scholarly journals On the Solutionn-Dimensional of the Product⊗kOperator and Diamond Bessel Operator

2010 ◽  
Vol 2010 ◽  
pp. 1-20
Author(s):  
Wanchak Satsanit ◽  
Amnuay Kananthai
Keyword(s):  
Wave Equation ◽  
Positive Integer ◽  
Nonlinear Equation ◽  
Unknown Function ◽  
Bessel Operator ◽  
Generalized Function ◽  
Inhomogeneous Wave ◽  
Inhomogeneous Wave Equation ◽  
Unknown Unknown

Firstly, we studied the solution of the equation⊗k◊Bku(x)=f(x)whereu(x)is an unknown unknown function forx=(x1,x2,…,xn)∈ℝn,f(x)is the generalized function,kis a positive integer. Finally, we have studied the solution of the nonlinear equation⊗k◊Bku(x)=f(x,□k−1LkΔBk□Bku(x)). It was found that the existence of the solutionu(x)of such an equation depends on the condition offand□k−1LkΔBk□Bku(x). Moreover such solutionu(x)is related to the inhomogeneous wave equation depending on the conditions ofp,q, andk.

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.

Time‐domain behavior of wide‐angle one‐way wave equations

Geophysics ◽  
1991 ◽  
Vol 56 (3) ◽  
pp. 382-384
Author(s):  
A. H. Kamel
Keyword(s):  
Wave Equation ◽  
Wave Equations ◽  
Source Function ◽  
Plane Waves ◽  
Wide Angle ◽  
Wave Processes ◽  
Inhomogeneous Wave ◽  
Dirac Delta ◽  
Inhomogeneous Wave Equation ◽  
Standard Tool

The constant‐coefficient inhomogeneous wave equation reads [Formula: see text], Eq. (1) where t is the time; x, z are Cartesian coordinates; c is the sound speed; and δ(.) is the Dirac delta source function located at the origin. The solution to the wave equation could be synthesized in terms of plane waves traveling in all directions. In several applications it is desirable to replace equation (1) by a one‐way wave equation, an equation that allows wave processes in a 180‐degree range of angles only. This idea has become a standard tool in geophysics (Berkhout, 1981; Claerbout, 1985). A “wide‐angle” one‐way wave equation is designed to be accurate over nearly the whole 180‐degree range of permitted angles. Such formulas can be systematically constructed by drawing upon the connection with the mathematical field of approximation theory (Halpern and Trefethen, 1988).

scholarly journals An inhomogeneous wave equation and non-linear Diophantine approximation

2008 ◽  
Vol 217 (2) ◽  
pp. 740-760 ◽  
Author(s):  
V. Beresnevich ◽  
M.M. Dodson ◽  
S. Kristensen ◽  
J. Levesley
Keyword(s):  
Wave Equation ◽  
Diophantine Approximation ◽  
Inhomogeneous Wave ◽  
Inhomogeneous Wave Equation ◽  
Non Linear

Diffraction Tomography I: The Wave-Equation

1980 ◽  
Vol 2 (3) ◽  
pp. 213-222 ◽  
Author(s):  
R. K. Mueller
Keyword(s):  
Wave Equation ◽  
Test Object ◽  
General Equation ◽  
Viscoelastic Medium ◽  
Perturbation Methods ◽  
Sound Propagation ◽  
Diffraction Tomography ◽  
Inhomogeneous Wave ◽  
General Wave ◽  
Inhomogeneous Wave Equation

A general wave equation for sound propagation in a viscoelastic medium is obtained. From this general equation an approximate inhomogeneous wave equation is derived by perturbation methods. Born's and Rytov's approximations are considered. The equation is finally brought into a form which provides transformation properties under rotation of the test object required for diffraction tomography.

Nonconforming Finite Elements Based on Nitsche-Type Mortaring for Inhomogeneous Wave Equation

2018 ◽  
Vol 26 (03) ◽  
pp. 1850028 ◽  
Author(s):  
Manfred Kaltenbacher ◽  
Sebastian Floss
Keyword(s):  
Wave Equation ◽  
Transmission Loss ◽  
Normal Component ◽  
Grid Method ◽  
Computational Time ◽  
Fe Method ◽  
Expansion Chamber ◽  
Inhomogeneous Wave ◽  
Inhomogeneous Wave Equation ◽  
Homogenization Approach

We propose the nonconforming Finite Element (FE) method based on Nitsche-type mortaring for efficiently solving the inhomogeneous wave equation, where due to the change of material properties the wavelength in the subdomains strongly differs. Therewith, we gain the flexibility to choose for each subdomain an optimal grid. The proposed method fulfills the physical conditions along the nonconforming interfaces, namely the continuity of the acoustic pressure and the normal component of the acoustic particle velocity. We apply the nonconforming grid method to the computation of transmission loss (TL) of an expansion chamber utilizing micro-perforated panels (MPPs), which are modeled by a homogenization approach via a complex fluid. The results clearly demonstrate the superiority of the nonconforming FE method over the standard FE method concerning pre-processing, mesh generation flexibility and computational time.

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