A Vox humana example

T₂₁(ω) +T₂₂(ω) Z_R(ω)

−1

Uˆ_in(ω). (C.1) The above can be evaluated once the radiation impedanceZ_R(ω)is known. The time do-main signalp_out(t)is obtained by the inverse Fourier-transform (implemented using IFFT) of the output pressurepˆ_out(ω)

p_out(t) =F⁻¹n ˆ p_out(ω)o

=F⁻¹n

H(ω)·Uˆ_in(ω)o

. (C.2)

The tool implements a very simple synthesizer that produces an ADSR² envelope which modulates the amplitude of the signal, so that the sound samplep_sample(t)is obtained as

p_sample(t) =A_ADSR(t)·p_out(t), (C.3)

withA_ADSR(t)denoting the ADSR function.

Since the simulation of a realistic attack transient would require much more effort (see e.g.

references [128] or [C15]), the ADSR is chosen for the sake of simplicity. By the simple fade-in the attack is masked and the listener can concentrate on the steady state characteristics of the sound.

C.3 A Vox humana example

As an example, theVox humanaresonator presented in Section 8.5.3 is modeled here. Figure C.1 shows the user interface ofReedResonatorSimwith theVox humanaproject file loaded. In the topmost region of the window, the geometry of the resonator is displayed; the values can be edited, conical sections can be added or removed. Next, the parameters of the shallot and the tongue can be configured. These values are needed to calculate the effective length of the shallot, as discussed in Section 8.4. In theCalculation and simulationpanel, the eigenfrequencies can be computed and the radiated sound can be simulated. Finally, in the bottom part of the window, stored results can be quickly compared to each other.

The calculated input admittance function of theVox humanaresonator is shown in Figure C.2.

In the left hand side (Figure C.2(a)) the input admittance without the shallot is shown, whereas the right hand side (Figure C.2(b)) displays the calculated input admittance of the complete

2Attack – Decay – Sustain – Release

C.3. AVOX HUMANAEXAMPLE 139

Figure C.1.The main window of the toolReedResonatorSimwith theVox humanaproject loaded

(a) Input admittance without shallot (b) Input admittance with shallot Figure C.2.Calculated input admittance of theVox humanaresonator

(a) Original sound spectrum (b) Sound spectrum with modified geometry

Figure C.3.Sound spectra of aVox humanapipe, simulated byReedResonatorSim

shallot–resonator system. As it can be seen, the frequencies of natural resonanance are affected by the shallot to a great extent: while the frequency of the first mode remains nearly the same, upper modal frequencies are shifted downwards significantly.

Figure C.3(a) shows the steady state sound spectrum resulting from the simulated excitation signal. The harmonic partials are identified by the regular, periodic sharp peaks, while maxima of the input impedance are found by observing the broad peaks in the baseline of the spectrum. An impedance maximum is present between the third and fourth partials, in correspondence with the first acoustic mode of the resonator–shallot system. As it can be seen, the first acoustic mode amplifies the fourth partial (second octave) in the sound to a great extent, while the third and the fifth harmonics are also enhanced. Since theVox humanashould amplify the third harmonic, it can be assessed that the original resonator design is suboptimal.

To achieve a better amplification of the third harmonic, the geometry of the resonator must be changed. However, it is not straightforward to tell how different parameters would affect the input impedance function. The objective ofReedResonatorSim is to provide a user friendly interface for modeling the effect of such changes. By modifying the values in theResonator geom-etrypanel and evaluating the input impedance function and the modal frequencies, the impact of each parameter can be assessed and different configurations can be compared with each other.

For example, by changing the length of the third section from 30.90 mm to 50.35 mm, the frequency of the first acoustic mode of the resonator–shallot system can be tuned to588 Hzwhich is exactly three times the fundamental frequency. The resulting steady state sound spectrum is shown in Figure C.3(b). By comparing the two spectra of Figure C.3, the remarkable difference is conspicuous. It can also be observed that the frequencies of higher modes are also shifted down, except for the second mode, whose frequency remained nearly the same. The steady state characteristics of the two sounds can also be compared in a subjective manner by playing the corresponding sound samples.

Appendix D

NiHu : A BEM/FEM toolbox for acoustics

This appendix briefly introducesNiHu¹, aC++- andMatlab-based toolbox for boundary and finite elements. This toolbox was utilized in the thesis for all acoustical finite element simulations.

Therefore, this appendix demonstrates only the finite element parts ofNiHu. The capabilities of the tool are illustrated by means of solving a two-dimensional academic problem.

D.1 The example problem

In this example an acoustical scattering problem of a plane wave reflected from an acoustically rigid cylinder is solved. It is assumed that the plane wave travels in the positivexdirection and has a frequency off = 400 Hz. The cylinder is located at the origin and its radius is chosen as R_cyl= 0.5 m. We define the boundaryΓas the surface of the cylinder, i.e.Γ ={(x, y)|x²+y²= R²_cyl}. The material properties of the fluid are chosen as average densityρ₀ = 1.2 kg/m³ and speed of soundc= 343 m/s. We are looking for the sound pressurep(x, ω), withω= 2πf, in the domain of interestx∈Ω, which is defined explicitly in the sequel.

The problem at hand involves scattering, hence the solutionp(x)is found as the superposition of the incidentp_inc(x)and the scatteredp_scat(x)pressure fields, as it was discussed in Section 4.1.5.

The same holds for the particle velocityv(x). As the cylinder is acoustically rigid, the normal particle velocityv_n vanishes on its surface, and thusv_n,scat(x) = −v_n,inc(x)ifx ∈ Γ. Since the incident pressure field and its normal derivative onΓare known analytically, only the reflected pressure field is sought. The latter is found by imposing a Neumann boundary condition onΓas

¯

v_n(x) =−v_n,inc(x), and solving the resulting boundary value problem.

The solution can be found analytically following [106, p. 401]. Assuming that the incident wave has unit amplitude, the solution in polar coordinates(r, φ)reads as

p_refl=

∞

X

m=0

A_mcos(mφ)H_m⁽²⁾(kr), (D.1) with

A_m=−m(−j)^m+1e^jγmsinγ_{m m}=

(1 if m= 0

2 if m≥1 (D.2)

and

tanγ0=−J1(kR_cyl)

N₁(kR_cyl) tanγm=−J_m−1(kR_cyl)−Jm+1(kR_cyl)

N_m+1(kR_cyl)−N_m−1(kR_cyl). (D.3)

1c Péter Fiala and Péter Rucz, 2009–2014. Some functions supporting PML are by Bence Olteán.

141

p_inc= e^−jkx

Rcyl

L_slab

Ω

LPML

Γ

(a) Arrangement of the example problem (b) The generated FE/PML mesh

Figure D.1.The arrangement and the discretized model of the example problem

The result of the finite element simulation will be compared to the above solution. The sim-ulation arrangement is depicted in Figure D.1(a). In the finite element model the simsim-ulation domainΩis defined asΩ ={(x, y)| −L_slab/2≤x, y≤L_slab/2 ∧ x²+y²≥R²_cyl}, with the choice ofL_slab = 3.5 m. Since the problem has open boundaries, the perfectly matched layer method is applied for emulating free field conditions on the outer boundary of the mesh. The thickness of the layer is chosen asL_PML= 0.4 m.