Questions tagged [wave-equation]

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

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D'alembert solution with Neumann and Dirichlet boundary conditions

How can I solve the wave equation with the D'alambert solution in a finite domain with one end of the string clamped and the other end free to move vertically? i.e. $u_{tt}=c^2u_{xx}, \quad 0<x<...
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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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Effect of boundary conditions on general solution

I am having problems integrating given boundary conditions on a wave-equation. The problem is as stated below. I am no expert in solving PDE's, so please forgive if I oversee something obvious or &...
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Regularity for the wave equation on the half line

I would like to know the regularity of the wave equation under simple conditions. Consider the wave equation on the half line: $$ \begin{cases} u_{tt}-\triangle u = 0, \quad (x,t) \in (0, \infty]\...
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A question about the derivation of Poisson formula with respect to 2-dimensional wave equation

In section 2.4 (wave equation) of Partial Differential Equation (Evans), the author used the descent method to derive the Poisson's formula for two-dimensional wave equation. For the $n=3$ case, the ...
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Compact supports of initial data implies compact supports for all $t$ in semi-linear wave equation $-\frac{\partial^2}{\partial t^2}u+\Delta u=u^3$.

Problem: Let $u(t,x,y)$ be a smooth real function defined on $\mathbb R \times \mathbb R^2$ where $t \in\mathbb R$ and $(x,y) \in\mathbb R^2$. We assume that it solves the following semi-linear wave ...
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Energy estimate of linear wave equation.

This is a question coming to me when I was reading Qian Wang's Lectures on Nonlinear Wave Equation. Link below: http://people.maths.ox.ac.uk/wangq1/Lecture_notes/nonlinear_wave_9.pdf Let $\square_g:=\...
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How to solve this Wave equation for a moving medium $[\frac{1}{c^2}(\frac{\partial}{\partial t} + u\cdot\nabla)^2 - \nabla^2]\phi=0$

$$[\frac{1}{c^2}(\frac{\partial}{\partial t} + u\cdot\nabla)^2 - \nabla^2]\phi=0$$ How to solve for $\phi \in R$ given $u \in R$ ? comment: ofcourse if $u=0$ the solutions are familiar ones (sinusoids)...
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Dispersive Waves on a Semi-Infinite String

I am having an insane amount of trouble figuring out this problem that I solved probably ten years ago. Googling leads to solutions that make use of D'Alembert's formula, but that doesn't work for ...
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Verifying Ill-Posedness of Wave Equation System

Consider the wave equation $u_{tt} = c^2 u_{xx}$ with $u(0, x) = f(x)$ and $u_{t}(0, x) = g(x)$. By defining $u_1 = u_t$ and $u_2 = u$, we obtain the following system: $$\begin{cases}\frac{\partial ...
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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 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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Find the phase shifts that minimize the amplitude of the sum of a few sine waves of different period and amplitude?

I need to find the phase shifts that minimize the amplitude of the sum of a few sine functions. Example: $y_1 = max(a_1 * sin(b_1x + c_1))$ $y_2 = max(a_2 * sin(b_2x + c_2))$ ... $y_n = max(a_n * sin(...
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How to find $\langle x\rangle$ and $\langle p\rangle$ of a particle in an infinite square well if wave function at $t=0$ is $Ax(a-x)$?

I have tried finding it and i got $\langle x\rangle = 4a/15$. The method I used is as follows: Integrate $x|\Psi(x,0)|^2$ from 0 to $a$. But I don't know if it is correct because we always do it by ...
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On coefficients of hyperbolic and harmonic solutions to the heat equation

In this post I show an example of finding the coefficients of the heat equation. The same principles applies for the Laplace equation and for the wave equation. However one question that turns to ...
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The wave equation in space, is it a first-order hyperbolic linear system?

A first order linear system is any PDE of the form $$q_t(x, t) + A q_x(x, t) = 0$$ Where $q(x,t):\mathbb{R}\times\mathbb{R} \to \mathbb{R}^n$ is a vector function, and $A$ is an $n\times n$ matrix ...
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Alternative derivation of d'Alembert's solution using the method of characteristics.

The Problem. Consider the initial value problem for the wave equation, $u_{tt} = c^2 u_{xx}$, $x \in \mathbb{R}$, $t > 0$, with initial conditions $u(x,0) = \phi(x), u_t(x,0) = \psi(x)$. The ...
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Stability of a Linear wave equation

The question is as follows: For solutions $\phi(x,t)= \phi_{0} e^{ikx-i \omega(k)t}$, find the dispersion relation, group and phase velocities, and then comment on the stability of the corresponding ...
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Finding the coefficient for the wave equation

I have the given problem, \begin{equation} \alpha u_{xx}=u_{tt} \end{equation} with conditions: \begin{equation} \begin{cases} u(0,t)=u(L,t)=0\\ u(x,0)=x \\ u_t(x,0)=0 \end{cases} \...
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Crank-Nicolson method for wave equation without a change in variable

