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Questions tagged [wave-equation]

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

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12 views

Solution of a PDE on a semi-infinite string using D'Alembert formula

I have to do the following assignment: Consider the wave equation on a semi-infinite interval $$\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2u}{\partial x^2} \hspace{1cm} 0<x<\...
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18 views

Wave equation with source

I've been trying to solve the following equation: $$u_{tt} - u_{xx} =f(x, t) $$ With the following contour and initial conditions: $u(0,t)=\phi_1(t)$ $u(L,t)=\phi_2(t)$ $u(x,0)=\psi_1(x)$ $u_t(x,0)=\...
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1answer
21 views

which of following option is correct for heat equation (CSIR june 2018)

If $u(x,t)$ is the solution to $$\frac{\partial u}{\partial t}=\frac{\partial^{2} u}{\partial x^2}~, \quad 0<x<1,~t>0 \\ u(x,0) = 1 + x + \sin(\pi x) \cos(\pi x)~, \quad u(0,t)=1,~u(1,t)=2$$ ...
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25 views

Finding wave solution for $\frac{\partial^2 f(x,t)}{\partial t^2} - g(t) \frac{\partial f(x,t)}{\partial t} - \frac{\partial^2 f(x,t)}{\partial x^2}$

I want to solve \begin{align} \frac{\partial^2 f(x,t)}{\partial t^2} - g(t) \frac{\partial f(x,t)}{\partial t} - \frac{\partial^2 f(x,t)}{\partial x^2} =0 \end{align} I am trying to find the general ...
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31 views

Coefficients in 1D wave equation PDE

So I was looking through my professor's solution for the following PDE problems and don't really understand where she gets these coefficients from. I've been doing the problems using integration and ...
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1answer
58 views

Show that energy for the wave equation does not increase with time

Define the energy $E(t)$ as $$E(t)= \frac{1}{2} \iiint u_t^2 + c^2(u_x^2 + u_y^2 + u_z^2)\,dx\,dy\,dz$$ where the integral extends over all of space. Show that the energy for the wave equation ...
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16 views

Derivative of a Partial derivative in wave equation

Given that: $$y=f(x,t)$$ Prove that: $$\frac{d}{dx} \frac{\partial{y}}{\partial{x}} = \frac{\partial^2{y}}{\partial{x}^2} $$ I have zero experience in PDE's , I just stumbled across this stuff while ...
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15 views

Laplace transform to solve the wave equation

The wave equation defined by the below $$ \frac{d^{2}u}{dx^{2}}+sin(at)sin(\pi x)= \frac{d^{2}u}{dt^{2}}$$ with boundary and initial conditions u(0,t)=0 , u(1,t)=0 , t>0 u(x,0)=0 ut(x,0)=0 , 0 0
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12 views

Shock-like phenomena modelled by the 1-D wave equation

I am attempting to determine the behaviour of characteristic lines that arise from my solution to the PDE: $\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}=0$ Assuming that $u(x, 0)$ ...
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1answer
98 views

Trying to solve wave equation with initial condition

Solve the initial value problem: $$ u_{tt} - u_{xx} = 0 \\ u(x,0) = 0 \\ u_t(x,0) = \begin{cases} \cos \pi (x-1) & \text{if } 1 < x < 2 \\ 0 & \text{otherwise} \end{cases} $$ Find ...
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1answer
21 views

Solving Wave Equation on $0 < x < L$ with initial condition containing both sine and cosine terms

I am trying to solve the wave equation as follows: $$u_{tt} = u_{xx} \,\,\,\,\,\,\,\, 0 < x < L$$ $$u(0,t) = u(L,t)$$ $$u_x(0,t) = u_x(L,t)$$ $$u(x,0) = sin\bigg(\frac{2\pi x}{L}\bigg) + cos\...
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1answer
29 views

Wave equation with Neumann BC on semi-infinite domain

This problem is from our recitation which I do not have solutions for, and I'm stuck on the very last part where I need to satisfy the $u_t(x,0)$ initial condition. The problem is: $$ \left\{ \...
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1answer
39 views

