Daniel P. Elford, L. Chalmers, F. Kusmartsev, G.M. Swallowe.
1
The results of numerical modelling of sonic crystals with resonant array elements are detailed. The acoustic band structure and transmission characteristics of such systems have been computed with the use of Finite Element Methods, specifically through the framework of the acoustics package in COMSOL Multiphysics. Acoustic wave propagation in two-dimensional periodic structures has been investigated since the late 1980’s. It was proposed that such a structure would forbid the transmission of a certain frequency range, known as a band gap. The ability for a sonic crystal to manipulate sound can be found in the interference of sound waves as they are scattered by the cylinders. Because of the periodic nature this interference is constructive or destructive interference, dependent on the frequency of the incoming sound wave (See Figure 1). Acoustic Band Structure Calculation A sonic crystal structure is assumed to be infinite and periodic in the direction x with the period a 1 and in the direction y with the period a 2 and described by two basis vectors: (a 1 , 0) and (0, a 2 ). According to the Floquet-Bloch theorem, the relation for the pressure distribution p for nodes lying on the boundary of the unit cell can be expressed as: where x is the position vector in the unit cell and k = (k x , k y ) is the Bloch wavevector. Considering the periodic boundary conditions above allows the reduction of the model to a single unit cell (See Figure 2). Next a phase relation is applied in the boundary of the unit to define boundary conditions between adjacent units. This phase relation is related to the wavenumber of the incident wave in the periodic structure. Fig 2: Unit cell with phase relation for a) Conventional sonic crystal b) Locally resonant sonic crystal c) Matryoshka locally resonant sonic crystal By defining the Bloch wavevector in the first Brillouin zone, the analysis of the first ten Eigenfrequencies and the corresponding Eigenvectors are computed. The Eigenvectors are related to the pressure distribution of the mode and is plotted against the three principal symmetry directions. Transmission Analysis COMSOL is utilised to calculate the pressure field behind a sonic crystal and to generate a pressure map of the system at fixed frequencies. The equation used to analyse the acoustic wave problems is expressed as a Helmholtz equation for a time harmonic pressure wave excitation, p = p 0 e iωt In the case of the cylinders in the sonic crystal system, sound-hard boundary conditions have been applied. The radiation boundary conditions at the exterior edges of the rectangular domain are considered to be perfectly absorbing. One of the domain edges is modelled as a radiation condition with pressure source. By solving the Helmholtz equation in COMSOL, the acoustic pressure field distribution can be obtained. By solving for a parametric sweep of frequency, a frequency spectrum displaying the attenuation properties of the sonic crystal can also be constructed. Fig 1: (Left) Interference occurs when the spacing of the crystal, a is comparable to the wavelength, λ of the incoming sound. Sound energy is reflected back. (Right) Eusebio Sempere's sculpture in Madrid has unique acoustic properties. The interaction between the resonances produces band gaps and gives rise to phenomena that can lead to acoustic attenuation. The continuum band of the surrounding effective medium interacts with resonance states by hybridization (mixtures of different waves states) giving rise to hybridization gaps. For low frequency broadband attenuation the most optimal configuration is the Matryoshka sonic crystal. This system forms numerous gaps in the lower frequency regime, whilst maintaining a reduced crystal size viable for noise barrier technology. This finding opens alternative perspectives for the construction of sound barriers in the low frequency range usually inaccessible by traditional means. Fig 3. Conventional Sonic Crystal – (Top Left) Acoustic band structure, (Bottom Left) Comparison of frequency spectrum to band structure, (Right) Acoustic Pressure Field Distribution. Fig 4. Locally Resonant Sonic Crystal – (Top Left) Acoustic band structure, (Bottom Left) Comparison of frequency spectrum to band structure, (Right) Acoustic Pressure Field Distribution. Fig 5. Matryoshka Locally Resonant Sonic Crystal – (Top Left) Acoustic band structure, (Right) Comparison of frequency spectrum to band structure, (Bottom Left) Finite Element computed Eigenmode pressure distribution. Daniel P. Elford, L. Chalmers, F. Kusmartsev, G.M. Swallowe. Department of Physics, Loughborough University, Loughborough, Leicestershire, LE11 3TU, UK. Bragg Gap Bragg Gap Hybridization Gap Publications: Daniel P. Elford et al. J. Acoust. Soc. Am. 130 (5), (Nov 2011). International patent application: PCT/EP2010/051370 (Feb 2010). Please scan with smart phone to find out more information about the commercialisation of this research at Loughborough University The general concept of a locally resonant sonic crystal is that it utilizes acoustic resonances to form additional band gaps that are decoupled from Bragg gaps. Conventional Sonic Crystal Results Matryoshka Locally Resonant Sonic Crystal Results Locally Resonant Sonic Crystal Results Email: [email protected]
Transcript of Daniel P. Elford, L. Chalmers, F. Kusmartsev, G.M. Swallowe.