I am trying to solve the 2D/3D wave equation using FEM (c = 1 for simplicity). I have a constraint that the solution should only be in terms of displacement. so $$\frac{\partial^2 \textbf u}{dt^2}-\...
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Need help - Robin condition for a 1d wave equation on the first quardant

Given $\alpha \neq 0$ and $$u_{tt}-c^2u_{xx} =0 \quad x,t>0,$$ $$u(x,0) = f(x), \quad x\geq 0,$$ $$u_t(x,0)=g(x), \quad x\geq 0,$$ $$u_x(0,t)+\alpha u(0,t)=0, \quad t\geq 0.$$ I know it’s a Robin ...
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Using D'Alemberts formula for a solution of a general wave equation (without specific I.C.)

So, I have the general wave equation \begin{equation} c^2u_{xx}=u_{tt} \end{equation} with given I.C. : $u(x,0)=g(x)$ and  $u_t(x,0)=h(x)$ I have to use D'Alemberts formmula on the solution. ...
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Energy Estimate for Nonlinear Wave Equation in $H^s$ Space

In Theorem 2.1 of Prof. Lawrie notes (which can be found here), he provided without proof the following result: Let $(f,g)\in H^s\times H^{s-1}$ and $h\in L^1([0,T], H^{s-1})$. Then the linear wave ...
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Solving a BVP with the wave equation by using the Laplace Transform

What follows is an example on how to solve a BVP using the Laplace Transform. I will copy and paste the example and its solution from the book that I am using and then I will ask my question. When ...
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1 answer
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Show that this is a solution of this wave equation

Given the wave equation $\frac{1}{c^2} \frac{\partial^2}{\partial t^2}u=\Delta u$ where $\Delta u$ is the Laplacian operator and a function $g$ that's two times continuously differentiable, show that $...
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Convolution of the fundamental solution of the heat equation

Let $$F(x,t)=\frac{1}{\sqrt{4\pi kt}} e^{-\frac{x^2}{4kt}}$$ be the fundamental solution to the heat equation and observe $$\int_{-\infty}^\infty F(x-y,t) dx=1$$ for arbitary $y$. Next, assume that $\...
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Is it true: the energy is not preserved in wave equation with transverse force

The wave equation with transverse force is given as following: $u_{tt}-c^2u_{xx}+ku=0$ $(k>0)$ Define the total energy $$E(t)=\frac{1}{2} \int_{-\infty}^{\infty}((u_t)^2+c^2(u_x)^2)dx$$ There are ...
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Uniqueness of Homogeneous Wave Equation with Initial Conditions

In Selberg's PDE lecture notes, to prove Theorem 1 (which is equivalent to proving uniqueness of homogeneous wave equation), he defined the energy function as $$ E(t) := \frac{1}{2}\int_{B_t} |\...
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In 3 dimensions, does it hold that $\int r^{-2} h\le C\int r^{-1}(1+r)^{-1} h$?

in theorem 9 from these notes notes, after computing the divergence of the flux, I get the coefficient of the $\phi^2$ term to be similar to $\frac 1{r(1+r)^{2+\delta}}$, so upon plugging into the ...
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Estimate $\Vert \Delta u \Vert_{2}$ for wave equation [closed]

We consider the wave equation \begin{equation}\label{1} \left\{ \begin{array}{ll} u_{tt}(x,t)-\Delta u(x,t)=0, x \in \Omega, t>0\\ u=0, \quad u \in \partial \Omega, t>0 \\ u(0,x)=u_{0}(x), \quad ...
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Finite Lenght Wave equation With Only Initial Conditions.

Let $u(x,t)$ be a solution of $$u_{tt}=u_{xx}; 0<x<1, u(x,0)=x(1-x), u_t(x,0)=0$$ Then $u(1/2,1/4)$ is $1$. $3/16$. $2$. $1/4$. $3$. $3/4$. $4$. $1/16$. If i apply D’Alembert formula for Wave ...
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Numerical solutions to the 3D wave equation

I am doing a research to explore the existing numerical schemes that are used to solve the $3$D wave equation. The standard form of the problem in $3$ dimensional setting is : $$\Delta u= \frac{1}{c^2}...
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1 answer
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Are P and S waves well-defined if we account for heterogeneity of Lamé parameters?

Consider the isotropic elastic wave equation. \begin{align} \rho \ddot{\boldsymbol{u}} = \nabla \cdot \boldsymbol{\sigma} + \boldsymbol{f} \end{align} where \begin{align} \boldsymbol{\sigma}(\...
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Wave equation $u_{xx}-u_{tt}=e^x+6t$.

Consider the Wave equation $u_{xx}-u_{tt}=e^x+6t,x\in\mathbb R,t>0$ with initial condition $u(x,0)=sin(x),u_t(x,0)=0$, then value of $u(\frac{\pi}{2},\frac{\pi}{2})$ is $1.$ $e^{\frac{\pi}{2}}(1+\...
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How to learn Strichartz estimates for wave equations?