Wave equation piecewise initial condition

I have the following homogeneous 1-dimensional wave equation with $c = 1$: $$u_{tt} - u_{xx} = 0$$ With initial data $$ u(x,0) = \phi(x) = \begin{cases} 1 & |x| \leq 1 \\ 0 & ...
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1answer
51 views

Proving uniqueness of solution to 1-D Wave equation without energy conservation

We have a homogeneous string of length L fastened at its ends, performing small transverse motion in a vertical plane. The tension in the string is assumed sufficiently large for gravitational forces ...
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1answer
43 views

Mode Decomposition With Wave Equation

I'm working through the paper (1) which uses the elastic wave equation and can't understand how they obtain a set of vectors. I have the elastic wave equation defined as (2.6)-(2.8): $$ \rho\frac{\...
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12 views

When Linearised Model of Finite Depth Waves is Not a Sufficient Model

I have been investigating the linearised model of water wave motion in a finite depth fluid. In my particular case the flow is Inviscid, Irrotational and Incompressible and surface tension effects are ...
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26 views

Wave PDE system with first order derivative in the right part

I have a following system of partial differential equations: $$ \frac{\partial^2 x}{\partial t^2} = \frac{\partial^2 x}{\partial s^2} + k\frac{\partial{y}}{\partial t} $$ $$ \frac{\partial^2 y}{\...
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1answer
45 views

Wave equation - use parallelogram rule to solve the problem

Given wave equation: $ u_{tt}-c^2u_{xx}=0 $, let u be a solution. Points A, B, C, D are vertices of parallelogram of two pairs of characteristic lines: $ x-ct=c1, x-ct=c2, x+ct=d1, x+ct=d2 $ Use ...
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26 views

Finite difference method for 2nd order Wave equation with mixed boundary conditions

I have matlab code of finite difference method for solving wave equation with Dirichlet B.C. I want to modify this algorithm in order to solve the problem with mixed B.C. of the form $u(0,t) = 0, \...
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1answer
26 views

How do I solve the non-homogeneous wave equation with homogeneous boundary and initial conditions?

I want to solve $$v_{tt}(x,t) - v_{xx}(x,t) = -\left(\frac{3}{4} \cos(t) + \frac{1}{4} \cos(3t)\right) \sin(x)$$ with boundary conditions $$v(0,t) = v(\pi,t) = 0$$ and initial conditions $$v(x,0) = ...
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1answer
45 views

Solving a PDE ( wave equation )

$$u_{tt}- \Delta u= e^{t}, 0<x<2\pi, 0<y<1, t>0$$ $$u_x(0,y,t)=u_x(2\pi,y,t)=0, 0<x<2\pi, t>0$$ $$u_y(x,0,t)=u_y(x,1,t)=0, 0<x<2\pi, t>0$$ $$u(x,y,0)=0, 0<x<2\pi,...
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13 views

Expected value of $(z- \langle z \rangle )^2$ in a potential well

today I am asking for help in checking my work. It seems like I am off by a constant for calculating the the expected value of $(z- \langle z \rangle )^2$ in a potential well with interval $\frac{-n \...
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0answers
31 views

wave equation D'Alembert's solution

Consider the wave equation $$u_{tt} = c^2u_{xx}$$ with $$u(x,0) = \begin{cases} \sin x, && 2\pi < x < 3\pi \\ 0, && x < 2\pi,\ x> 3\pi \end{cases} $$ What initial ...
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2answers
270 views

Robin BC in the 1D wave equation

The problem of interest is as follows: the quantity of interest: $u(x,t)$ the wave equation: $\partial_2^2u(x,t)-c^2\partial_1^2u(x,t)=0$ where $c>0$ one Robin boundary condition at $x=0$: $\...
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1answer
76 views

A partial solution to the wave equation

I have been presented with the one-dimensional wave equation: $$\frac{\partial^2 y}{\partial x^2}=\frac1{c^2} \frac{\partial^2 y}{\partial t^2}$$ for a particular string of length $L$ that is fixed ...
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35 views

