J. Redondo^{a},
R. Pico^{a} and B. Roig^{b}

^{ }

^{a }*Grupo de Dispositivos y Sistemas
Acústicos y Ópticos, DISAO. Escuela Politécnica Superior
de Gandía Carretera Nazaret Oliva S/N, Grao de Gandia, **Spain** fredondo@fis.upv.es
*

^{b}* Departamento de Matemática
Aplicada. Escuela Politécnica **Superior** de Gandia. Carretera Nazaret Oliva
S/N, Grao de Gandia, Spain*

**RESUMEN:** En el presente trabajo se estudia la evolución temporal de la
propagación de las ondas sonoras en el interior de un instrumento de
viento mediante el método de diferencias finitas en el dominio temporal
(FDTD). El modelo de instrumento adoptado se compone de un resonador con
simetría axial (cuerpo del instrumento) excitado por medio de pulsos
periódicos de aire. Para simular el mecanismo de excitación del
instrumento se emplean fuentes transparentes en la entrada del resonador. La técnica
de cálculo empleada en el método de diferencias finitas reduce
las oscilaciones espureas ya que utiliza dos mallas al tresbolillo. Esta
técnica ha sido utilizada en las últimas décadas en otros
ámbitos de la Acústica, tales como la Acústica de Salas
[1] o la Acústica Ambiental [2]. La simulación numérica
del instrumento permite analizar la influencia de la geometría interna
(agujeros, forma de la terminación) en la propagación sonora del
instrumento, así como, la evolución temporal del régimen
de oscilación.

**ABSTRACT:** In the present work the time evolution of the
travelling sound waves inside a wind instrument is studied by means of a
Finite-Difference Time Domain (FDTD) method. The adopted model for the
instrument consists of a resonator with axial symmetry (body of the instrument)
excited by periodic air pulses. In order to simulate the excitation mechanism
transparent sources should be used at the input of the resonator. The considered calculation technique
reduces the spurious oscillations due to the use of staggered grids. This
technique has been successfully applied in the last decades in several fields
within Acoustics, such as Room Acoustics [1] or Environmental Acoustics [2].
The numerical simulation of the instrument makes possible to analyze the
influence of the internal geometry (orifice, termination shape, and so on) on
the sound propagation and the time evolution of the oscillation regime.

In the field of woodwind acoustics, different
numerical methods have been employed to simulate the behaviour of acoustical
bores. Some of this works are applied in sound synthesis by physical modelling
of woodwind, such as digital waveguide modelling (DWM) [3] and the multi
convolution algorithm (MCA) [4] or the wave digital modelling (WDM) [5]. The approach is based on a
one-dimensional model of wave propagation in the bore. In this work, the
Finite-difference time-domain method (FDTD) is used to evaluate the acoustic
response of a simplified instrument consisting of a cylindrical pipe excited at
the entrance.

**2. TIME DOMAIN SIMULATION
USING FINITE DIFFERENCE**

The finite-difference time-domain (FDTD) method was
first introduced by Yee in 1966 [6] in order to study the scattering of
electromagnetic waves. Since that moment a counterpart for acoustics has been
developed in several fields, mainly in room acoustics. In the case of musical
instruments authors have mainly use different numeric techniques like Finite
Element Methods (FEM) and Boundary Element Methods (BEM). In most of the cases
frequency domain methods were considered. In last decade some extensions of
those methods to the time domains has been used in order to analyze transient
states.

The first order acoustic
equations in a homogeneous medium without losses can be written as follows,

_{} (1)

_{} (2)

where *p* is the pressure field, _{} is the vector particle
velocity field, _{} is the mass density of the
medium and _{} is the compressibility of
the medium.

In the case of axial symmetric
sound fields, equations (1) and (2) can be written in cylindrical coordinates
as follows,

_{} (3)

_{} (4)

_{} (5)

The spatial and time
derivatives of any function can be approximated by central finite difference.
For instance,

_{} (6)

where D*r* is the spatial interval between considered points.
Defining three staggered grids (see figure 1) and after some algebra equations
(3)-(5) can be expressed as follows:

_{} (7)

_{} (8)

_{} (9)

Figure 1 – |
So, the pressure, |

Assuming a locally reactive boundary, the particle velocity
on the boundary can be expressed as follows:

_{} (10)

where _{}** **is the vector normal to the
boundary and Z is the acoustic impedance of the boundary. As an example we will
express the boundary condition in the defined staggered grids defined above for
a flat surface between _{} and_{},

_{} (11)

