Questions tagged [wave-equation]

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

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30 views

Wave Equation: what do Characteristic Curves mean?

This is my question: Compute the characteristic curves of the following wave equation $$ \frac{\partial^{2} u}{\partial t^{2}}-a^{2} \frac{\partial^{2} u}{\partial x^{2}}=0 $$ and draw them ...
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5 views

Can impulses offset in time be added in wave propogation?

I'm following this post to understand convolution. The post uses the example of stones being dropped on water to illustrate the uses of convolution. The post uses the idea of a fundamental solution i....
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8 views

harmonics and wave equation

I would like to ask if there how I could see from solution (by the method of separation) of 1-D wave equation that e.g. a string creates harmonics and not just overtones as e.g. a drumhead. What shall ...
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1answer
20 views

Solve Wave Equation Initial-Boundary-Value-Problem

I am trying to solve the following problem and this is my working so far. I'm struggling to get to the general solution for $X(x)$ as I'm not sure of the $\lambda$ value. Please could someone point ...
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1answer
23 views

Superposition of waves to find a general solution

I am in trouble to understand the "spirit" of the following discussion that my teacher wrote on his notes. Consider $$\ddot q_k=-\omega_0^2q_k-a(2q_k-q_{k+1}-q_{k-1})\qquad\qquad\qquad (1)$$ with $k\...
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pls help me with materials explaining concretely the FEM for this problem [closed]

$\min\limits_{(u,y) \in U \times X} J(u,y) =\frac{1}{2}\parallel y - y_{d}\parallel^2_{L^2(\Omega)} + \frac{\alpha}{2}\parallel u \parallel_{L^2(Q)}$ Such that \begin{equation} y_{tt}-\bigtriangleup y ...
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32 views

Solving $U_{xx} - 3U_{xt}-4U_{tt}=0$ with $U_{x}(x,0)=x^2, U_{t}=e^x$ Using D'Alembert's solution

So we have the following problem $$ U_{xx} - 3U_{xt}-4U_{tt}=0 \tag{1}\\ U(x,0)=x^2 \\ U_{t}=e^x $$ So Solving this by factorising and using a change of variables we get $$U_{xx} - 3U_{xt}-4U_{tt}=\...
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why this boundary condition is given?

In my book it is given the following BV problem: $u_{xx}=u_{tt},\qquad-\infty<x<\infty$ (1) $u(-\infty,t),u(+\infty,t)<\infty$ (2) $u(x,0)=f(x),\qquad u_t(x,0)=g(x)$ (3) I ...
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28 views

Wave equation solution $u(x,t)$ for $u_0(x)=x$ and $v_0(x)=0$

Let a string with linear density $\rho$ and tension $k$ Its left and right hand ends $ [-\pi,\pi]$ are held fixed at height zero (Maybe not at $t=0$, but it are for $t>0$). Initial velocity $...
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6 views

How to solve following wave equation?

Use the CTCS scheme with X2-X1=1/3 and t2-t1=1/5 to solve the wave equation Utt(x,t)=Uxx(x,t) with the boundry conditions: U(0,t)=U(1,t)=0 0<=t<=0.6 U(x,0)=(3^1/2)sin(pi....
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33 views

Solution of the Wave Equation by Separation of Variables

We know for $\left\{\begin{array}{cccc}\dfrac{1}{c^2}u_{tt}&=&u_{xx}\hspace{0.25cm}x\in [0,L]&\\ u(0,t)&=&u(L,t)=0\\ u(x,0)&=&f(x)\\ u_t(x,0)&=&g(x) \end{array}\...
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23 views

Wave equation PDE estimate of the second derivative

Let $\phi \in L^2((a,b) \times (0,T) )$ such that $$\phi_{xx} = \phi_{tt} + a \phi$$ where $a \in L^{\infty}((a,b) \times (0,T) )$. I wanted to prove the following estimate: $$||\phi_{xx}||_{L^2(a,b;...
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2answers
50 views

