Questions tagged [wave-equation]
For questions related to solutions and analysis of the wave equation.
1,009
questions
0
votes
0answers
5 views
question about superposition linear wave motion [closed]
given Y= A sin (kx-omega t + psi ), determine the result of superposition 2 wave if different amplitude and frequency, but another parameters is same
-2
votes
0answers
30 views
Question about the solution of the differential equations. [closed]
Let $f$ be a $C^1$-function from $R$ to $R$. For$x\in R$ and $t\in R_+$, one consider the equation$$\partial_t u_x + \partial_xf(u_x)-\epsilon\partial_{xx}u_x=0 \tag{1} $$
where $\epsilon$ is a small ...
0
votes
0answers
25 views
Schwarzschild metrics form a smooth one-parameter family of smooth metrics.
In my lecture notes, they claim that the family of Schwarzschild metrics form a smooth one-parameter family of smooth metrics, but the proof of this is left to the reader as an exercise and I am ...
0
votes
0answers
32 views
What is the period of the solution of a wave equation boundary value problem
In my studies of numerical PDEs, I was given this problem
We consider a vibrating string that satisfies the wave equation $u_{tt}=u_{xx}$ on the unit interval with boundary conditions $u(0,t)=0$, $u(...
1
vote
2answers
33 views
Solving a wave equation with boundary conditions on a circle?
How might one solve a wave equation which had boundary conditions on a circle?
i.e. given $$(\partial^2_x+\partial^2_y)\phi(x,y)=0$$
And known values, $f$ on a circle:
$$\phi(\cos(\theta),\sin(\theta))...
1
vote
0answers
22 views
Assume $u \in L^\infty ([0,T];H^{s-1})$, $u_t \in L^\infty ([0,T];H^{s-1})$ then can we show that $u \in C([0,T];H^{s-1})$
In Sogge's "Lectures on nonlinear wave equations" it states that :Assume $u \in L^\infty ([0,T];H^{s-1})$, $u_t \in L^\infty ([0,T];H^{s-1})$ then $u \in C([0,T];H^{s-1})$ .
Here $u_t$ is ...
1
vote
0answers
22 views
Newton's Law $F=ma$ in its longitudinal and transverse components (Wave equation derivation)
In Partial Differential Equations by Walter Strauss, Ch 1.3 Example 2, in deriving the wave equation the author states that Newton's law $F=ma$ in its longitudinal (x) and transverse (u) components is
...
0
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0answers
18 views
Wave equation on half space with nonzero Dirichlet boundary condition
We consider the wave equation for $u(t,x)$
$$
\partial_t^2 u - \Delta u = 0 \, \text{ in } (0 , \infty) \times \mathbb R^n_+,
$$
where $\mathbb R^n_+ := \{ (x_1, \cdots , x_n ) \in \mathbb R^n: x_n &...
1
vote
1answer
65 views
Vector field commutator and wave equation
I am studying this review on black holes https://arxiv.org/abs/0811.0354 by Dafermos and Rodnianski, and I try to prove the proposition E.0.1. I am currently stuck on the following equation :
$$
{\...
1
vote
1answer
14 views
Solving general non-homogenous wave equation with homogenous boundary conditions
I have been given the following PDE to solve:
$$
u_{tt}-u_{xx}=g(t)\sin x \;\;\; (t,x)\in (0,\infty)\times(0,\pi)\\u(0,x)=u_t(0,x) \;\;\;x\in(0,\pi)\\
u(t,0) = u(t,\pi) =0 \;\;\; t>0
$$
This ...
2
votes
1answer
62 views
Uniqueness of coefficients in plane wave representation of wave equation's solution
3D wave equation
$$
\Delta p(\boldsymbol{r},t) - \frac{1}{c^{2}} \frac{\partial^{2} p}{\partial t^{2}}(\boldsymbol{r},t) = 0 \tag{1}
$$
By applying the spatial Fourier transform to this PDE, we obtain ...
