NUMERICAL MODELING OF SOUND PROPAGATION IN ANECHOIC CHAMBER WITH QUADRATIC RESIDUE DIFFUSER

HTML
DOWNLOAD

Wielkość: px

Rozpocząć pokaz od strony:

Download "NUMERICAL MODELING OF SOUND PROPAGATION IN ANECHOIC CHAMBER WITH QUADRATIC RESIDUE DIFFUSER"

Roman Czajkowski
7 lat temu
Przeglądów:

1 ELEKTRYKA 2014 Zeszyt 1 (229) Rok LX Lukáš KOUDELA, Petr POLCAR, Oldřich TUREČEK University of West Bohemia in Pilsen, Czech Republic NUMERICAL MODELING OF SOUND PROPAGATION IN ANECHOIC CHAMBER WITH QUADRATIC RESIDUE DIFFUSER Summary. This paper discusses numerical solution of acoustic transient field in the anechoic chamber with quadratic residue diffuser. The goal is to obtain the time evolution of acoustic pressure when the source of signal is Gaussian monocycle pulse with defined parameters. The mathematical model is described by the partial differential equation supplemented with boundary conditions and the numerical solution is realized by a fully adaptive higher-order finite method implemented into codes Hermes and Agros2D developed at the University of West Bohemia. Keywords: Gauss monocycle pulse, acoustic transient field, higher-order finite element method, Schroeder quadratic residue diffuser MODELOWANIE NUMERYCZNE PROPAGACJI DŹWIĘKU W KOMORZE BEZECHOWEJ ZA POMOCĄ DYFUZORA TYPU QRD Streszczenie. W artykule omówiono numeryczne rozwiązywanie pola akustycznego w stanie nieustalonym w komorze bezechowej za pomocą dyfuzora QRD. Celem było otrzymanie przebiegu ciśnienia akustycznego, jeśli sygnałem wymuszającym był impuls typu gaussowskiego (pierwsza pochodna funkcji Gaussa) o zdefiniowanych parametrach. Model matematyczny został opisany równaniem różniczkowym cząstkowym uzupełnionym warunkami brzegowymi, zaś rozwiązanie numeryczne zostało uzyskane za pomocą w pełni adaptacyjnego algorytmu wykorzystującego metodę elementów skończonych wyższego rzędu implementowanego w aplikacji Agros2D bazującej na bibliotece Hermes i stworzonej na Uniwersytecie Zachodnioczeskim. Słowa kluczowe: impuls typu gaussowskiego, pole akustyczne w stanie nieustalonym, metoda elementów skończonych wyższego rzędu, dyfuzor Schroedera typu QRD 1. INTRODUCTION The numerical modelling of acoustic transient phenomena represents one of the useful tools for the solving in the discipline of room acoustics because it is much cheaper than the commonly used measurement. It can provide the same results and it is also possible to obtain

8 L. Koudela, P. Polcar, O. Tureček other problematic measurable quantities, but the main contribution of numerical modelling is primarily in the development and design of new types of diffusors.

2 8 L. Koudela, P. Polcar, O. Tureček other problematic measurable quantities, but the main contribution of numerical modelling is primarily in the development and design of new types of diffusors. In the field of room acoustics the sound heard in most environments is the sound combination of the direct part (e.g. from loudspeaker) and the part created by the reflections form the walls, floor and ceiling. The aim is to use the reflections for improving the sound parameters in the environment. Part of sound striking the surface could be (instead of transmission and absorption) reflected by a large flat surface (see Fig. 1, left part). When the surface is appropriately modified into a diffusion surface (see Fig. 1, right part) a significant portion of the reflected sound could be dispersed. Fig. 1. Comparison of reflected sound from the large flat surface (left part) and from the diffusion surface (right part) [1] Rys. 1. Porównanie dźwięku odbitego od dużej płaskiej powierzchni (strona lewa rysunku) i od powierzchni rozpraszającej (prawa strona rysunku) An acoustic diffuser is an element that disperses a significant portion of reflected sound wave. It is commonly used to adjust the sound level distribution in the concert halls, theatres or other areas that require perfect acoustics. In this paper the numerical modelling of Schroeder quadratic residue acoustic diffuser (QRD) placed in the anechoic chamber is carried out. The possibilities of geometry shapes of 1D Schroeder diffuser are depicted in Fig. 2. Fig. 2. Possibilities of geometry shapes and realizations of quadratic residue diffuser (Figure sources: [1]) Rys. 2. Możliwości geometrii - QRD dyfuzor (źródła rysunków: [1])

3 Numerical modeling of sound propagation 9 2. MATHEMATICAL MODEL The continuous mathematical model of the acoustic field is given by the non-stationary partial differential equation with appropriate boundary conditions. The corresponding wave equation derived from the Newton s Second Law, from the Continuity Equation and from the Equation of State for adiabatic processes is considered in the form [2, 5] p div grad p 0, (1) 2 2 c t where c is the speed of sound wave propagation, ρ stands for the specific mass, p is the acoustic pressure and t represents time. The equation (1) is supplemented with Dirichlet and Neumann boundary conditions respecting the reflective surfaces, axis of symmetry, source of acoustic signal and finally the impedance matched boundary Gauss monocycle pulse For the numerical calculation of acoustic transient field it is necessary to specify the shape of the source signal that excites the field. The function of the Gauss monocycle pulse is commonly used in the field of room acoustics and in the solved task is defined as the Dirichlet boundary condition. The time dependence is described using the equation [2] 2 f 2 t 1/ f t = Ae p0. (2) The symbol A (m 3.s 1 ) is the amplitude of the pulse and f 0 (Hz) denotes the pulse bandwidth Diffusion coefficient Diffusion coefficient determines a measure of the uniformity of the reflected sound. The purpose of this coefficient is to enable the design of diffusers and allow acousticians to compare the performance of surfaces for room design and performance specifications. The diffusion coefficient d ψ could be calculated using the term [1] d 2 L / i n i1 n n L /10 /10 2 i Li i1 i1, (3) where L i is a set of sound pressure levels in decibels in a polar response, n is the number of receivers in semicircle area with even angular spacing and ψ is the angle of incidence. n

