Questions tagged [wave-equation]
For questions related to solutions and analysis of the wave equation.
0
votes
0answers
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
$$\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2u}{\partial x^2} \hspace{1cm} 0<x<\...
0
votes
0answers
18 views
Wave equation with source
I've been trying to solve the following equation:
$$u_{tt} - u_{xx} =f(x, t) $$
With the following contour and initial conditions:
$u(0,t)=\phi_1(t)$
$u(L,t)=\phi_2(t)$
$u(x,0)=\psi_1(x)$
$u_t(x,0)=\...
0
votes
1answer
21 views
which of following option is correct for heat equation (CSIR june 2018)
If $u(x,t)$ is the solution to
$$\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$$
...
1
vote
0answers
25 views
Finding wave solution for $\frac{\partial^2 f(x,t)}{\partial t^2} - g(t) \frac{\partial f(x,t)}{\partial t} - \frac{\partial^2 f(x,t)}{\partial x^2}$
I want to solve
\begin{align}
\frac{\partial^2 f(x,t)}{\partial t^2} - g(t) \frac{\partial f(x,t)}{\partial t} - \frac{\partial^2 f(x,t)}{\partial x^2} =0
\end{align}
I am trying to find the general ...
0
votes
0answers
31 views
Coefficients in 1D wave equation PDE
So I was looking through my professor's solution for the following PDE problems and don't really understand where she gets these coefficients from. I've been doing the problems using integration and ...
0
votes
1answer
58 views
Show that energy for the wave equation does not increase with time
Define the energy $E(t)$ as
$$E(t)= \frac{1}{2} \iiint u_t^2 + c^2(u_x^2 + u_y^2 + u_z^2)\,dx\,dy\,dz$$
where the integral extends over all of space.
Show that the energy for the wave equation ...
0
votes
0answers
16 views
Derivative of a Partial derivative in wave equation
Given that:
$$y=f(x,t)$$
Prove that:
$$\frac{d}{dx} \frac{\partial{y}}{\partial{x}} = \frac{\partial^2{y}}{\partial{x}^2} $$
I have zero experience in PDE's , I just stumbled across this stuff while ...
0
votes
0answers
15 views
Laplace transform to solve the wave equation
The wave equation defined by the below
$$ \frac{d^{2}u}{dx^{2}}+sin(at)sin(\pi x)= \frac{d^{2}u}{dt^{2}}$$
with boundary and initial conditions
u(0,t)=0 , u(1,t)=0 , t>0
u(x,0)=0 ut(x,0)=0 , 0
0
0
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0answers
12 views
Shock-like phenomena modelled by the 1-D wave equation
I am attempting to determine the behaviour of characteristic lines that arise from my solution to the PDE:
$\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}=0$
Assuming that $u(x, 0)$ ...
2
votes
1answer
98 views
Trying to solve wave equation with initial condition
Solve the initial value problem:
$$
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}
$$
Find ...
1
vote
1answer
21 views
Solving Wave Equation on $0 < x < L$ with initial condition containing both sine and cosine terms
I am trying to solve the wave equation as follows:
$$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\...
2
votes
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:
$$
\left\{
\...
3
votes
1answer
39 views
Wave equation piecewise initial condition
I have the following homogeneous 1-dimensional wave equation with $c = 1$:
$$u_{tt} - u_{xx} = 0$$
With initial data
$$ u(x,0) = \phi(x) = \begin{cases}
1 & |x| \leq 1 \\
0 & ...
0
votes
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 ...
2
votes
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):
$$
\rho\frac{\...
0
votes
0answers
12 views
When Linearised Model of Finite Depth Waves is Not a Sufficient Model
I have been investigating the linearised model of water wave motion in a finite depth fluid. In my particular case the flow is Inviscid, Irrotational and Incompressible and surface tension effects are ...
0
votes
0answers
26 views
Wave PDE system with first order derivative in the right part
I have a following system of partial differential equations:
$$
\frac{\partial^2 x}{\partial t^2} = \frac{\partial^2 x}{\partial s^2} + k\frac{\partial{y}}{\partial t}
$$
$$
\frac{\partial^2 y}{\...
0
votes
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 ...
0
votes
0answers
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, \...
1
vote
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 $$v(x,0) = ...
2
votes
1answer
45 views
Solving a PDE ( wave equation )
$$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,...
0
votes
0answers
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 \...