The results of numerical modelling of sonic crystals with resonant array elements are detailed. The acoustic band structure and transmission characteristics of such systems have been computed with the use of Finite Element Methods, specifically through the framework of the acoustics package in COMSOL Multiphysics.
Acoustic wave propagation in two-dimensional periodic structures has been investigated since the late 1980’s. It was proposed that such a structure would forbid the transmission of a certain frequency range, known as a band gap.
The ability for a sonic crystal to manipulate sound can be found in the interference of sound waves as they are scattered by the cylinders. Because of the periodic nature this interference is constructive or destructive interference, dependent on the frequency of the incoming sound wave (See Figure 1).
Acoustic Band Structure Calculation
A sonic crystal structure is assumed to be infinite and periodic in the direction x with the period a1 and in the direction y with the period a2 and described by two basis vectors: (a1, 0) and (0, a2). According to the Floquet-Bloch theorem, the relation for the pressure distribution p for nodes lying on the boundary of the unit cell can be expressed as:
where x is the position vector in the unit cell and k = (kx, ky) is the Bloch wavevector. Considering the periodic boundary conditions above allows the reduction of the model to a single unit cell (See Figure 2).
Next a phase relation is applied in the boundary of the unit to define boundary conditions between adjacent units. This phase relation is related to the wavenumber of the incident wave in the periodic structure.
Fig 2: Unit cell with phase relation for a) Conventional sonic crystal b) Locally resonant sonic crystal c) Matryoshka locally resonant sonic crystal
By defining the Bloch wavevector in the first Brillouin zone, the analysis of the first ten Eigenfrequencies and the corresponding Eigenvectors are computed. The Eigenvectors are related to the pressure distribution of the mode and is plotted against the three principal symmetry directions.
Transmission Analysis
COMSOL is utilised to calculate the pressure field behind a sonic crystal and to generate a pressure map of the system at fixed frequencies. The equation used to analyse the acoustic wave problems is expressed as a Helmholtz equation for a time harmonic pressure wave excitation, p = p0eiωt
In the case of the cylinders in the sonic crystal system, sound-hard boundary conditions have been applied. The radiation boundary conditions at the exterior edges of the rectangular domain are considered to be perfectly absorbing. One of the domain edges is modelled as a radiation condition with pressure source.
By solving the Helmholtz equation in COMSOL, the acoustic pressure field distribution can be obtained. By solving for a parametric sweep of frequency, a frequency spectrum displaying the attenuation properties of the sonic crystal can also be constructed.
Fig 1: (Left) Interference occurs when the spacing of the crystal, a is comparable to the wavelength, λ of the incoming sound. Sound energy is reflected back. (Right) Eusebio Sempere's sculpture in Madrid has unique acoustic properties.
The interaction between the resonances produces band gaps and gives rise to phenomena that can lead to acoustic attenuation. The continuum band of the surrounding effective medium interacts with resonance states by hybridization (mixtures of different waves states) giving rise to hybridization gaps.
For low frequency broadband attenuation the most optimal configuration is the Matryoshka sonic crystal. This system forms numerous gaps in the lower frequency regime, whilst maintaining a reduced crystal size viable for noise barrier technology. This finding opens alternative perspectives for the construction of sound barriers in the low frequency range usually inaccessible by traditional means.
Fig 3. Conventional Sonic Crystal – (Top Left) Acoustic band structure, (Bottom Left) Comparison of frequency spectrum to band structure, (Right) Acoustic Pressure Field Distribution.
Fig 4. Locally Resonant Sonic Crystal – (Top Left) Acoustic band structure, (Bottom Left) Comparison of frequency spectrum to band structure, (Right) Acoustic Pressure Field Distribution.
Fig 5. Matryoshka Locally Resonant Sonic Crystal – (Top Left) Acoustic band structure, (Right) Comparison of frequency spectrum to band structure, (Bottom Left) Finite Element computed Eigenmode pressure distribution.
Daniel P. Elford, L. Chalmers, F. Kusmartsev, G.M. Swallowe. Department of Physics, Loughborough University, Loughborough, Leicestershire, LE11 3TU, UK.
Bragg Gap
Bragg Gap
Hybridization Gap
Publications: Daniel P. Elford et al. J. Acoust. Soc. Am. 130 (5), (Nov 2011). International patent application: PCT/EP2010/051370 (Feb 2010).
Please scan with smart phone to find out more information about the commercialisation of this
research at Loughborough University
The general concept of a locally resonant sonic crystal is that it utilizes acoustic resonances to form additional band gaps that are decoupled from Bragg gaps.
Conventional Sonic Crystal Results
Matryoshka Locally Resonant Sonic Crystal Results
Locally Resonant Sonic Crystal Results
Email: [email protected]