0
votes
0answers
43 views
Vibrating Circular Membrane with Vibrating Boundary
Looking online, I have found many solutions to variations of 'vibrating circular disc problem', that is solving the following PDE
\begin{align*}
\begin{cases}
u_{tt} - c^2\nabla^2
u = 0\\
u(R, \varphi,...
0
votes
0answers
12 views
Propagating wave spherical spreading (or geometric spreading) question
The intensity (power per unit area) of a spherical wave falls off as $1/4\pi r^2$.
My question: Does this mean the wave amplitude falls off as $1/2\pi^{1/2} r$ ?
I understand the $1/r$ and the $1/2$ ...
1
vote
0answers
21 views
Wave equation near two boundaries in n dimensions
Consider a function $u(t, x, y)$ defined on $t \in (t_i, t_f) \ , x > 0 \ , y \in \mathbb{R}^{n - 2}$ satisfying the wave equation (here $\Delta_y$ is the Laplace operator w.r.t. the variable $y$)
\...
1
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0answers
13 views
Implementing Crank-Nicolson scheme for 1-D wave equation
I am trying to implement the Crank-Nicolson scheme directly for the second order wave equation by replacing $E^n$ terms in spatial derivative with $(E^(n-1)+2E^(n)+E^(n+1))/4$. I am sure I have ...
1
vote
1answer
32 views
D'Alemberts formula with both initial value and derivative equal to 0
Trying to solve a PDE problem consisting of these equations:
$$ u_{tt} - c^2 u_{xx} = 0 \quad \quad(0 < x < L, t > 0)$$
$$u(0,t), \quad u(L,t)=Bsin(\omega_{0}t) \quad (t >0)$$
$$u(x,0) = 0,...
0
votes
0answers
28 views
Eigenfunction for solving wave equation
$u(x,t)=\sum_{n=1}^\infty \cos\frac{(2n-1)\pi x}{2L}(c_{1n} cos\frac{(2n-1)\pi t}{2}+c_{2n} sin\frac{(2n-1)\pi t}{2})$, $n=1,2,...$ $x\in[0,L]$
The ICs are
$u(x,0)=p(0)x$, $u_t(x,0)=p'(0)x$
Applied ...
0
votes
2answers
42 views
How to transform a non-homogeneous Neumann boundary condition into a homogeneous for the wave equation? [closed]
I have a initial/ boundary value problem for standard wave equation
$$
\frac{\partial^2u}{\partial t^2}=c\frac{\partial^2u}{\partial x^2},
$$
where one of the boundary conditions is non-homogeneous: ...
0
votes
2answers
46 views
1D Wave PDE with Nonzero Initial and Boundary Conditions
I'm not sure how to start this PDE since the initial and boundary conditions are nonzero. May someone point me in the right direction?
This is the problem:
$$u_{tt} = u_{xx}$$
$$u(x,0) = \frac{1}{2+ \...
0
votes
0answers
28 views
Show that a function is in $L^2(\mathbb{R_{+}^3})$
Let u $\in C^2([0, \infty] \times\mathbb{R^3})$ be the solution to the heat equation
$$
u_{tt}-\nabla^2 =0 \\
u(0,x)=0\\
u_t(0,x)=h(x) \in C_0(\mathbb{R^3})
$$
Suppose there exists a constant $C$ ...
0
votes
1answer
26 views
What does this sine wave say exactly?
I am currently starting to learn about sine waves and I was just wondering why this sine waves amplitude is the way it is? Like why exactly does it start at 2, peak at negative 1 and peak at 3 instead ...
1
vote
1answer
40 views
Wave equation with piecewise (discontinuous) initial condition
I have the equation $u_{tt}-u_{xx}=0$, with initial conditions $u(0,x)=f(x)=0$ and $u_t(0,x)=g(x)=\begin{cases} 1, & \mbox{if } x>0 \\ 0, & \mbox{if } x \leq 0 \end{cases}$. I have to solve ...