0
votes
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 ...
11
votes
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 $x=0$: $\...
5
votes
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 ...
2
votes
0answers
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, &...
1
vote
0answers
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.
...
3
votes
0answers
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 ...
1
vote
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:
$$ f(x) = sin(\...
0
votes
0answers
9 views
How to prove the linear convergence of the implicit euler method when applied to the semi discretised wave equation?
I'm solving the wave equation ($u_{tt} + c^2 \Delta u = f$) with dirichlet boundary conditions and a given starting deflection $u_0$ and a starting velocity $v_0$ in one or two dimensions. I ...
1
vote
1answer
57 views
Use the Laplace transform to solve $ u_{tt}(x, t) − c^2u_{xx}(x, t) = 0$
Use the Laplace transform to solve the following initial boundary value problem for the wave equation $ u_{tt}(x, t) − c^2u_{xx}(x, t) = 0$
$u(x, 0) = 0$,
$u_{t}(x, 0) = 0 ∀x > 0$ , and
$u(0, t) ...
7
votes
1answer
106 views
What is a soliton?
I have read about solitons and they seem to be a big deal as a phenomenon. The definition I find from wikipedia is that solitons are characterized by the following three properties:
They are of ...
0
votes
0answers
44 views
Is that a correct question?
In page no. 132 of the book Linear Partial differential equations for scientists and engineers by Tyn Myint-U, there is following example. Here $|\sin x|$ is not differentiable at $x =n\pi$, $n\in\Bbb ...
0
votes
1answer
59 views
How does this solution to the wave equation evolve?
If we consider the wave equation $u_{tt}-c^2u_{xx}=0$ on the one-dimensional half space (i.e. $x>0$), and suppose the solution for $t<0$ takes the form of a left-moving wave $u=f(x+ct)$, how ...
1
vote
2answers
59 views
Solution of a nonhomogeneous wave equation
Consider the equation:
$$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$$
How ...
3
votes
0answers
74 views
Solve wave equation with non-constant wave speed using method of characterstics?
I am trying to get a better understanding of wave pulses in a domain with a non-constant wave speed. I am trying to solve either one of the two equations:
$$\frac{\partial^2u}{\partial t^2}-c(x)^2\...
4
votes
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 ...
0
votes
0answers
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 $\...
0
votes
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\...
0
votes
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 ...
4
votes
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 ...
1
vote
0answers
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 ...
1
vote
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 ...
1
vote
0answers
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,
$$\begin{gather}
\frac{\partial}{\partial z}\...
0
votes
1answer
21 views
How do theoretical convergence speeds translate into real life when using FEM?
I'm currently solving the wave equation in a 1d and a 2d domain using the finite element method in space and the leapfrog or crank-nicolson method in time. Theoretically, I expect a convergence of $O(...
0
votes
1answer
32 views
how to verify a solution to the 3D wave equation
Is $E = (A\sin(k(x-ct)),0,0)$ a solution to the wave equation $c^2 \nabla^2E=\frac{\partial^2E}{\partial t^2}$?
What is $\frac{\partial E}{\partial t}$?
4
votes
0answers
26 views
Appropriate boundary conditions of the wave equation
Here, I have some acoustic waves generated in a compressible fluid by small oscillations of a cylinder with boundary at $r=a$.
In this problem, I am only interested in the solution outside of the ...
2
votes
1answer
40 views
Partial Differential Equations: D'Alembert Formula
Solve the IVP:
$u_{tt}-c^{2}u_{xx}=0,$
$u(0,x)=e^{x},$
$u_{t}(0,x)=sin(x)$
with D'Alambert formula in the simplest explicit form.
Using the formula I obtained the following solution:
$u(x,t)=1/2[...
0
votes
1answer
91 views
Wave equation on the half line with inhomogeneous Dirichlet boundary condition
I have difficulties solving the following exercise:
Consider the IBVP on the half line $(0,\infty)$ (with $T \in (0,\infty)$:
$u_{xx}-u_{tt}=0$, on $(0,T)\times(0,\infty)$
$u(0,x)=u_{0}(x), x>0$,
...
0
votes
0answers
32 views
How to numerically solve the 1-D wave equation with linear restoring force?
I have been trying to numerically solve the equation of a string vibrating on an elastic foundation in order to see its dispersive behaviour with no success.
The equation is of motion is
$$ \rho \...
