Questions tagged [wave-propagation]
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Analytical Equation of the gaussian 1D wave equation with periodic Boundary condition
I am trying to validate the 1D analytical wave equation with a numerical solution with periodic boundary conditions. I have implemented the periodic boundary condition for the numerically calculated ...
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Solving $u_{tt} = u_{xx}$ with spectral methods and homogeneous Dirichlet/Neumann BC
Suppose we want to approximate a function $u(x,t)$ with $(x,t)\in[-1,1]\times[0,4]$ that satisfies the wave equation $u_{tt}=u_{xx}$ in the inside and BC $u(-1,t)=u_x(1,t)=0$ with initial value given ...
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(Regular) Coulomb wave function
I'm looking for a way to implement the regular Coulomb wave function in python. This function is a solution to
\begin{align}
\frac{\text{d}^2\,u}{\text{d}z^2}+\left(1-\frac{2\eta}{z}-\frac{\ell(\ell+1)...
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Unstable FDTD for axisymmetrical wave propagation
I am writing a 2.5D axisymmetric FDTD solver for Maxwell equations. It operates on three fields on staggered grids ($E_r$, $E_z$, $B_\varphi$).
As a test problem, I am considering a cylindrical ...
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Looking for non-trivial examples of solutions to 3D wave equations?
We have developed a (new) numerical scheme to solve the classical wave equation in 3 dimensions and we aim to publish the results.
We can read in the aim and scope of the journal of computational and ...
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Interpolation of 1D solution from an original grid to a new grid
I have a solution of a 1D wave on a grid (tangent hyperbolic variation) and now I want to interpolate the obtained solution to a new grid with the same number of points as the previous grid but the ...
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How to define a stretched coordinate perfectly matched layer (PML) parameters for maximum absorbtion?
I have written a MATLAB code for solving Maxwell's equations (electromagnetic wave propagation) in 3D with perfectly matched layer (PML) boundaries. I am using a stretched coordinate PML, but I see ...
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Numerical solution of 2D wave equation using Fourier transform and finite differences
This is the $2$-dimensional wave equation
$$ u_{tt} = u_{xx} + u_{yy} $$
with initial condition $u(x,y,0)=f(x,y)$ and $u_{t}(x,y,0) = 0$.
The inverse Fourier transform used is
$$ u(x,y,t) = \iint \hat{...
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Can you describe the Galerkin numerical method to solve the wave equation?
How would you describe the Galerkin method to solving the 3D wave equation
$$u_{tt}= c^2\Delta u$$
to someone who wants to implement it immediately?
More precisely, we want to solve the Cauchy problem
...
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stability of a numercial scheme for a hyperbolic system?
This is related to my question here https://math.stackexchange.com/questions/4447383/lax-wendroff-scheme-stability-analysis-for-a-linear-system-of-conservation-laws .
Consider the numerical scheme ...
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schrodinger eq time propagation with dissipation using split step operator
I am looking in ways to include energy dissipation while propagating a coherent wavepacket in a 1d TDSE. for example I use the split step method: exp[Δt(D+V)]≈exp[ΔtV/2]exp[ΔtD]exp[ΔtV/2], and ...
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Open boundary condition for 1d wave equation with variable wave speed using finite differences
I have implemented a finite difference solver for the 1d wave equation with variable wave speed:
$$ u_{tt} = c(x)u_{xx}, \hspace{10mm}c(x) = \dfrac{6 -x^2}{2} \hspace{5mm} $$
on $-2 \leq x \leq 2, t &...
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Beam propogation method for a waveguide. How to get single mode?
I am simulating a waveguide using diffractio python library (https://diffractio.readthedocs.io/en/latest/readme.html). The idea is to create a single mode waveguide using wave propogation method.
...
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How do I identify negative group speeds?
This question is a continuation of one of my other questions.
I've been trying to show that collocated (non-staggered) grids can suffer from negative group speeds in the linearized shallow water ...
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A simple wave for the linear shallow water equations
I'm looking for a simple, right-traveling wave for the linear shallow water equations (1D). My question: what are the initial conditions (velocity $U_0(x)$ and/or average water height, average ...
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FEM port Boundary definition for electromagnetics and wave guides
We are currently in the process of implementing ports in our EM FEM simulation SW.
We have come across the definition of boundary conditions for the ports, and we do not understand the equation for ...