Wave equation $u_{tt}=u_{xx}$ with range $-\frac{x}{2}<t$ and “initial conditions” at $u(x,-\frac{x}{2})$ and $u_x(x,-\frac{x}{2})$

Wave equation $u_{tt}=u_{xx}$ with range $-\frac{x}{2}<t$ and $-\infty<x<\infty$. "initial conditions": \begin{align} u\big(x,-\frac{x}{2}\big) &= \begin{cases} 1, & |x|<1, \\ 0, &...
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28 views

Non-Dimensionalizing Separation of Variables PDE

I have a problem where I'm considering a cylinder with a closed end at the bottom (x = 0) and open end at the top (x = L). The problem is looking at pressure disturbances throughout the air column. ...
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56 views

Shock and mass conservation law derivation (Rankine-Hugoniot)

Suppose we are looking at the non-linear system $u_t+uu_x=0$. Some of the waves from this system are drawn below where I included $a$ and $b$ as interval-bounds around where the most happens. There is ...
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2answers
50 views

Solving wave equation using Fourier series

So I have the problem as follows: $$ u_{tt} = c^2 u_{xx}(x,t), 0<x<1 , t>0 $$ $$ u(0,t) =0, u(1,t) = 0 , t >0 $$ $$ u(x,0)= f(x) , u_t(x,0) = g(x) 0 < x < 1$$ For: $$ f(x) = sin(\...
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9 views

How to prove the linear convergence of the implicit euler method when applied to the semi discretised wave equation?

I'm solving the wave equation ($u_{tt} + c^2 \Delta u = f$) with dirichlet boundary conditions and a given starting deflection $u_0$ and a starting velocity $v_0$ in one or two dimensions. I ...
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1answer
57 views

Use the Laplace transform to solve $ u_{tt}(x, t) − c^2u_{xx}(x, t) = 0$

Use the Laplace transform to solve the following initial boundary value problem for the wave equation $ u_{tt}(x, t) − c^2u_{xx}(x, t) = 0$ $u(x, 0) = 0$, $u_{t}(x, 0) = 0 ∀x > 0$ , and $u(0, t) ...
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1answer
106 views

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 ...
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44 views

Is that a correct question?

In page no. 132 of the book Linear Partial differential equations for scientists and engineers by Tyn Myint-U, there is following example. Here $|\sin x|$ is not differentiable at $x =n\pi$, $n\in\Bbb ...
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1answer
59 views

How does this solution to the wave equation evolve?

If we consider the wave equation $u_{tt}-c^2u_{xx}=0$ on the one-dimensional half space (i.e. $x>0$), and suppose the solution for $t<0$ takes the form of a left-moving wave $u=f(x+ct)$, how ...
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2answers
59 views

Solution of a nonhomogeneous wave equation

Consider the equation: $$u_{tt}-u_{xx}=\cos(2t)\cos(3x)-2t, 0<x<\pi, t>0$$ $$u_x(0,t)=0, t\geq 0$$ $$u_x(\pi,t)=2\pi t, t\geq 0$$ $$u(x,0)=\cos^2(x), x\in [0,\pi]$$ $$u_t(x,0)=1+x^2$$ How ...
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74 views

Solve wave equation with non-constant wave speed using method of characterstics?

I am trying to get a better understanding of wave pulses in a domain with a non-constant wave speed. I am trying to solve either one of the two equations: $$\frac{\partial^2u}{\partial t^2}-c(x)^2\...
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1answer
54 views

Wave Equation with Initial Conditions on Characteristic Curves

I am trying to solve the initial value problem: $$\begin{cases} u_{tt}-u_{xx} =0\\ u|_{t = x^2/2} = x^3, \quad |x| \leq 1 \\ u_{t}|_{t = x^2 / 2} = 2x, \quad |x| \leq 1 \\ \end{cases} $$ I'm unsure ...
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8 views

Convergence of the HHT-$\alpha$ method for the second order wave equation?

