scholarly journals A convergent low-wavenumber, high-frequency homogenization of the wave equation in periodic media with a source term

2021 ◽  
pp. 1-34
Author(s):  
Shixu Meng ◽  
Othman Oudghiri-Idrissi ◽  
Bojan B. Guzina
Keyword(s):  
Wave Equation ◽  
High Frequency ◽  
Source Term ◽  
Periodic Media

scholarly journals High-frequency homogenization for travelling waves in periodic media

2016 ◽  
Vol 472 (2191) ◽  
pp. 20160066 ◽  
Author(s):  
Davit Harutyunyan ◽  
Graeme W. Milton ◽  
Richard V. Craster
Keyword(s):  
Wave Equation ◽  
High Frequency ◽  
Travelling Waves ◽  
Travelling Wave ◽  
Coupling Constants ◽  
Convergence Theory ◽  
Periodicity Cell ◽  
Periodic Media ◽  
Carrier Wave ◽  
Convergence Type

We consider high-frequency homogenization in periodic media for travelling waves of several different equations: the wave equation for scalar-valued waves such as acoustics; the wave equation for vector-valued waves such as electromagnetism and elasticity; and a system that encompasses the Schrödinger equation. This homogenization applies when the wavelength is of the order of the size of the medium periodicity cell. The travelling wave is assumed to be the sum of two waves: a modulated Bloch carrier wave having crystal wavevector k and frequency ω 1 plus a modulated Bloch carrier wave having crystal wavevector m and frequency ω 2 . We derive effective equations for the modulating functions, and then prove that there is no coupling in the effective equations between the two different waves both in the scalar and the system cases. To be precise, we prove that there is no coupling unless ω 1 = ω 2 and ( k − m ) ⊙ Λ ∈ 2 π Z d , where Λ =(λ 1 λ 2 …λ d ) is the periodicity cell of the medium and for any two vectors a = ( a 1 , a 2 , … , a d ) , b = ( b 1 , b 2 , … , b d ) ∈ R d , the product a ⊙ b is defined to be the vector ( a 1 b 1 , a 2 b 2 ,…, a d b d ). This last condition forces the carrier waves to be equivalent Bloch waves meaning that the coupling constants in the system of effective equations vanish. We use two-scale analysis and some new weak-convergence type lemmas. The analysis is not at the same level of rigour as that of Allaire and co-workers who use two-scale convergence theory to treat the problem, but has the advantage of simplicity which will allow it to be easily extended to the case where there is degeneracy of the Bloch eigenvalue.

scholarly journals Godunov type scheme for the linear wave equation with Coriolis source term

2017 ◽  
Vol 58 ◽  
pp. 1-26 ◽  
Author(s):  
Emmanuel Audusse ◽  
Stéphane Dellacherie ◽  
Minh Hieu Do ◽  
Pascal Omnes ◽  
Yohan Penel
Keyword(s):  
Wave Equation ◽  
Source Term ◽  
Linear Wave ◽  
Linear Wave Equation

Unsteady disturbances of streaming motions around bodies

1989 ◽  
Vol 209 ◽  
pp. 385-403 ◽  
Author(s):  
H. M. Atassi ◽  
J. Grzedzinski
Keyword(s):  
Wave Equation ◽  
Body Surface ◽  
Source Term ◽  
The Body ◽  
Entire Body ◽  
Two Dimensional ◽  
Mean Flow ◽  
Inhomogeneous Wave ◽  
Inhomogeneous Wave Equation ◽  
The Mean

For small-amplitude vortical and entropic unsteady disturbances of potential flows, Goldstein proposed a partial splitting of the velocity field into a vortical part u(I) that is a known function of the imposed upstream disturbance and a potential part ∇ϕ satisfying a linear inhomogeneous wave equation with a dipole-type source term. The present paper deals with flows around bodies with a stagnation point. It is shown that for such flows u(I) becomes singular along the entire body surface and its wake and as a result ∇ϕ will also be singular along the entire body surface. The paper proposes a modified splitting of the velocity field into a vortical part u(R) that has zero streamwise and normal components along the body surface, an entropy-dependent part and a regular part ∇ϕ* that satisfies a linear inhomogeneous wave equation with a modified source term.For periodic disturbances, explicit expressions for u(R) are given for three-dimensional flows past a single obstacle and for two-dimensional mean flows past a linear cascade. For weakly sheared flows, it is shown that if the mean flow has only a finite number of isolated stagnation points, u(R) will be finite along the body surface. On the other hand, if the mean flow has a stagnation line along the body surface such as in two-dimensional flows then the component of u(R) in this direction will have a logarithmic singularity.For incompressible flows, the boundary-value problem for ϕ* is formulated in terms of an integral equation of the Fredholm type. The theory is applied to a typical bluff body. Detailed calculations are carried out to show the velocity and pressure fields in response to incident harmonic disturbances.

scholarly journals Modelling the asymmetries of the Sun’s radial p-mode line profiles

2020 ◽  
Vol 635 ◽  
pp. A81 ◽  
Author(s):  
J. Philidet ◽  
K. Belkacem ◽  
R. Samadi ◽  
C. Barban ◽  
H.-G. Ludwig
Keyword(s):  
Wave Equation ◽  
Spectral Density ◽  
Power Spectral Density ◽  
Source Term ◽  
Acoustic Noise ◽  
Spatial Extent ◽  
Velocity Spectrum ◽  
Line Profiles ◽  
Stochastic Excitation ◽  
Power Spectral

Context. The advent of space-borne missions has substantially increased the number and quality of the measured power spectrum of solar-like oscillators. It now allows for the p-mode line profiles to be resolved and facilitates an estimation of their asymmetry. The fact that this asymmetry can be measured for a variety of stars other than the Sun calls for a revisiting of acoustic mode asymmetry modelling. This asymmetry has been shown to be related to a highly localised source of stochastic driving in layers just beneath the surface. However, existing models assume a very simplified, point-like source of excitation. Furthermore, mode asymmetry could also be impacted by a correlation between the acoustic noise and the oscillating mode. Prior studies have modelled this impact, but only in a parametrised fashion, which deprives them of their predictive power. Aims. In this paper, we aim to develop a predictive model for solar radial p-mode line profiles in the velocity spectrum. Unlike the approach favoured by prior studies, this model is not described by free parameters and we do not use fitting procedures to match the observations. Instead, we use an analytical turbulence model coupled with constraints extracted from a 3D hydrodynamic simulation of the solar atmosphere. We then compare the resulting asymmetries with their observationally derived counterpart. Methods. We model the velocity power spectral density by convolving a realistic stochastic source term with the Green’s function associated with the radial homogeneous wave equation. We compute the Green’s function by numerically integrating the wave equation and we use theoretical considerations to model the source term. We reconstruct the velocity power spectral density and extract the line profile of radial p-modes as well as their asymmetry. Results. We find that stochastic excitation localised beneath the mode upper turning point generates negative asymmetry for ν <  νmax and positive asymmetry for ν >  νmax. On the other hand, stochastic excitation localised above this limit generates negative asymmetry throughout the p-mode spectrum. As a result of the spatial extent of the source of excitation, both cases play a role in the total observed asymmetries. By taking this spatial extent into account and using a realistic description of the spectrum of turbulent kinetic energy, both a qualitative and quantitative agreement can be found with solar observations performed by the GONG network. We also find that the impact of the correlation between acoustic noise and oscillation is negligible for mode asymmetry in the velocity spectrum.

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