Use the Laplace Transform to solve the wave equation

We are given $u_{tt}-u_{xx}=0$, $\forall t>0,x \in \mathbb R$ subject to $u(x,0)=sin\pi x, u_t(x,0)=0$, $\forall x\in \mathbb R$ Use the Laplace Transform to solve the wave equation. -- This ...
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45 views

Let a string with linear density $\rho$ and tension $k$. Find $u(x,t)$

Let a string with linear density $\rho$ and tension $k$ Its left and right hand ends $ [-\pi,\pi]$ are held fixed at height zero (Maybe not at $t=0$, but it are for $t>0$). Initial velocity $...
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0answers
13 views

spherical Bessel - Legendre relation, incident wave

I was reading the partial wave expansion for incident and scattered wave. I cannot understand two things: 1. Why in this process it indicates that the relation indicated in the following picture is ...
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1answer
16 views

Simplification of Three Integrals Partial Differential Equations [closed]

I am using Kirchoff's formula to solve wave equation in three dimensions, and I have the following three integrals: $\int_{\partial B(x_0,t)}t*|y-x_0|^2dS(y)$ $\int_{\partial B(x_0,t)}|y-x_0|^6dS(y)$...
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1answer
38 views

Mathematical references that deal with Kirchhoff's integral theorem?

I've seen Kirchhoff's integral theorem applied to the wave equation and similarly to the Helmholtz equation in optics books, but I am interested in gaining a mathematician's point of view on this ...
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9 views

non-homogeneous wave equation variable change

in the case of the non-homogenous wave equation with the general form of: Utt-c^2*Uxx=h(x,t) if we define m = x-ct and n = x+ct, then how we conclude the following equation? -(4*c^2)Umn=h(m,n) ...
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1answer
152 views

Help solving shallow water equations initial value problem?

The question is as followed: "Consider the initial value problem (IVP) for the linearised shallow water equations (1) $\frac{\partial h}{\partial t} + H_0 \frac{\partial u}{\partial x} = 0$ ; (2) ...
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I do not understand what this PDE problem is looking for?!

This is an exercise in Qing book. the given hint is not helping me with what the problem is looking for and how to begin. I would appreciate any help. ps. I am studying uniqueness of solutions of ...
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Wave equation: function 2L-periodic.

I have to prove that any function that is even with respect to x=0 and x=L is necessarily 2L-periodic. We are studying wave equations but I don´t know how to prove it. Can you give me a clue?
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1answer
50 views

Transforming PDE $u_{tt}+ (u^2)_{xx} + u_{xxxx} = 0$ into travelling wave ODE

I was wondering on how can we transform a pde to an ode using $$\partial_{tt}u+ \partial_{xx}u^2+\partial_{xxxx} u= 0$$ $$u=f(w)=f(x-vt)$$ how can I transform it into the ode $$f'' +v^2f+f^2=aw+b$$...
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Wave equation, kirchhof furmula

Given that $u(x,t)\in C_{x,t}^{2,2}(R^3\times(0,\infty))$ solution of $\frac{\partial^2 u}{\partial t^2}$-$\Delta u$=0 ,t>0 $x\in R^n$ $u(x,0)=1$ $u_t(x,0)=0$ for all : $r_0\leq x\leq r_1$ Prove u(0;...
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11 views

Prove the existence of cut off function

I want to prove the existence of cut off function: "a smooth "cut-off" function $\Gamma$ such that $0\leq \Gamma(x) \leq 1$, $\Gamma(x)=1$ for $x \in B_{R}(0)$ and $\Gamma(x)=0$ for $x \notin B_{2R}(0)...
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1answer
39 views

Desmos help - making a finite length line with origin on a sine wave move along it and vary angle orthogonal to sine wave line

I'm trying to model something that seems fairly simple, but it's trickier than I thought on https://www.desmos.com. On desmos I created a function: $f(x)=y$ $y=(0 \leq x \leq 2)\sin( \pi x+b )$   ...
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1answer
62 views