What is the convergence speed of the HHT-$\alpha$ method when applied to the second order wave equation (1d or 2d)? I know that the Newmark-Methods hav a convergence speed of 2 for $\beta = 1/4$ and $\...
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0answers
10 views

Inner products harmonic oscillator

How do I compute these inner products; $(\Delta x)^2 = \frac{\int_R \!x^2\psi_m\psi_mdx}{\int_R \!\psi_m\psi_mdx}$ $(\Delta p)^2 = \frac{\int_R \!\psi_m\frac{d^2}{dx^2}\psi_mdx}{\int_R \!\psi_m\...
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0answers
40 views

Solving the Wave Equation With Boundary Conditions $u(0,t)=0, \ u_x(1,t)=0$

Using separation of variables, construct the general solution of the wave equation $$u_{tt}=u_{xx},$$ for $0<x<1, \ t>0$ with boundary conditions $u(0,t)=0$ and $u_x(1,t)=0$ for $t>0$ and ...
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1answer
51 views

One Dimensional Wave Equation with Piecewise Initial Conditions

The problem I am trying to solve is: $$ \begin{cases} u_{tt} - c^{2} u_{xx} = 0 \\ u(x, 0) = g(x) \\ u_{t}(x,0) = h(x) \end{cases} $$ where $h(x) = 0$ and $g(x) = \begin{cases} 0 \ : x < 0 \\ 1 ...
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25 views

Are there discontinous solutions of the wave equation in two dimensions?

Can you construct a discontinous solution for the wave equation $u_{tt} - c^2 \Delta u = f $ with homogenous dirichlet boundary conditions on the domain $\Omega = [0,1]^2$? Background: I programmed a ...
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1answer
70 views

Separation of variables for nonhomogeneous equations

I’m trying to learn PDE from An introduction to partial differential equations, Pinchover and Rubinstein. On page 114 section 5.4 it explains the use of separation of variables for nonhomogeneous ...
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37 views

Trace of diffusion equation matrix

I'm studying waves moving through mediums and I come across this type of equation a lot while describing acoustic waves speed and stress on the material, $$\begin{gather} \frac{\partial}{\partial z}\...
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1answer
21 views

How do theoretical convergence speeds translate into real life when using FEM?

I'm currently solving the wave equation in a 1d and a 2d domain using the finite element method in space and the leapfrog or crank-nicolson method in time. Theoretically, I expect a convergence of $O(...
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1answer
32 views

how to verify a solution to the 3D wave equation

Is $E = (A\sin(k(x-ct)),0,0)$ a solution to the wave equation $c^2 \nabla^2E=\frac{\partial^2E}{\partial t^2}$? What is $\frac{\partial E}{\partial t}$?
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0answers
26 views

Appropriate boundary conditions of the wave equation

Here, I have some acoustic waves generated in a compressible fluid by small oscillations of a cylinder with boundary at $r=a$. In this problem, I am only interested in the solution outside of the ...
2
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1answer
40 views

Partial Differential Equations: D'Alembert Formula

Solve the IVP: $u_{tt}-c^{2}u_{xx}=0,$ $u(0,x)=e^{x},$ $u_{t}(0,x)=sin(x)$ with D'Alambert formula in the simplest explicit form. Using the formula I obtained the following solution: $u(x,t)=1/2[...
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1answer
91 views

Wave equation on the half line with inhomogeneous Dirichlet boundary condition

I have difficulties solving the following exercise: Consider the IBVP on the half line $(0,\infty)$ (with $T \in (0,\infty)$: $u_{xx}-u_{tt}=0$, on $(0,T)\times(0,\infty)$ $u(0,x)=u_{0}(x), x>0$, ...
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0answers
32 views

How to numerically solve the 1-D wave equation with linear restoring force?

I have been trying to numerically solve the equation of a string vibrating on an elastic foundation in order to see its dispersive behaviour with no success. The equation is of motion is $$ \rho \...