4s, also known asacoustic ohms.
3.2. WAVE PROPAGATION IN CYLINDRICAL DUCTS 25
3.2.2 Lossy wave propagation
The wave equation (3.17) and the Helmholtz equation (3.21) were derived above assuming ideal materials without any sources of internal losses. In reality, however, no material can be consid-ered ideal, thus, the propagation of acoustic waves in all types of material is inherently lossy.
The losses are conveniently quantified by theabsorption coefficient, denoted byα, which affects the propagation wave numberkby introducing the lossy wave numberk0as
lossy wave number
k0=k−jα. (3.31)
Assuming thatαis positive it can be seen that the solution to the one-dimensional Helmholtz equation (3.28) becomes the superposition of waves with exponentially decaying amplitude along the direction of propagation. The sources of losses can be classified as follows.
Intrinsic losses are characteristic to the medium in which sound propagates. These losses may be subdivided into three basic types. (1) Friction losses occur whenever there is relative motion between adjacent control volumes of the fluid, i.e. during shear deformations or compressions and expansions that accompany the transmission of a sound wave. (2)Heat conduction lossesare present whenever thermal energy is conducted between regions of dif-ferent temperatures. (3)Molecular lossesrefer to processes in which the kinetic energy of the molecules is converted into potential, rotational, or vibrational energy.
Boundary losses occur at the interface of the fluid and its solid boundaries have two basic types.
(4) Viscous lossesare due to the friction between the boundary and the fluid. (5)Thermal lossesarise from heat transfer between the wall and the fluid.
The intrinsic losses due to viscosity (1) and thermal conduction (2) are incorporated by the classical absorption coefficientαc, given as
classical absorption coefficient αc= ω2η
2ρ0c3 4
3 +γ−1 Pr
, (3.32)
withηrepresenting the dynamic viscosity of the fluid,Prdenoting the Prandtl number, defined asPr = ηcP/κ, wherecP is the specific heat at constant pressure andκstands for the thermal conductivity of the fluid. In dry air at atmospheric pressure and the temperature ofT0 = 20◦C the classical absorption coefficient isαc/f2≈1.4·10−11m−1s2.
The deduction of equation (3.32) is found in [84, chapter 8]. In the applications presented in this thesis boundary losses are dominant over intrinsic losses, therefore they are examined in more detail in Section 3.2.3.
3.2.3 Viscous and thermal losses at rigid walls
In the previous section we have assumed that the wall is completely rigid and hence the normal derivative of the pressure in the radial direction is zero atr=a. In real life conditions this can never be achieved, however, in lot of practical cases the vibration of the walls can be neglected.
There are no real walls immune to viscous and thermal effects, which makes the latter impor-tant in practical applications, such as modeling the walls of wind instruments. In the following paragraphs the viscous and thermal losses at rigid walls are briefly introduced in order to finally attain the wall absorption coefficient that characterizes wall losses in a unified manner.
Viscous boundary layer
If friction is present between the fluid and the walls, a so-calledno-slip boundary conditionshould be prescribed at the walls, meaning that the difference of the velocity of the wall and the particle
velocity of the fluid at the wall vanishes. For a non-moving wall this means that the particle velocity is zero at the walls, i.e.v =0ifr =a. To fulfill the no-slip condition the total velocity fieldvis written as a superposition of aprimary plane wavevzand asecondary wavev0, so that
v(r, z, t) =vz(z, t) +v0(r, z, t). (3.33) By solving the Navier – Stokes equation with the no-slip condition after some simplifications (see [84, chapter 8]) the secondary wave is obtained as
viscous
skin depth v0=−vze−(1+j)(a−r)/δv with δv= r 2η
ρ0ω. (3.34)
δvdenotes theviscous boundary layer thickness, also known as the viscous skin depth.
Thermal conduction
The temperature field corresponding to a planar wave given by the particle velocityvz(z, t) = v+e−j(kz−ωt)is given as
T(z, t) =T0+T0(γ−1)v+
c e−j(kz−ωt). (3.35)
With the assumption of an isothermal wall, analogous to the viscous boundary layer, the temper-ature field is also found as the superposition of a primary and a secondary wave. After solving the diffusion equation, the skin depth of the thermal boundary layerδtis found as
thermal
skin depth δt=
r 2κ
cPρ0ω. (3.36)
The wall absorption coefficient
For the deduction of the wall absorption coefficientαwthe nondimensional ratiosrvandrt are introduced as
By means of the boundary layer thicknesses the series impedanceZvand shunt admittanceYtof a tube section of unit length can be given (see e.g. [25]) as
Zv= jωρ0
S 1
1−Fv and Yt= j ωS
ρ0c2[1 + (γ−1)Ft]. (3.38) The complex valued parametersFvandFtare defined as
F∗= 2 r∗j3/2
J1 r∗j3/2
J0 r∗j3/2, (3.39)
with usingrvandrtasr∗for calculatingFvandFt, respectively.