4 10 L. Koudela, P. Polcar, O. Tureček 3. TYPICAL ARRANGEMENT The discussed acoustic diffuser is placed in front of the planar signal source in the anechoic chamber. Fig. 3 shows the principal arrangement with the geometrical dimensions. The dimensions of the impedance matched room (considered anechoic chamber at the University of West Bohemia in Pilsen) are 2.75 meters by 4.73 meters. The source of signal has a square platform with dimensions of 0.1 meter and the signal spreads from one side. Around the diffuser 47 microphones are placed along the radius of 2 meters where the values of acoustic pressure are investigated. Fig. 3. Left: geometrical arrangement of the solved area (anechoic chamber) with placed source of signal and QRD diffuser, right: QRD diffuser in detail Rys. 3. Po lewej: geometria rozwiązywanej powierzchni (komora bezechowa), pokazana lokalizacja źródła sygnału i dyfuzor QRD, po prawej: QRD dyfuzor pokazany szczegółowo The illustrative example is solved in the air with the following parameters: the specific mass density ρ = kg m 3 and speed of sound c = 343 m s 1 (for the ambient temperature 20 C). 4. NUMERICAL SOLUTION The numerical solution of equation (1) was performed by a fully adaptive higher-order finite element method (hp-fem) implemented in the C++ library Hermes and used in Agros2D application developed by the group at the University of West Bohemia [3, 4]. The convergence of the numerical solution is defined by the CFL number. The typical value of the CFL number in case of the 2 nd order elements is about 0.1 for reaching the convergence, see [5]. In this task lower CFL value 0.05 was used, which ensured more accurate results with sufficient computer requirements.

Numerical modeling of sound propagation 11 4.1. Results of numerical solution Time evolution of acoustic pressure in the solved area is depicted for several time steps in Fig. 4. Principle function of acoustic diffuser is seen there.

5 Numerical modeling of sound propagation Results of numerical solution Time evolution of acoustic pressure in the solved area is depicted for several time steps in Fig. 4. Principle function of acoustic diffuser is seen there. The incident sound wave along the axis of symmetry is reflected from the diffusion surface and is dispersed into the surrounding area. Moreover, the reflected sound is also diffused in time domain. Fig. 4. Time evolution of acoustic pressure for Gauss monocycle pulse bandwidth f 0 = 600 Hz (from left 10 ms, 13 ms, 15 ms and 19 s) Rys. 4. Przebieg czasowy ciśnienia akustycznego dla pasma częstotliwości impulsu gaussowskiego f 0 = 600 Hz (od lewej 10 ms, 13 ms, 15 ms i 19 ms) The calculated diffusion coefficient using the equation (3) is depicted in Fig 5. This means, that for this type of diffuser shape, the diffusion coefficient decreases depending on the frequency. This result represents non-normalized diffusion coefficient and for the further use it must be recalculated according to [1]. Fig. 5. Calculated diffusion coefficient using the quadratic residue diffusor Rys. 5. Obliczony za pomocą dyfuzora QRD współczynnik rozproszenia

6 12 L. Koudela, P. Polcar, O. Tureček CONCLUSION This contribution discusses the results of numerical modeling of acoustic transient field in the room with quadratic residue acoustic diffuser. The knowledge of time evolution of acoustic pressure reflected from the QRD diffuser is crucial for the design and evaluation of function of this diffuser shape. ACKNOWLEDGEMENT This work was supported by the Ministry of Industry and Trade of The Czech Republic under the project No. FR-TI4/569: Development of a new generation of acoustic diffusers and their modeling. BIBLIOGRAPHY 1. Cox J. T., D Antonio P.: Acoustic Absorbers and Diffusers Theory. Design and Application, Taylor & Francis Group Rossing T. D.: Springer Handbook of Acoustics. Springer Science + Business Media, LLC New York Solin P. et al.: Hermes Higher-order Modular Finite Element System, 4. Karban P. et al.: Agros2D Multiplatform C++ Application for the Solution of PDEs, 5. Koudela L., Jansa J., Karban P.: Numerical modeling of Transient Acoustic Field Using Finite Element Method. In proceedings of Computer Applications in Electrical Engineering 2013, Poznan 2013, p Ing. Lukáš Koudela, Ing. Petr Polcar, Ph.D., Ing. Oldřich Tureček, Ph.D. University of West Bohemia in Pilsen Univerzitni 26, Pilsen, Czech Republic koudela@kte.zcu.cz plcarp@kte.zcu.cz turecek@kte.zcu.cz

Podobne dokumenty

Proposal of thesis topic for mgr in. (MSE) programme in Telecommunications and Computer Science

Proposal of thesis topic for mgr in (MSE) programme 1 Topic: Monte Carlo Method used for a prognosis of a selected technological process 2 Supervisor: Dr in Małgorzata Langer 3 Auxiliary supervisor: 4