**2.2 Perfectly Matched
Layers**

Due to some cumulative errors using equation (10) for
boundary conditions with impedance matching the acoustic impedance of the
medium (*r
**c*), the
absorption of the boundaries is limited. For practical reasons it is impossible
to achieve level differences between reflected and incident waves bigger that
60 dB. Therefore a huge grid should be considered in order to separate in time
the boundary spurious reflections from the significant reflections inside the
integration area. The bigger the integration area is, the slower the
calculations result. To avoid the need for huge integration areas a technique
called “Perfectly Matched Layers” (PML) has been proposed [7-8].
This technique consists on defining a lossy medium in the zones near to the
boundaries. This implies the inclusion of additional terms, including two
attenuation factors (g_{r} and g_{z}) in the different spatial directions
considered, i.e:

_{} (12)

_{} (13)

_{} (14)

_{} (15)

where the sound pressure *p*
has been split into two additive components *p*_{r}
and *p*_{z}. These components
have no physical meaning and are defined only because of mathematical reasons.

_{} inside
the integration area (16)

_{}, *j = *1,…,
*m* at
the *j*-th layer of the PML (17)

*g** _{z}* is defined in an
analogous way.

**2.3 Excitation (source
inclusion)**

Once defined the integration scheme is important to consider
which of the possible excitation techniques should be used. Excitation
techniques can be divided attending to the field that is excited, pressure or
particle velocity, and attending to the way in which this excitation is
included. In the second case one can simply impose a spatial excitation at time
*t* = 0 or to include a time
perturbation on at least one point of the grids. This means that the time
evolution of one (or more) point is not governed by equations given above but
from a discrete time function, i.e.:

_{} (18)

In order to avoid numerical dispersion the use of softly
varying functions, such as Ricker wavelets [9], is strongly recommended.

Another common problem to solve is the elimination of
unwanted reflections caused by the sound source (excitation point). In order to
avoid those spurious reflections from the source “Transparent
sources” must be performed [10]. However, in our case the excitation will
be performed at a boundary region, so, there is no need for the use of such a
kind of sources.

**2.5 Music Instruments
simulations**

As commented above our purpose is to use an FDTD scheme to evaluate the acoustic behaviour of the body of a wind instrument. As starting point we have considered a clarinet. This instrument has geometry near to a cylinder. So we will work with a cylinder excited by a Ricker wavelet at one extreme. The surrounding will be rounded by a multilayer PML to avoid reflections. Figure 2 illustrate schematically the calculation area.

Figure 2 – *Geometry of the calculation area (the tube has a length of 50 cm and an
inner diameter of 28 mm)*

The maximum frequency to be evaluated is 8000 Hz. Frequencies above
that are non significant and should not be considered. So, the minimum wavelength to evaluate
will be approximately 4 centimetres. In order to consider enough points per
wavelength we will take a value of 1 millimetre for the spatial step (D*r* and D*z*). Given that, the
maximum time step is limited by the Courant number (*s*),

_{} (19)

We will take the maximum value of D*t*, i.e. 1’47·10^{-6}
sec.

Figure 3 – *Relative sound pressure level for Ricker wavelets excitation with
central frequency: a) 8 kHz, b) 4 kHz, at different time steps*

Musical
instruments which have no flare at the open end of their air column include
organ pipes and flutes. For such cases, the sound emitted by these instruments
is determined by the characteristics at the open end of the cylindrical pipe. Figure 3 shows sound pressure level at
different time steps. (Animations corresponding to that plots can be download
at http://ttt.gan.upv.es/~fredondo/).

For the sake of brevity we only present here the results for
the reflection function of the termination, i.e, the relationship between the
wave propagating backwards and the wave propagating forward. In the frequency
domain these two waves are related by the following equation

_{} (20)

where r(f), is the
reflection function. Figure 4 shows the results obtained for the reflectance of
a flanged pipe.

Figure 4 – *Reflection coefficient of
the termination of the instrument*

These results
are in good agreement with the theoretical model proposed by Levine and
Schwinger in [11], in which they obtained a rigorous and
explicit solution of the reflectance taking into account for the radiation of
sound from a circular pipe.

In this paper it is demonstrated that FDTD methods are
useful for simulating the acoustic behaviour of a tube, as a first
approximation, to analyze the transient characteristics of a wind instrument.
In future works we will include in our simulations several terminations of the
tube to analyze its effect on the sound radiated by the instrument.

This work has been supported by the “Ministerio de
Ciencia y Tecnología” (CICYT **Mat2003-04068**).

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