Wave equation with Neumann boundary condition

I have the following problem: $$ \begin{array}{ll} &u_{tt}(x,t)=4u_{xx}(x,t),&x>0, t>0\\ &u_x(0,t)=-\cos(t),&t>0\\ &u(x,0)=e^{-x},&x>0\\ &u_t(x,0)=2e^{-x},&...
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1answer
26 views

Finding wave equation based off a certain scenario

Problem: May 7th is a full moon (meaning we see all of the moon). A full moon happens every 29.5 days (I know, it’s kind of weird that it’s a half day, just go with it); halfway in between is a new ...
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24 views

Wave equation with open boundary

I want to solve the wave equation with the following boundary and initial conditions $$\begin{cases} u_{tt} & =c^2u_{xx}\\ u(x,0) & =0\\ u_t(x,0) &= 0 \\ u(0,t) & = A\sin(\omega t)\\ ...
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21 views

Build D.E. that $\Psi_{\gamma,\kappa,\rho}(t)=e^{{\gamma}it}e^{(\kappa A+\rho A^*)}$ solves

Consider $$\Psi_{\gamma,\kappa,\rho}(t)=e^{{\gamma}it}e^{(\kappa A+\rho A^*)}$$ for real parameters $\gamma,\kappa,\rho>0$, real variable $t\in(0,1)$ and $A=\frac{1}{\log(t)},$ $A^*=\frac{1}{\log(...
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1answer
39 views

Ferris Wheel Trig Problem

Your ride on a clockwise Ferris wheel begins at the top of the ride, and your height is described by the function $$h(t)=4\cos{\left(\frac{\pi}{18}t\right)}+50,$$ where $h$ is in feet, and $t$ is in ...
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61 views

Holmgren's uniqueness theorem applied to the wave equation in a cylinder

Consider the wave equation in $\mathbb{R}^{3+1}$ dimensions with the following boundary conditions: $$\begin{align} \partial_t^2 \psi &= \Delta \psi \ \text{ in }\ \mathbb{R} \times B(0,1) \\ \...
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19 views

Laplace's equation on Lorentzian manifold

In general relativity one wants to find "harmonic" functions $u$ on (a neighborhood $U$ in) a Lorentzian manifold $(M, g)$. In arbitrary coordinates $\{x^\mu\}$ the equation $\Delta_g u = 0$ reads $$...
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16 views

Area of Spherical Cap to solve a PDE

there StackExchange, In order to solve a PDE I need to solve for the area of a spherical caps and what I get doesn't reduce to anything nice or that makes sense in the context of the IVP of the PDE ...
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29 views

How to characterize the linearized operator around a solution of a PDE by conserved quantities?

Let $$\phi_{tt}-\phi_{xx}= \phi-\phi^3 ,\: (t,x)\in \mathbb{R}\times \mathbb{R}. \qquad \qquad (1)$$ I know that the linearized operator around a solution $\phi$ is given by $$\mathcal{L}=\frac{\...
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14 views

3D Wave Equation with IVP

Find the solution of the wave equation $u_{tt} - \Delta_x u =0$ $\in \mathbb{R}^3$ satisfying the initial conditions; $u(x,0)=0$, $u_t(x,0)=g(x)$ where $$ g(x)=\left\{ \begin{array}[ll] 11 & \text{...
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1answer
34 views

Monotone increasing solutions of the wave equation

If I have the following PDE; $${{\partial_t^2 u(t,x)}} - {\partial_x^2 u(t,x)} = 0 \\ u(0,x) = g(x) \\ {\partial_t u(0,x)} = h(x) $$ Is is true if $g$ and $h$ are monotone increasing then $u(x,t)$ ...
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20 views

d'Alembert solution PDE

Consider a wave described by the one-dimensional wave equation ∂ttu = ∂xxu with zero initial velocity and initial configuration given by φ(x) := {$2 − 2|x|$ if $|x| ≤ 1$ and $ 0$ Otherwise ...
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1answer
39 views

Linearized operator around a solution by conserved quantities

Let $$u_{tt}-u_{xx}= u-u^3 ,\: (t,x)\in \mathbb{R}\times \mathbb{R}.$$ I know that this equation admits wave traveling solution of the form $u(x,t)=\varphi(x-ct)$, where $ c\in \mathbb{R}$ is a ...
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27 views

Why does a sine wave speed up and slow down?

