This section briefly presents the methodology applied throughout this thesis. As it was men-tioned in Chapter 1, the thesis focuses on two main aspects. On the one hand, specific design problems of labial and lingual organ pipes are addressed and novel procedures are introduced to overcome the limitations of current scaling methods. On the other hand, the methodology for modeling organ pipes is developed by extending some of the techniques summarized in Sec-tion 2.3. An approach that combines theory, numerical modeling, and validaSec-tion by means of measurements is chosen in both aspects.
Following the general methodology presented in Section 2.3.1, the techniques discussed in the thesis also rely on the separation of the sound generation mechanism into a nonlinear excitation and a linear acoustic resonator part. Although
resonator design
most of the cited literature concentrate more on the nonlinear excitation part and give less focus to the resonator, Chapters 5–8 of this thesis are dedicated entirely to the design issues of resonators of specific pipe families.
Most of the resonator forms investigated in this thesis are axisymmetric. In these cases, one-dimensional acoustic models (see Chapter 3) are applied in order to determine the characteristic properties of the resonators. The one-dimensional model is also applied for the optimization of the resonator geometry, e.g. in case of chimney pipes (see Chapter 5). Development
optimization of scaling
methods allowing sound design requires the usage of heuristic and unconstrained global opti-mization methods, such as the Nelder – Mead technique [109].
numerical techniques
Modeling irregularities, like tuning slots (see Chapter 7) or the radiation impedance from the open conical pipe end (see Chapter 8), involves the usage of numerical techniques, such as finite or boundary element methods (see Chapter 4). In order to reduce the size of the com-putational model and increase the flexibility of the simulations, postprocessing techniques for deriving equivalent parameters from the calculated acoustic fields are applied. Hence, the result-ing equivalent acoustic elements could be inserted into various one-dimensional models.
hybrid models
These so-calledhybrid models combine the accuracy attainable by three-dimensional models and the efficiency of one-dimensional techniques.
edge tone simulation
In Chapter 9 the simulation of the free air jet and edge tone in a pipe foot model is discussed.
The simulation is carried out by solving the Navier – Stokes set of equations numerically, by means of thefinite volume method. Three-dimensional flow models with over a million degrees of freedom involve the usage of highly parallelized simulation runs performed on a supercom-puter grid.
validation techniques
The validation of the results obtained using theoretical models or simulations is an important step of the methodology applied throughout the thesis. Results attained either by analytical or numerical techniques are validated by means of measurements, whenever it is possible. When the sound quality of certain pipe designs have to be assessed, comparative listening tests are performed with the help of experienced organ builders and voicers.
Chapter 3
One-dimensional modeling techniques in linear acoustics
Thischapter presents the background of unidimensional modeling techniques for the simulation introduction of acoustic wave propagation in ducts and pipes. Despite the limitations of one-dimensional
models, they are often applied in physics-based simulations of the sound generation of various musical instruments thanks to their simplicity and efficiency required by a computationally ef-fective (real-time or near real-time) simulation model. Furthermore, unidimensional models can be extended by results from three-dimensional simulations (see Chapter 4), resulting in hybrid models and overcoming some of the limitations of the unidimensional framework.
The objective the aim of
this chapter of this chapter is to derive the relations of one-dimensional models applied in
the subsequent chapters starting from the basic relations and clarifying the assumptions and ne-glects. Since the physical background and the derivation of the wave equation is covered in a large number of textbooks, this chapter does not intend to give a complete and detailed review, instead the reader is referred to the cited literature for further discussion. Nevertheless, the chap-ter aims to provide the reader with the background on which the following chapchap-ters are built.
The chapter focuses on the modeling of the propagation of pressure waves in cylindrical tubes, which is the simplest and the most important one-dimensional acoustical system for simulating the resonators of organ pipes.