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Divergence on wave equation simulation
I'm currenly working on my own PDE solver for non-linear simulations in python. I've done succesfully simulations for KdV and Fisher's equation, but now I'm playing with second order derivatives in ...
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Results blow up when number of intervals is increases (Yee algorithm FDTD, dielectric sphere)
I have been trying to write a program that analyses EM wave scattering by a dielectric sphere for a project.
The reference is Sadiku's book Numerical Methods in electromagnetics Edition 3.
Now the ...
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Acoustic Simulation, how are boundaries handled?
I don't have a background in numerical modeling so this question is rather broad.
What I am interested in is modeling the propagation of an ultrasonic acoustic wave in 3d space. The basic 3d wave ...
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Gauss-Lobatto quadrature and nodal points for FEM
By using the Legendre-Gauss-Lobatto (LGL) quadrature formula (QF) and LGL nodal points one achives a diagonal mass-matrix for finite element problems. (More specifically, the spectral element method.)
...
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Split of complex parts in weak form
I am working on a numerical model to simulate the acoustic and elastic wave propagation in frequency domain via the Finite Element Method. Basically, the problem is to solve the Helmholtz equation in ...
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Solve wave equation with discontinuous coefficients numerically?
I would like to solve the following equation
$$\frac{\partial^2 y}{\partial t^2} - c^2(x,t)\frac{\partial^2 y}{\partial x^2}=0,$$
for $y=y(x,t)$ numerically. The wave speed, $c(x,t)$, is of the form
$$...
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What is the meaning of the Helmholtz wave equation?
I am trying to build understanding on the Helmholtz wave equation $\Delta p + k^2 p = 0$, where $p$ is the deviation from ambient pressure and $k$ the wave number, in order to use it in numerical ...
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Methodology Suggestion for Wave-propagation Problem using Finite Elements
I want to simulate the propagation of a sinusoidal plane wave in a rectangular domain using Finite Elements Method. First, the wave should propagate through a fluid medium, then it will encounter a ...
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Partial differential equation FEM application
I have a PDE which looks like Helmholtz wave equation on one dimensional domain.
$$\dfrac{d^2u(x)}{dx^2}+\pi^2u(x)=f(x)$$
where $-\infty <x<\infty $
Also, $f(x)= 1$ for $-0.25<x<0.25$, I ...
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Circumferencial waves on a cylinder/sphere
I was wondering how we can introduce $e^{ik.x}$ terms associated with circumferencially propagating waves? In this case $\hat{e}_\theta$ is the direction of wave propagation. However, I was not able ...
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Simulating pressure waves at an impedance boundary
I am trying to simulate pressure waves crossing a boundary from one medium to another (e.g., water to air) in Matlab. The code that I have got so far, which is largely taken from Wikipedia on Partial ...
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If not MATLAB, what software/programming language should I use to simulate/animate wave functions in various potentials + more? (example given)
I want to integrate programming into my learning in math and science in a very specific way. I want to create visualizations and simulations of concepts I am learning. When I learn a numerical method ...
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Numerical solution of zero-potential time-dependent Schrödinger equation in 1D
I want to solve numerically the one-dimensional time-dependent Schrödinger equation $$i \psi_t(x,t)=-\frac{\hbar}{2m} \psi''(x,t)$$
My issue is that I don't have the physical background to understand ...
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Modified Equation and Stability for Centred Finite Differences for Wave Equation
I am trying to use the modified equation to derive the stability condition for the finite difference approximation
$$
\frac{u(x,t+\Delta t) - 2 u(x, t) + u(x, t -\Delta t)}{\Delta t^2} = c^2 \frac{...
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What is the right way to set up two random tensor fields which have an identical average diffusivity
I want to compare some properties of traveling waves through two randomly diffusive media. The traveling waves follow the fisher equation:
$$\frac{du}{dt} = \nabla(\mathbf{D}_{\gamma} \nabla u) + u(1-...
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Incorporating a potential barrier in a wave-packet simulation (Fourier Transform method)
I'm trying to simulate the scattering of a wave-packet at a potential barrier in Python. I'm using a Fourier Transform method (not sure if its the same as the Split-Step method), where I apply Fourier ...
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Finite difference for 1D wave equation: why the spike initial data results in a noisy output?
I am using a second-order finite difference in space and time approximation for the 1D wave equation.
No source but initial data: $I(x)=\mathrm{e}^{-400 (x-0.5)^2}$.