I was playing with the resource linked below, which allows you to see how the value of sine changes as the angle changes. I understand that the hypotenuse side must move faster than the adjacent side ...
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2answers
106 views

How to find the conserved quantities of $\phi^4$ model?

Consider \begin{equation}\label{1} \partial^2_t\phi-\partial^2_x\phi=\phi -\phi^3,\: \ (x,t) \in \Bbb{R}\times \Bbb{R} \tag{1} \end{equation} the $\phi^4$ model, often used in quantum field theory and ...
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0answers
12 views

1D Wave on the Half-Plane with piecewise defined conditions

I'm studying for an exam and I'm completely lost for one particular problem. We are asked to solve by the method of characteristics \begin{align*} u_{tt}-4u_{xx}& = 0 \qquad x>0,\, t>0\\ u(x,...
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12 views

Are the boundary conditions of this PDE inconsistent?

Recently, I was attempting to solve the following boundary-value problem for the wave equation with a student that I tutor: $$\frac{\partial^2 u}{\partial x^2}=\frac{1}{4}\frac{\partial^2 u}{\partial ...
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1answer
15 views

Neumann functions of integer order - are they undefined by definition?

I have the following definition for Neumann functions (in terms of Bessel functions) $$ Y_n(z) = \frac{J_n(z) \cos(n\pi) - J_{-n}(z)}{\sin(n \pi)}. $$ Now my problem is concerned only with $n \in \...
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20 views

Solving PDE using separate variables method

Given $u_{tt} - u_{xx} = 0$ where $0 < x <3$ with boundary conditions $u(0,t) = u(3,t) = 0$ To solve it, separate the variables $$u(x,t) = T(t)X(x)$$ and get $$\frac{T''}{T} = \frac{X''}{X} = - ...
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18 views

Numerical solution to a nonlinear wave equation

I am currently studying a nonlinear wave equation taking the form $$ v_{tt} - v_{rr} + v + \frac{1}{r^6}v^7 = 0 $$ Some might recongnize this from W. Strauss' book on PDEs. In this example, he goes on ...
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0answers
26 views

Inhomogeneous wave equation with Dirichlet condition

I'm trying to solve the problem: $\begin{cases} \frac{\partial u}{\partial tt} -\frac{\partial u}{\partial xx} =g(t)sin(x)\\ u(x,0)=\frac{\partial u}{\partial t}(x,0)=0 \\ u(0,t)=u(\pi, t)=0 \end{...
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0answers
12 views

Feedback on python plots of plane waves incident on a cylinder

I'm currently working on a project to do with scattering of acoustic waves around a cylinder and I want to plot my solutions in python and I'd really like to get some feedback on my plots, I'm really ...
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12 views

Wave equation interpretation for different physical quantities

I've been working on a project to do with scattering of acoustic waves and one of the first steps towards finding my solutions was to derive the wave equation: $$ \nabla^2 U = \frac{1}{c^2} \frac{\...
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12 views

Wave equation with piecewise-defined initial condition

I want to confirm whether the solution I have is right. So I'm tasked with solving the equation $u_{tt} = u_{xx}$ with the initial conditions: $$ u(x, 0) = \begin{cases} cos(\frac{\pi x}{2\mu }), \...
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1answer
51 views

A circular vibrating membrane

Our Lecturer gaves us the following problem: $u_{tt}=u_{rr}+\frac{u_r}{r}$ , $0<r<a,0<t$ (1) with Initial Conditions (IC): $u(r,0)=f(r)$, $u_t(r,0)=0$ and Bounadry Conditions (BC):...

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