Questions tagged [wave-equation]
For questions related to solutions and analysis of the wave equation.
912
questions
-1
votes
1answer
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 ...
0
votes
0answers
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....
0
votes
0answers
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 ...
0
votes
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 ...
-1
votes
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\...
-1
votes
0answers
11 views
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 ...
1
vote
0answers
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
$$U_{xx} - 3U_{xt}-4U_{tt}=\...
0
votes
0answers
15 views
why this boundary condition is given?
In my book it is given the following BV problem:
$u_{xx}=u_{tt},\qquad-\infty<x<\infty$ (1)
$u(-\infty,t),u(+\infty,t)<\infty$ (2)
$u(x,0)=f(x),\qquad u_t(x,0)=g(x)$ (3)
I ...
2
votes
0answers
28 views
Wave equation solution $u(x,t)$ for $u_0(x)=x$ and $v_0(x)=0$
Let a string with linear density $\rho$ and tension $k$ Its left and right hand ends $ [-\pi,\pi]$ are held fixed at height zero (Maybe not at $t=0$, but it are for $t>0$).
Initial velocity $...
0
votes
0answers
6 views
How to solve following wave equation?
Use the CTCS scheme with X2-X1=1/3 and t2-t1=1/5 to solve the wave equation Utt(x,t)=Uxx(x,t) with the boundry conditions:
U(0,t)=U(1,t)=0 0<=t<=0.6
U(x,0)=(3^1/2)sin(pi....
0
votes
0answers
33 views
Solution of the Wave Equation by Separation of Variables
We know for
$\left\{\begin{array}{cccc}\dfrac{1}{c^2}u_{tt}&=&u_{xx}\hspace{0.25cm}x\in [0,L]&\\
u(0,t)&=&u(L,t)=0\\
u(x,0)&=&f(x)\\
u_t(x,0)&=&g(x)
\end{array}\...
0
votes
0answers
23 views
Wave equation PDE estimate of the second derivative
Let $\phi \in L^2((a,b) \times (0,T) )$ such that $$\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;...
0
votes
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 ...
0
votes
2answers
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 $...
0
votes
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 ...
0
votes
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:
$\int_{\partial B(x_0,t)}t*|y-x_0|^2dS(y)$
$\int_{\partial B(x_0,t)}|y-x_0|^6dS(y)$...
2
votes
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 ...
0
votes
0answers
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)
...
1
vote
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) ...
0
votes
0answers
32 views
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 ...
0
votes
0answers
10 views
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 donĀ“t know how to prove it. Can you give me a clue?
1
vote
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
$$f'' +v^2f+f^2=aw+b$$...
0
votes
0answers
14 views
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;...
0
votes
0answers
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)...
0
votes
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:
$f(x)=y$
$y=(0 \leq x \leq 2)\sin( \pi x+b )$
Ā
...
1
vote
1answer
62 views
Wave equation with Neumann boundary condition
I have the following problem:
$$
\begin{array}{ll}
&u_{tt}(x,t)=4u_{xx}(x,t),&x>0, t>0\\
&u_x(0,t)=-\cos(t),&t>0\\
&u(x,0)=e^{-x},&x>0\\
&u_t(x,0)=2e^{-x},&...
0
votes
1answer
26 views
Finding wave equation based off a certain scenario
Problem: May 7th is a full moon (meaning we see all of the moon). A full moon happens every 29.5 days (I know, itās kind of weird that itās a half day, just go with it); halfway in between is a new ...
0
votes
0answers
24 views
Wave equation with open boundary
I want to solve the wave equation with the following boundary and initial conditions
$$\begin{cases}
u_{tt} & =c^2u_{xx}\\
u(x,0) & =0\\
u_t(x,0) &= 0 \\
u(0,t) & = A\sin(\omega t)\\
...
0
votes
0answers
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(...
0
votes
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 ...
2
votes
0answers
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:
$$\begin{align}
\partial_t^2 \psi &= \Delta \psi \ \text{ in }\ \mathbb{R} \times B(0,1) \\
\...
1
vote
0answers
19 views
Laplace's equation on Lorentzian manifold
In general relativity one wants to find "harmonic" functions $u$ on (a neighborhood $U$ in) a Lorentzian manifold $(M, g)$. In arbitrary coordinates $\{x^\mu\}$ the equation $\Delta_g u = 0$ reads
$$...
0
votes
0answers
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 ...
0
votes
0answers
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
$$\mathcal{L}=\frac{\...
0
votes
0answers
14 views
3D Wave Equation with IVP
Find the solution of the wave equation $u_{tt} - \Delta_x u =0$ $\in \mathbb{R}^3$ satisfying the initial conditions; $u(x,0)=0$, $u_t(x,0)=g(x)$ where
$$
g(x)=\left\{
\begin{array}[ll]
11 & \text{...
0
votes
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 $u(x,t)$ ...
0
votes
0answers
20 views
d'Alembert solution PDE
Consider a wave described by the one-dimensional wave equation
āttu = āxxu
with zero initial velocity and initial configuration given by
Ļ(x) := {$2 ā 2|x|$ if $|x| ā¤ 1$ and $ 0$ Otherwise
...
-1
votes
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 ...
0
votes
0answers
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 ...
4
votes
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 ...
1
vote
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,...
0
votes
0answers
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:
$$\frac{\partial^2 u}{\partial x^2}=\frac{1}{4}\frac{\partial^2 u}{\partial ...
0
votes
1answer
15 views
Neumann functions of integer order - are they undefined by definition?
I have the following definition for Neumann functions (in terms of Bessel functions)
$$
Y_n(z) = \frac{J_n(z) \cos(n\pi) - J_{-n}(z)}{\sin(n \pi)}.
$$
Now my problem is concerned only with $n \in \...
0
votes
0answers
20 views
Solving PDE using separate variables method
Given $u_{tt} - u_{xx} = 0$ where $0 < x <3$ with boundary conditions $u(0,t) = u(3,t) = 0$
To solve it, separate the variables
$$u(x,t) = T(t)X(x)$$
and get $$\frac{T''}{T} = \frac{X''}{X} = - ...
0
votes
0answers
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 ...
1
vote
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{...
0
votes
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 ...
0
votes
0answers
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:
$$
\nabla^2 U = \frac{1}{c^2} \frac{\...
0
votes
0answers
12 views
Wave equation with piecewise-defined initial condition
I want to confirm whether the solution I have is right. So I'm tasked with solving the equation $u_{tt} = u_{xx}$ with the initial conditions:
$$
u(x, 0) = \begin{cases}
cos(\frac{\pi x}{2\mu }), \...
0
votes
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):...