I am a student planning to learn some knowledge about Strichartz estimates for wave equations. My goal is to understand the Strichartz estimates or a priori estiamtes of weak solutions for the linear ...
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Initial Value Problem for Modified Wave Equation with Cosine Term

I'm trying to solve this initial value problem for a wave equation with a cosine term. I've been solving other wave IVPs using d'Alembert's formula but I'm not sure how to deal with the extra term in ...
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Is there a wave equation which straightens/rounds the lines between troughs and crests?

I'm looking for a periodic wave-shape that can transform from something like a sine wave to a zigzag. I'm particularly interested in: the straightening/rounding of the curve between the crests and ...
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1 answer
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How to find the phase shift given two graphs?

I am currently stuck on a problem in my physics book but its the math that I am having trouble with. I am trying to find the phase shift of a sine function of the form $$ s(x,t) = A\,\sin \left\{ 2\pi ...
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Proof: If a solution of the nonhomogeneous wave equation exists, then it is unique.

So I was going through the proof of a proposition which states that, 'If a solution of the nonhomogeneous wave equation exists, then it is unique.' It is the very first proof in the following pdf: The ...
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is it possible to solve this equation for x?

Is it possible to solve this equation for x? $$\frac{-f+e^{tix}(f \cdot cos(f t)+x \cdot i \cdot sin( f t))}{2 \pi (x^2 - f^2)} = n$$ im writing a program for a fourier analysis, and I need to ...
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Modeling a Bowl of Water's Displacement

As far as I understand, if an object (think something small and heavy) is dropped into a bowl of water, the displacement of every point on the surface of the water at every time after that will be ...
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Phase difference between two waves

I'm facing much confusion about a rather silly problem recently. Suppose I have two functions $\sin(x+\phi_1)$ and $\sin(x+\phi_2)$. Defining $\theta_1=x+\phi_1$ as the phase of the first and $\...
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How would one describe $k$ iterations of $\cos(n)$?

What function would one use to describe $k$ iterations of $\cos(n)$? I'm pretty sure that the function would be a damped sine wave (as can be seen in the curve fit equation I wrote in the third row), ...
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Behavior of solution to the wave equation as $t\to \infty$

The equation $u_{tt}-9u_{xx}=0$ has the initial data $$u(x,0)=f(x) = \begin{cases} 1, &\vert x\vert\leq2 \\ 0, & \vert x \vert>2 \end{cases}$$ $$u_t(x,0)=g(x)=\begin{cases}1,&\vert x\...
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Two-weak shock solution of Riemann problem

Trying to find the shock solution for a system of conservation laws: $$ u_{t}+ \frac{1}{2}(u^2+v^2)_{x}=0$$ $$v_{t}+(uv)_{x}=0$$ for the left state $\mathbf{u}_{L}=\mathbf{u}_{0}$ and right state $\...
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3 votes
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Partial Differential Equations With Two Solution Paths

When solving a $2$D Heat Equation, suppose I separate the solution into time and space, i.e., $f_1(t,\ T(t),\ T_t(t),\ ...) = f_2(x,\ y,\ Z(x,\ y),\ Z_x(x,\ y),\ Z_y(x,\ y),\ ...) = \lambda$, and then ...
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2 votes
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Drum PDEs, Double Fourier Expansions, and Synthesis

In studying the $2$D Wave Equation, the application often encountered is the displacement of a drum. The main solution to the PDE is a double Fourier summation, either a double Fourier Sine Series, a ...
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Rankine-Hugoniot for system of autonomous conservation laws

We have a system of autonomous conservations laws: $$ \boldsymbol{u}_t + (\boldsymbol{f}(\boldsymbol{u}))_x=0 $$ with shock solutions of Riemann problem, left state $ \boldsymbol{u}_L=\boldsymbol{u}_0 ...
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General non-homogeneous PDE

I want to solve the following: $$ u_{tt} - c^2u_{xx} = F(x)\cos(\omega t) \\ u(0, t) = u_x(\pi, t) = u(x, 0) = u_t(x,0) = 0$$ First of all I looked for a solution of the form $ u(x, t) = X(x)\cdot T(t)...
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General integral solution to separable partial differential equation

Given a partial differential equation such as the wave equation: $$ \frac{\partial^2 \psi}{\partial^2 x} = \frac{1}{c^2}\frac {\partial^2 \psi}{\partial^2 t},$$ we look for separable solutions of the ...
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Exact solutions to a kind of 3D wave equation

Consider the mapping ${\bf{u}}: [0,T] \times \mathbb{R}^3 \mapsto \mathbb{R}^3$, defined by ${\bf{u}} = {\bf{u}}(x,y,z,t)$. I am looking for general solutions ${\bf{u}}$ of the following PDE $$ \frac{\...
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