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.
3.3 Acoustic circuits and transmission lines
In the previous sections, only simple systems consisting of one or two elements were examined.
For such simple systems it was straightforward to infer the proper boundary conditions and the relations describing the interactions of the elements of the system. However, when having more complex systems at hand, it is useful to describe and visualize the systems by means of acoustic circuits, in which the connections relating the elements of the system can easily be followed.
This section introduces the concentrated and distibuted parameter elements of acoustic cir-cuits. Theelectrical–mechanical–acoustical analogiesare useful for finding the correspondence be-tween circuits of the three types. The transfer matrix description of acoustic transmission lines is of great potential when systems consisting of multiple waveguide elements are considered. The theory of acoustic circuits is an essential tool for the analysis of acoustic transmission lines, such as the air columns of wind instruments of various kinds.
3.3. ACOUSTIC CIRCUITS AND TRANSMISSION LINES 29
3.3.1 Lumped two-pole elements
Lumped or concentrated parameter two-pole elements of a circuit can be described by a single parameter, the impedanceZ(ω)that is the ratio of pressure difference between the poles and the acoustic volume velocity flowing through the element. The three basic types of the lumped two-pole elements are introduced in the sequel.
The simplest lumped acoustic element is a short tube of lengthLwith both ends open. The impedance of the tube is found under the assumption that the air inside it moves without com-pression, as a rigid piston. This gives
impedance
The same result is also obtained as a first order approximation of (3.45) as tankL ≈ kL with kLπ/2. The acoustic impedance of such small tube is analogous to the mechanical impedance of a massM =ρ0SL, and is similar to an inductance in an electrical circuit. Based on this analogy a small tube can be referred to as anacoustical inductance.
The second element that can be described by a lumped parameter is a small cavity of volume V. The change of the volume−δV occupied by the air inside the cavity as a response to a pressure perturbationpis found as
p=−BδV
V with B=γp0=ρ0c2. (3.49)
B is known as thebulk modulusof air. Due to the principle of continuity, the change of volume δV is expressed as the time integral of volume flow into the cavity, and the acoustic impedance of the cavity is obtained as (see e.g. [67, pp. 152–153])
impedance
with substituting V = SL. It is seen that (3.50) is the first order approximation of (3.46) as cotkL ≈ (kL)−1with kL π/2. The acoustic impedance of such a cavity is analogous to the mechanical impedance of a spring with complianceC =V /ρ0c2. The corresponding element of an electrical circuit is a capacitance, thus, a small cavity is anacoustical capacitance.
The third element, the acoustical resistanceis modeled as acoustic flow throgh a plate with small perforations. It is assumed this time that the radius of the perforationsapare smaller than the boundary layer thicknesses, that isap < δvandap < δt. Then, the motion is dominated by viscothermal losses and the volume velocity perturbations are in phase with that of the pressure.
The acoustic impedance of this element is given as
acoustical resistance
Zresist=R, (3.51)
withRdenoting the value of the acoustical resistance depending on the number and size of the aperture channels.
Using a combination of these three elements, a lumped element with arbitrary impedance can
be constructed. When limits of
applicability an acoustic circuit is constructed from a physical system care must be taken
at the range of applicability of concentrated parameter elements. It was discussed above that the impedances of acoustical inductances and capacitances are the first order approximations of an open and a closed tube, respectively. Therefore, the error of the approximation depends on the ratio of the (characteristic) length of the element and the wavelength,L/λ. As a rule of thumb, L/λ <1/8is taken usually (see e.g. [27, part XIII]) as the upper frequency limit of regarding tubes or cavities as lumped elements.
3.3.2 Acoustic transmission lines
Contrary to lumped elements, transmission lines can not be characterized by a sole parameter, therefore they are often referred to asdistributed parameter orwaveguideelements. Such an ele-ment, for example, is a duct with a length comparable to the wavelength. Transmission lines will be the essential building blocks of the acoustic circuits utilized for modeling organ pipes. Trans-fer quantities of acoustic circuits are often of special interest in musical acoustic applications. In these cases the so-calledtransfer matrix descriptionis a useful tool for characterizing the system.
The transfer matrix approach
Generally, the acoustic transfer matrix of a two-port (four-pole) circuit is defined as the matrix relating the input and output sound pressures and volume velocities of the system, that is transfer
Note that the matrix elements have different dimensions in the above description. The quantities T11andT22are nondimensional, whereasT12has impedance dimension andT21has admittance dimension. In general, the transmission matrixT(ω)is dependent on the frequency.
The greatest advantage of the transfer matrix approach is that systems with different transfer matrices can easily be joined connecting the input port of one system to the output system of the other. AssumingN systems connected in a serial manner and denoting the transfer matrix of the ith system byTi, the complete system is characterized as
pˆin
The total transfer matrix is then expressed asTtot=QTi.
Another quantity of our interest is usually the transfer impedance or admittance of the sys-tem, relating one of the input quantities with the other output quantity. If the termination impedanceZLis known, the transfer quantities or the input impedance can readily be expressed from the transfer matrix description (3.52).
The transfer matrix of a cylindrical duct2
The transfer matrix of a cylindrical duct can be calculated based on the fundamental laws of mass and momentum conservation. Taking the one-dimensional versions of equations (3.11) and (3.12), making use of the constitutive equation (3.13) we get
1
Introducing the quantitiesZandY as Z =jωρ0
S and Y = jωS
ρ0c2, (3.55)
multiplying byS, and usingUˆ =Sˆvwe get the system of linear differential equations
∂
2This section is written based on S. Adachi:Personal letter to the author(2010).
3.4. SOUND RADIATION FROM OPEN TUBES 31 be applied, with the definitions given in equations (3.38) and (3.39). If a termination impedance ZL(ω)is assumed atz=z2, the input impedance of the tube is found as
Zin(ω) = p(zˆ 1)
U(zˆ 1) =ZL(ω)T11+T12
ZL(ω)T21+T22 =Z00ZL(ω) cosk0L+ jZ00sink0L
jZL(ω) sink0L+Z00cosk0L, (3.58) withTijdenoting the elements of the transfer matrix in equation (3.57). As expected, the result is identical to the relation found in (3.44).
3.4 Sound radiation from open tubes
While infinite termination impedance is a reasonable approximation of a closed pipe, assuming zero termination impedance is a quite rough and inaccurate approximation for an open pipe end. In this section radiation impedances of different openings of a cylindrical organ pipe are examined. The radiation impedance from the open end, the mouth, and other irregular openings are discussed.
3.4.1 Flanged and unflanged pipe ends
Radiation by a rigid piston in an infinite baffle can be calculated analytically by means of the Rayleigh integral(see e.g. [61, pp. 122–124]). The latter gives the pressure field over an infinite plane as
whereG(x,y)denotes theGreen’s functionfor the Helmholtz equation with a Dirac-delta source δ(x)located at the pointy:
∇2G(x,y) +k2G(x,y) =−δ(x−y). (3.60) The Green’s function in three dimensions is given as
G(x,y) = e−jk|x−y|
4π|x−y|. (3.61)
The acousticalradiation impedanceof the piston is found by evaluating the Rayleigh integral for the piston of radiusamoving with unit velocity (see [116, p. 92])
radiation
withH1denoting the first order Struve function of the first kind (see [122, pp. 288–289]) and the superscript (pi) referring to the piston case. When radiation from a pipe is examined, it is often useful to regard the imaginary part of the radiation impedance as alength correction, i.e. an extra length increasing the effective length of the given tube compared to an ideally open (ZR = 0)