0
votes
1answer
24 views
Lax Equation formulation of the Focusing Non-Linear Schrodinger Equation.
For a project I am writing at university, I have a section where I am writing about rational solutions to the NLSE, which use Lax pairs as the starting point for the derivation. I therefore wish to ...
0
votes
0answers
9 views
Solve the following wave equation on half line via separation of variables.
Use separation of variables and then Fourier integral to solve the following wave equation:
$$u_{tt}=c^2u_{xx}, \ \ 0<x<\infty, t>0,$$ $$u(0, t)=0,\ t\geq 0$$ $$u(x, 0)=0, \ u_t(x, 0)=g(x)=\...
2
votes
0answers
44 views
wave equation in $\mathbb{R}^3$ with radial symetric initial data
Let $u$ be a solution of the wave equation $-u_{tt}+\nabla^2u =0$ with the initial conditions $u(0,x)=0$ and $u_t(0,x)=h(x) = h_1(\vert x \vert)$
Show that $$u(x,t)=\int_{\Vert x \Vert -t}^{\Vert x \...
1
vote
0answers
88 views
Calculating the bending moment ($M$) of a beam?
I am trying to learn how to work out the bending moment for progressively complex systems to model a beam. I am assuming only one axis of motion (transverse = $y$).
For the basic equations of motion, $...
0
votes
2answers
52 views
Find the general term.
I tried to solve the wave equation:
After some calculation I reached at this step where I have to find the constant alpha.
For even value of n alpha is 0
For n = 1, 3, 9, 11, 17, 19 . . . . . . ...
0
votes
0answers
16 views
Characteristic wave equation - Cauchy problem
I was trying to find some reference to solve the following wave equation:
\begin{equation} \label{eq:vL-LS2020}
\left\{\begin{aligned}
(\partial_t^2 -\Delta) v & = 0 && \text{in}\; \Omega \...
0
votes
0answers
27 views
Gauss-Lobatto quadrature and nodal points for FEM
By using the Legendre-Gauss-Lobatto (LGL) quadrature formula (QF) and LGL nodal points one achives a diagonal mass-matrix for finite element problems. (More specifically, the spectral element method.)
...
1
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1answer
60 views
D'alembert's approach for boundary value problems
I have the following problem
$\left\{\begin{array}{l}
u_{t t}=u_{x x}+2 u, \quad t>0, \quad 0<x<\pi \\
\left.u\right|_{t=0}=0,\left.\quad u_{t}\right|_{t=0}=0 \\
\left.u_{x}\right|_{x=0}=1,\...
0
votes
1answer
46 views
Solution of Wave Equation initial conditions
A vibrating string fixed at $ x=0 $ and $ x=L $ undergoes oscillations described by the wave equation $ \frac{\partial^2u}{\partial x^2} = \frac{1}{c^2} \frac{\partial^2u}{\partial t^2} $ where $ u(x,...
0
votes
0answers
52 views
Wave equation under instantaneous blow inconsistent?
Consider an infinite string, at rest for $t<0$. It receives an instantaneous transverse blow at $t=0$ which imparts an initial velocity of $V\delta(x-x_0)$ where $V$ is constant. Derive the ...
1
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0answers
36 views
Solving the wave equation on the unit disk
I am having trouble solving my first wave equation on the unit disk. The problem that we are posed is to consider a membrane covering the unit disk:
$$
D =\{(x,y) \in \mathbb{R}^2: x^2+y^2\leq 1 \}
$$
...
2
votes
1answer
31 views
Decomposition of wave number
Wave number can be easily obtained in one dimensional situation.
$$k=\omega/c$$
In the book, Fundamentals of Acoustics, the method of separation of variables is applied to solve two dimensional wave ...
