Questions tagged [wave-equation]
For questions related to solutions and analysis of the wave equation.
1,283 questions
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Why is the amplification factor in a finite difference method for PDEs constant regardless of the time?
I'm trying to prove the Courier condition of $r=\frac{c \Delta t}{\Delta x}\leq 1$ for a hyperbolic PDE of the form $\dfrac{\partial^2 y}{\partial t^2}=c^2\dfrac{\partial^2 y}{\partial x^2}$
Through ...
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Why this particular change of variables in this PDE?
I am trying to study some methods of resolution of PDEs, for my exam of mathematical methods for physics. Currently I am reading “A guide to mathematical methods for physicists” (volume 2) by Petrini, ...
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How to implement a source term on the boundary of a finite volume (or discontinuous galerkin) method?
I've written a 3D linear acoustic discontinuous galerkin method that works on an unstructured tetrahedral mesh. This models pressure, $p$ and velocity, $\mathbf{u}=(u,v,w)$ and has fully reflective ...
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How to reduce the $(2+1)D$ radially symmetric linear wave equation to the $(1+1)D$ linear wave equation?
I cannot verify the result in this question. It asks for a solution to the 2D linear wave equation, assuming the initial conditions are radially symmetric about some point,
\begin{equation}
ru_{tt} = (...
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Is the following linear wave equation $ \partial_\alpha\bigl(a^{\alpha\beta}\,\partial_\beta\phi\bigr) = F $ globally well-posed?
Consider the general linear wave equations of the form
$$
\partial_\alpha\bigl(a^{\alpha\beta}\,\partial_\beta\phi\bigr) = F,
\qquad
(\phi,\partial_t\phi)\bigl|_{t=0} = (\Phi,\Phi')
$$
where $a^{\...
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Wave equation with Dirichlet boundary conditions and constant derivative on part of the boundary
A solution of the one dimensional wave equation, with homogeneous Dirichlet boundary conditions, can have constant normal derivative at one boundary?
Precisely, given $T>0$ and $Q=(0,T)\times (0,1)$...
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Confusing notation and derivation of Higher Order Commutator Identity
I'm working on the following problem 3.2 from these lecture notes https://home.uni-leipzig.de/gajic/download/nlw_lecturenote_final%20(1).pdf on nonlinear wave equations. Let $k \in \mathbb{N}_{\geq 1}$...
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Question on D'Alembert's formula
I am trying to obtain D'Alembert's formula for the solution of the Cauchy problem for the 1-dimensional wave equation (with initial conditions $u(x,0)=\varphi(x)$ and $u_t(x,0)=\psi(x)$) starting from ...
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Wave equation and Lorentz transformation
Consider the linear wave equation $$u_{tt}-\Delta u = 0$$ with compactly supported initial data $f,g\in C_c^\infty(\mathbb R^3)$, i.e. $u(0,x) = f(x)$, $u_t(0,x) = g(x)$. Now consider a lorentz ...
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meaning of the solution of the wave equation
Consider $u(x,t)$ the solution of
$
u_{tt} = \Delta u, \: t> 0, \: x \in \mathbb{R}^d
$
with initial conditions for the $u(t=0,x) = u_0(x)$ and $u_t(t=0,x)=v_0(x)$.
What does $u(x,t)$ represent ...
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Exercise on wave equation
I am trying to do the following exercise on wave equation:
Let $\alpha\neq -1$ be constant. Determine the solution of the following problem:
\begin{equation*}
u_{tt}-u_{xx}=0, u(x,0)=\varphi(x), u_t(...
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Local smoothing conjecture on Riemannian manifolds
In this talk: https://www.youtube.com/watch?v=14qX6L95NOI , at about 38min, Larry Guth said that the (2+1) dimension case of the local smoothing conjecture is false due to Minicozzi-Sogge (I believe ...
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Finite speed of propagation of two wave equations
I attempt to prove the finite speed of propagation property of the following wave equation using the energy method，
$$
\begin{aligned}
u_{tt}-\Delta u&=uv,\\
v_{tt}-a^{2}\Delta v&=uv,\,(0<a\...
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Pressure field from a sound source located exactly at the interface between two media
Background
I have asked this question on physics stack exchange, but was encouraged to migrate it over here.
There are loosely related questions here, here, here and others, but I did not understand ...
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How to find the center of compression of a longitudinal wave?
The following plot is generated at $t=0$ and with $A=1$, $k=1$ and $v$ (unspecified)
$$s(x,0)= x + \sin x$$
based on the equation of motion of a single particle.
$$s(x,t)= x + A \sin k(x-vt)$$
As ...
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Stability Analysis of the Wave Equation using Crank-Nicolson Method - No Clear Time Step Restriction?
I am analyzing the stability of the following wave equation:
$$
\frac{\partial^2 \phi}{\partial t^2} = c^2 \frac{\partial^2 \phi}{\partial x^2}.
$$
I applied the Crank-Nicolson method directly to the ...
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Wave equation, partial differential equation formulation
I want to formulate the problem of partial differential equation into a mathematical form.
Suppose a string of length $L$ is tied to both ends. Then the string
is released from rest with initial ...
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Fourier Analysis, Stein and Shakarchi, Chapter 6 Exercise 10
Problem
Let $u(x, t)$ be a solution of the wave equation, and let $E(t)$ denote the energy of this wave
$$E(t) = \int_{\mathbb{R}^d}\left|\frac{\partial u}{\partial t}(x, t)\right|^2dx + \sum_{j = 1}^...
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How to show the orthogonality of associated Laguerre Polynomials in the Hydrogen Atom Radial Equation
In "Topics in Atomic Physics", Springer, by Charles E.Burkhardt and Jacob J. Leventhal, Chapter 4, Section 4.5.
To normalize the radial equation of hydrogen atom, it is necessary to use the ...
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I'm having trouble with solving this wave equation on the interval $[0, \pi]$
The wave equation is the follwing:
$U$$tt$$-$$c^2U$$xx$$=0$, $x\in[0, \pi]$
$U$$x$$(0,t)=0$, $U(\pi, t)=0$
$U(x,0)=0$, $U$$t$$(x,0)=-12c· cos(\frac{3}{2x})$
I began by expressing the solution in the ...
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Confusion about the sign of separation constant to use when solving PDE-IBVPs
This is just a general question regarding solving PDE-IBVPs for the standard Heat, Wave, Laplace equations with the Method of characteristics (in 1D and 2D respectively). Could someone please explain ...
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Equivalence of single/double integral forms of 1D wave equation with a source
My question is about the wave equation on the infinite line with zero initial data and a source $F(x, t)$. I've seen two forms of the solution for this, and I would like to know how they are ...
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Trouble in understanding a thing or two about wave equations in a PDE course.
I was studying the book "Linear Partial Differential Equations for Scientists and Engineers" by Lokenath Debnath and Tyn-Myint U.
There was a chapter dedicated to wave equations and I am ...
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Geometrical name for the group of circles of decreasing diameter whose centers are displaced along a straight line, as observed in Doppler Effect
The wavefront received by a receiver from a moving source, or vice versa, is shown in the literature as circles of decreasing diameter whose centers are displaced along the direction of motion. I have ...
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1D Wave Equation Rigor Question
Suppose we want to solve the following wave equation: \begin{cases}
u_{tt} = c^2 u_{xx} \text{ for } 0 < x < 1, t > 0 \\
u(x, 0) = x(1-x) \text{ for } 0<x<1 \...