The viscous and thermal skin depths in air at atmospheric pressure and T0 = 20◦Cat the frequency of f = 20 Hzare obtained asδv ≈ 4.9×10−4m andδt ≈ 5.8×10−4m. Therefore, in the case of cylindrical ducts of real musical instruments, such as organ pipes, it is generally valid to assumeaδvandaδt. With this assumption thewall absorption coefficientαwcan be expressed using the asymptotic approximation of Benade [25] (see also [67, chapter 8])
wall
3.2. WAVE PROPAGATION IN CYLINDRICAL DUCTS 27
Figure 3.1.Frequency dependence of coefficients of intrinsic (αc) and wall (αw) losses in cylindrical ducts at temperatureT0= 20◦C
Finally, the total absorption coefficientαis obtained as the sum of the classical and the wall
absorption coefficients as total
absorption coefficient
α=αc+αw. (3.41)
For the example of a typical cylindrical organ pipe in an 8’, 4’, or 2’ stop, having a fundamental frequency of roughly65to260 Hz,ais in the range of centimeters. In this case, it can be seen by evaluating the expressions given in equations (3.32) and (3.40) that the wall losses are dominant over the intrinsic losses in the whole frequency range of musical relevance (i.e. where the strong harmonic components are present), as illustrated by Figure 3.1. Therefore, it is reasonable to neglect intrinsic losses in the applications of this thesis.
A more detailed elaboration of intrinsic and wall propagation losses can be found in the books by Zwikker & Kosten [154] or Kinsleret al.[84].
3.2.4 Finite cylindrical pipes
Obviously, the pipes that appear in real musical instruments are of finite length. This means that pressure waves are reflected at the (open or closed) ends of the duct. A property of key importance for the characterization of such ducts is theinput impedance, introduced in the sequel.
Let us consider a finite cylindrical tube that extends from z = 0to z = Lterminated by the acoustic impedanceZL(ω)at z = L. Since the unidimensional Helmholtz equation (3.28) is valid inside the pipe under the cutoff frequency, the resulting pressure field of the pipe is the superposition of two counterpropagating planar waves, as obtained by the d’Alembert form solution (3.29).
The reflection coefficient at the far end (z=L) of the tubeRL(ω)is expressed as the ratio of the complex amplitudes of the counterpropagating plane waves at the locationz =L. Making use of the fact thatp(L, ω)/ˆ Uˆ(L, ω) =ZL(ω), it is found that
reflection coefficient RL(ω) =p−e+jkL
p+e−jkL =ZL(ω)−Z0
ZL(ω) +Z0 (3.42)
withZ0 = ρ0c/S denoting the acoustic plane wave impedance of the tube. Consequently, the ratio of the complex amplitudesp−andp+is attained as
p−
p+ =RL(ω)e−2jkL. (3.43)
The input impedance of the pipeZinis defined as the ratio of the complex amplitude of the pressure and volume velocity atz= 0. This can be expressed by rewriting the complex
exponen-tials using trigonometric functions to get input
impedance Zin(ω) = p(0, ω)ˆ
Uˆ(0, ω) =Z0
ZL(ω) + jZ0tankL
Z0+ jZL(ω) tankL. (3.44) For the characterization of an acoustic system the input impedance is of key importance. Its reciprocal, theinput admittance Yin = 1/Zin is also often used for the same purpose. Since the system responds with maximal pressure to unit input volume velocity whenZin(ω)has a local maximum, the natural resonance and anti-resonance frequencies of the system can also be found as the local extrema of the input impedance or input admittance functions.
Two extremal cases regarding the termination impedanceZLare of special interest. The first case is when the pipe is terminated by zero impedance. This corresponds to zero pressure at the pipe end,p(L, ω) = 0. In this case the system can be consideredideally openand from equation (3.44) the input impedance is attained as
Zin(open)= jZ0tankL. (3.45)
The second case corresponds to the pipe being terminated by a rigid wall, meaning thatZL →
∞and implying zero volume velocity at z = L. In this case the system is considered ideally closed(often referred to asstopped) and its input impedance can be expressed by taking the limit ZL→ ∞of equation (3.44) to get
Zin(closed)=−jZ0cotkL. (3.46)
Let us assume a simple model of a labial organ pipe. It was discussed in Chapter 2 that the air jet drives the pipe at the pipe mouth. The driving jet produces a pressure fluctuation on the air inside the tube. The frequencies of natural resonance of the pipe are found where the pressure oscillations induce maximal response inside the tube (see [105, chapter 23]), i.e. when the acoustic input impedance of the pipeZinis minimal. Hence thefrequencies of natural resonancefn =c/λn
of the system are found by the corresponding wavelengthsλnas
λn=
4L
2n if ZL= 0 4L
2n−1 if ZL→ ∞
n= 1,2, . . . (3.47)
Therefore, the cylindrical pipe open at both ends acts as a half-wave resonator, whereas the pipe with one end open and the other end closed is a quarter-wave resonator.