Thischapter is structured as follows. Section 3.1 presents the governing equations of linear structure acoustics and derives the wave equation from the fundamental relations of continuum mechanics
and thermodynamical laws. The properties of the propagation of pressure waves in cylindrical ducts are discussed in Section 3.2. The theory of acoustic transmission lines and lumped parame-ter elements is presented in Section 3.3. Section 3.4 discusses the phenomenon of sound radiation from openings of different shapes. Finally, in Section 3.5, a simple model of a straight cylindrical labial pipe is assembled and examined.
3.1 The governing equations
This section derives the wave equation for ideal gases such as air. The discussion starts from the fundamental laws of continuum mechanics. Then, the wave equation is obtained by means of linearization and using the equation of state for adiabatic processes. Finally, the Helmholtz equa-tion is attained by transforming the wave equaequa-tion into the frequency domain. Our discussion follows the references [67], [84], [96], and [106].
19
3.1.1 Laws of continuum mechanics
The governing equations of linear acoustics are based on the basic laws of continuum mechan-ics. For the derivation of the fundamental equations the Eulerian approach [21] is used. In our discussion a domainΩin thed-dimensional space is consideredΩ⊆Rd. It is assumed that the domainΩis limited by the boundaryΓ. It is also assumed that the normal vectornpointsoutward ofΩin all points ofΓ.
Conservation of massmeans that no mass is generated or destroyed during the motion. Thus, the change of mass in the domainΩmust be equal to the amount of mass flowing into the region through the boundaryΓ. The total amount of mass is expressed as M = R
Ωρ(x, t) dx, with ρdenoting density,xandtrepresenting spatial and time coordinates, respectively. The rate of mass flow into the region is given as−R
Γρ(x, t)v(x, t)·n(x) dx, withvdenoting the flow velocity vector. Hence, the law of mass conservation is expressed as
∂M Changing the order of integration and differentiation, converting the right hand side into a vol-ume integral making of the divergence theorem and rearranging the equation yields
Z
In addition to the global validity of the conservation of mass we require that equation (3.2) is also valid for any arbitrarily small neighborhood of each material point (often referred to ascontrol volume), which implies the differential form of equation (3.2) as
conserva-tion of mass
∂ρ
∂t +∇ ·(ρv) = 0. (3.3)
The principle ofbalance of momentummeans that the time rate of change of momentum equals the resultant forcefRacting on the body. Denoting the momentum vector byPthe conservation law for the momentum is expressed as
∂P where the first term on the right hand side is the momentum flux through the boundaryΓ. Ap-plying the divergence theorem on the momentum flux, then expanding the resulting terms and substituting from equation (3.3) gives
The vector of the resultant force contains volume forcesband external forcesσ. By neglecting friction (assuming zero viscosity) external forces incorporate only pressure forcespnacting in the normal direction, and hencefRis expressed as
fR=
In acoustics, body forcesb—that incorporate the effect of gravity for example—are usually neg-ligible compared to the pressure forces and thus the resultant force is expressed by the pressure forces only, which can be transformed into a volume integral by means of the divergence theorem
fR=−
3.1. THE GOVERNING EQUATIONS 21 Substituting equation (3.7) into (3.5) leads to the integral form
Z
Similar to the conversation of mass (3.2), momentum is also required to be balanced in any ar-bitrarily small control volume. This leads to the local form of conservation of momentum, also known as theEuler equation, which reads as
Euler equation ρ∂v
∂t +ρ(v· ∇)v+∇p= 0. (3.9)
3.1.2 Linearization
Commonly, problems of linear acoustics refer to small perturbations of the ambient quantities.
The latter are denoted here by the subscript0. The time dependent total quantities are the super-position of the time independent ambient quantities and the small perturbations, with the latter denoted by˜·.
p(x, t) =p0+ ˜p(x, t) ρ(x, t) =ρ0+ ˜ρ(x, t) v(x, t) =v0+ ˜v(x, t).