Velocity $c=1$, $nx=501$, $nt=...
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Why is my simulation of a first-order wave equation not stable?
According to the equation
$$ \frac{\partial y}{\partial t} = -a\frac{\partial y}{\partial x} $$
I simulated this in python. I used center differentiation, and I determined step size based on Von-...
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How can I solve the wave equation for a circular rod in cylindrical coordinates using finite differences?
I have a problem with the stability of finite difference method for the wave equation in cylindrical coordinates.
the equation is:
$$
\frac{\partial^2 \omega_n}{\partial r^2}+\frac{1}{r}\frac{\...
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Introducing EigenModes from 2D FEM into 3D FEM
This particular FEM question concerns waveguides and FEM 3D simulation. To excite a waveguide with waveport (TE10 and so on), we typically have to solve for eigenvalues ($k$) of helmholtz equation ...
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shallow water equation maccormack method
I am trying to make a code for 1D shallow water equation (nonlinear without source terms) using the MacCormack method for sinusoidal wave propagation. My issue is that the wave fluctuates and does not ...
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Trying to plot 1D wave equation for benchmarking
I am trying to plot a reference solution for the 1D wave equation using python.
The above link states the following: For a rod fixed at the right end and free at the left end and subjected to a ...
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Simulation of a lens, insufficient points
I am simulating the propagation of a light pulse using the equation
$$\frac{\partial}{\partial z}A=\frac{1}{2\cdot k_0}\nabla^2_rA$$
with
$$k_0=\frac{2\pi}{\lambda_0}$$
The propagation with a step ...
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How do we implement Parameter free generalised Moment limiter in 1D Case in Discontinuous Galerkin methods?
I am referring to this paper:-
"A Parameter-Free Generalized Moment Limiter for High-
Order Methods on Unstructured Grids " by Michael Yang and Z.J. Wang.
http://dept.ku.edu/~cfdku/papers/AIAA-2009-...
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Discrete wave simulation - absorbing boundaries?
I wrote a simple 2D wave simulation using the following equations:
$$\frac{\partial^2 u}{\partial t^2}=c^2\nabla^2u$$
Where $\nabla^2$ is the discrete laplace operator using a Von Neumann neighborhood ...
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Time step relationship with number of elements or material properties
When looking at the output file of my solver, I have been told that the time-step taken by the solver depends on parameters like the total number of elements and their relative size in my geometry, or ...
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CFL condition in polar coordinates
In this question, I suggested that the Couran-Friedrichs-Lewy (CFL) condition for the wave equation in polar coordinates reads
$$C = 2c\frac{\Delta t}{\Delta r \Delta \phi} \leq C_\max \enspace ,$$
...
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Degree of freedom for elastic wave propagation problem
I am solving a elastodynamics (vector valued elastic wave) equation.
I create the 2D mesh in Gmsh discretised into triangular elements of second order. Therefore, it is my understanding that the ...
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Regarding solution vector of the wave equation
I am simulating the wave equation using FEM. For a 2D wave equation, when I visualise my output in Paraview, I see a separate solution in 'x' and 'y' direction for each node on the mesh. Therefore, if ...
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Damping for Dynamic Problem using FEM
I came across this form of damping implemented in an elastodynamics problem.
The stress tensor without the damping would look like:
$ \sigma = 2 \mu \epsilon + (\lambda \, \text{tr} (\epsilon)) I $
...
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How to use non-dimensional form in open source codes instead of Units
I am using an open source FEM platform, which requires you to convert your equation system to non-dimensional form. So, there are no units specified for the parameters in the problem. If you use ...
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Using backward vs central finite difference approximation
I am solving the simple 2nd-order wave equation:
$$ \frac {\partial ^2 E}{\partial t^2} = c^2 \frac {\partial ^2 E}{\partial z^2} $$
Over a domain of (in SI units):
$ z = [0,L=5]$m, $t = [0,t_{max} ...
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Perfectly matched layer simulation with two vibrating sources
I'm doing PML simulation, and while I was looking for correct geometry for my layers of propagation, I got a question.
My device has two vibration sources, and this vibration will propagate through ...
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Seismic Wave modelling: Elastic Wave or Acoustic Wave?
I am modelling a seismic wave equation using FEM. In the few papers that I read, I understand the following: (Kindly correct me if you disagree)
A shear (secondary wave - no change of volume) is more ...
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