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Questions tagged [wave-equation]

For questions related to solutions and analysis of the wave equation.

The wave equation is a linear second order PDE that describe sound waves, light waves and water waves. It is defined by

\begin{equation*} \frac{\partial ^2 u}{\partial t^2}=c^2\frac{\partial^2 u}{\partial x^2} \end{equation*}

and can be derived from the mathematical model of a string vibrating in a two-dimensional plane where each elements are pulled in opposite directions.

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Intuitive explanation of the difference between waves in odd and even dimensions

Motivation: In odd dimensions, solutions to the wave equation: $$u_{tt}(x,t)=\nabla^2 u(x,t), \qquad u_t(x,0)=0, \qquad u(x,0)=f(x)$$ where $t \geq 0$ and $x \in \mathbb{R}^n$, have the nice property that the value of $u(x,t)$ only depends on the…
Sune Jakobsen
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Wave equation: predicting geometric dispersion with group theory

Context The wave equation $$ \partial_{tt}\psi=v^2\nabla^2 \psi $$ describes waves that travel with frequency-independent speed $v$, ie. the waves are dispersionless. The character of solutions is different in odd vs even number of spatial…
Sal
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Interpretation of an integral transform from the wave equation to the heat equation

I'm having troubles with understanding the physical meaning of a certain transform. If $u$ is a solution to the wave equation $$\partial_t^2u-\Delta u=0\ \mathrm{in}\ \mathbb{R}^n\times(0,\infty)\\u=g,\ \partial_tu=0\ \mathrm{on}\…
Maximilian M.
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Proving that $\int u^2 dx < Ce^{-at}$ where $u$ solves a linear pde

Let the unit sphere in $\mathbb{R^n}$ be $B_1(0)$ and let $u$ be the smooth solution of $$ \begin{cases} u_{tt} + a^2(x) u_t - \Delta u = 0 & B_1(0) \times (0,\infty)\\ u(x,t) = 0 & \partial B_1(0) \times (0,\infty) \\ u(x,0) = g, \; u_t(x,0) = h &…
Merkh
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Nonhomogeneous wave equation

In PDE Evans, 2nd edition, page 80 ... 2.4.2 Nonhomogenous problem We next investigate the initial value problem for the nonhomogeneous wave equation \begin{cases} u_{tt} - \Delta u = f & \text{in } \mathbb{R}^n \times (0,\infty) \\ u=0, u_t=0 &…
Cookie
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Representation theoretic explanation of Huygens' Principle

Huygens' principle -- Wave equations in $\mathbb{R}^n$ seem similar for different $n$, but behave quite differently depending on the parity of $n$: waves in odd dimensional spaces never look back, while waves in even dimensional spaces linger. A…
Student
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Robin BC in the 1D wave equation

The problem of interest is as follows: the unknown: $u(x,t)$ the wave equation: $\partial_2^2u(x,t)-c^2\partial_1^2u(x,t)=0$ where $c>0$ one Robin boundary condition at $x=0$: $\partial_1u(0,t)=\alpha u(0,t)$ where $\alpha>0$ since the Robin…
pluton
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Why don't elliptic PDE's have a time coordinate?

Usually second-order linear PDE's are classified as elliptic, parabolic, or hyperbolic (or ultrahyperbolic) depending on the eigenvalues of the coefficient matrix. The three cases correspond to the three most famous second-order PDE's: Elliptic -…
Jim Belk
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PDE : Mixture of Wave and Heat equations

Today I was given the following equation : $$\frac{1}{c^2}u_{tt} + \frac{1}{D}u_t = u_{xx}$$ with initial conditions : $u(x,0) = 1$ if $|x|
user88595
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Terry Tao didn't list the "dozens and dozens of wave equations out there" in his talk, but is there such a list somewhere?

In the new Simons Foundation video Terence Tao - From Rotating Needles to Stability of Waves: Local Smoothing... (November 29, 2023) after 01:15, Tao says: In mathematics we describe waves by partial differential equations, and there are dozens and…
uhoh
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Waves in spaces of even dimension

In an article by Jim Holt, "Geometric creatures," he says: In a space with an odd number of dimensions, ... sound waves move in a single sharp wave front. But in spaces with an even number of dimensions, ... a noise-like disturbance will…
Joseph O'Rourke
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Why there is no d'Alembert solution in 2D?

In 1D the wave equation $$\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial r^2}$$ can be satisfied with a wave $$u(r,t) = f(r-ct).$$ In 3D the wave equation $$\frac{\partial^2 u}{\partial t^2} = c^2…
Yrogirg
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Solving the wave equation, with boundary conditions, in the sense of distributions (Generalized functions)

After learning some distribution theory, I find that in my book, all PDEs given as examples are in free space (without any boundary conditions). I wonder if distribution theory can be used to tackle PDEs with boundary conditions. To be more…
Ma Joad
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What is a soliton?

I have read about solitons and they seem to be a big deal as a phenomenon. The definition I find from wikipedia is that solitons are characterized by the following three properties: They are of permanent form They are localized within a…
Jan Lynn
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Show that the spherical wave equation reduces to the Laplacian equation near the origin

Consider the acoustic wave equation: $$\left(\frac{1}{c^2}\frac{\partial^2}{\partial t^2}-\nabla^2\right)p=0$$ where $p$ is the pressure perturbation in the medium. A particular solution of this equation, for the case of spherical waves,…
Élio Pereira
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