(3.10)
It is also assumed that the ambient velocity of the fluid is zero, i.e.v0=0. After substituting and neglecting second order terms, the law of conservation of mass (3.3) reads as
linearized mass con-servation
∂ρ˜
∂t +ρ0∇ ·v˜= 0. (3.11)
Similarly, the Euler equation (3.9) is linearized and simplified as
linearized
In the linearized equations (3.11) and (3.12) there are three unknowns depending on the spa-tial and time coordinates: p˜,ρ˜andv˜. Therefore, in order to obtain a unified wave equation, a third relation is needed, which is discussed in the sequel.
3.1.3 The constitutive equation
In fluids sound propagates throughpressure wavesonly. Other fluctuations of the quantitiesp,ρ andv, caused by effects such as eddies—although detectable by acoustic sensors such as micro-phones or the human ear—are not regarded as sound, but perturbations out of the scope of the linear acoustic framework. In linear acoustics the fluctuations are considered to cause negligi-bly small heat flow and appear quickly, so that the temperature cannot equalize. Therefore the propagation of pressure waves can be characterized as anadiabatic process.
The propagation speed of pressure waves, also known as thespeed of sounddepends on the material properties of the fluid. Hence, the speed of sound is one of the characteristic properties of the fluid, which can be deduced from other characteristics of the material, depending on its state of matter. The speed of soundcis introduced as a constant relating the pressure and density fluctuations as
In this thesis only sound propagation in air is considered, thus, the speed of sound is only derived here forideal gases. To obtain the relation (3.13) our constitutive equation is the equation
of state, which reads for ideal gases in case of an adiabatic process as pρ−γ = const, with γ denoting the ratio of specific heats. Substituting from (3.10) and linearizing gives
c= rγp0
ρ0
. (3.14)
γcan be regarded as independent of the temperature in a relatively wide range, whereas the constantsp0andρ0 are interrelated with the ambient temperatureT0. The relation between the temperature, pressure and density is given by theideal gas law, that isp=ρRT. The individual gas constant for dry air isR = 287.058 J/kg K. The atmospheric pressure isp0 = 101 325 Pa. The dependence of the ambient density and the speed of sound on the temperature is also derived from the ideal gas law. If the temperature scale is relatively narrow, it is generally suffi-cient in engineering applications to assume linear dependence of the aforementioned quantities.
Using the results of (3.15) we get
ρ0= 1.2922·(1−0.0047·T0[◦C]) [kg/m3] and (3.16a) c= 331.45·(1 + 0.0018·T0[◦C]) [m/s], (3.16b) withT0[◦C]denoting the ambient temperature expressed in Celsius degrees.
3.1.4 The wave equation
In order to relate the equations (3.11) and (3.12), the constitutive equation (3.14) is used. The wave equation for linear acoustic problems is found by subtracting the time derivative of the linearized version of the conservation of mass (3.11) from the divergence of the linearized Euler equation (3.12). Thus, thehomogeneous wave equationfor linear acoustic problems is attained in the form
The operator ∇2 is usually referred to as the Laplace operator and is often denoted by∆. The wave equation (3.17) was derived using the pressure perturbations as the variable. Since equa-tion (3.14) relates pressure and density perturbaequa-tions, the same equaequa-tion is also valid for the density perturbationsρ.˜
The wave equation (3.17) is a partial differential equation (PDE), thus, in order to obtain its solution for a specific problem, boundary and initial conditions have to be specified. In the fol-lowing sections special cases of acoustic wave propagation problems are examined, or—in other words—the solutions of the acoustic wave equation are sought in specific configurations. Based on the three-dimensional wave equation (3.17) further, problem-specific relations are derived.
For the sake of convenience, the˜·notation of the time dependent perturbations is omitted in the following and the symbolsp,ρandvrefer to the perturbations unless otherwise noted.
3.1.5 The Helmholtz equation
In a significant number of cases we are interested in steady-state processes, where a frequency domain analysis is preferred for simplicity. In these cases time-harmonic perturbations with an
3.2. WAVE PROPAGATION IN CYLINDRICAL DUCTS 23