1
vote
1answer
67 views
Wave equation and falling bar
I want to understand the following: consider an (elastic) bar of one dimension (length only) with length $L=1$m floating at some height $h=2m$:
Vertically
Horizontally
I know that the equation ...
0
votes
0answers
20 views
Using Fourier transform to simplify linear system
I was looking at the question which has the elastic wave equation
$$
\rho\frac{\partial^2}{\partial t^2}\bigg(\begin{array}{c}
u_1\\
u_2\end{array}\bigg)
=
\frac{\partial}{\partial x_3}\bigg(\begin{...
0
votes
0answers
29 views
NTSC Colour Decoder Sampling
I've been working on this on and off for a number of years but I need help with the final maths part as i find it difficult to find the resources i need to learn and i'm not good a learning through ...
1
vote
0answers
27 views
manipulation on wave equation
Given $a_{ij}:\mathbb{R^n}\to \mathbb{R}$ and suppose that $a_0>0$ which $\sum_{ij}a_{ij}\xi_i\xi_j \geq a_o|\xi|^2$, $\forall x\in \mathbb{R^n}, \xi\in \mathbb{R^n}$. Considering
$$\frac{\partial ^...
0
votes
0answers
28 views
vibrating circular membrane that doesn't satisfy wave equation
I'm studying a "vibrating circular membrane" problem and I've attached some screenshots of one possible mode of vibration.
The equation for this mode of vibration is $$ \varphi(t,x,y)=\big(...
1
vote
0answers
26 views
What are the solutions of this differential operator?
I am thinking about the ‘complex’ transport equation:
$$\partial_t + a\cdot \nabla = 0$$
where $a \in \mathbb{C}^n$ and its elements are either real or pure imaginary. The simplest example for this ...
1
vote
0answers
30 views
What determines the stability and velocity of a 2D wave equation using finite difference method?
I have been able to master the 1D wave equation in finite differences, but I am struggling to get a similarly functioning 2D wave equation.
For background I will review my understanding of the 1D ...
0
votes
1answer
23 views
Amplitude of wave question
Suppose the following wave has amplitude 1.
$y(x,z,t)=Ab\cos(px)e^{iwt+ikz}$
Find a relationship between $A$ and $b$. Then rewrite $y$ in terms of that relationship.
My thoughts.
The amplitude is the ...
2
votes
1answer
89 views
1D Wave Equation Problem Separation of Variables
I need to solve the following 1D Wave Equation problem using Separation of Variables, but I cannot figure it out.
\begin{align}
u_{tt} &= u_{xx}\\
u_x(t,0) &= u_x(t,1) = 0\\
u(0,x) &= x(1-...
1
vote
0answers
62 views
Compact Support for a PDE
Suppose that $u$ is a $C^2$ solution to the wave equation in $\mathbb{R}^n\times\mathbb{R}$
Show that if $u(\cdot,0)$ and $u_t(\cdot,0)$ have compact support in $\mathbb{R}^n$, then $u(\cdot,t)$ has ...
1
vote
1answer
81 views
Exponential stability of damped wave equation
I like to proof the exponential stability of a damped wave equation with the following form.
Let $G \subset \mathbb{R}^n$ be bounded with smooth margin. For $\gamma > 0$ we are given the damped ...
2
votes
0answers
35 views
Showing classical solution of a wave equation IVP is identically equal to zero
Let $u:\mathbb{R}\times (0,+\infty) \to \mathbb{R}$ is a classical solution to the problem \begin{align}u_{tt}-u_{xx}+u&=0~;~x\in \mathbb{R}, t>0 \\u(x,0)&=\varphi(x)=0~;~x\in \mathbb{R}\\...
2
votes
1answer
40 views
Weird non-homogeneous wave equation
I've been trying to solve the Cauchy problem \begin{align}&u_{tt}-u_{xx}+u=0~; x \in \mathbb{R},t>0 \tag{1}\\&u(x,0)=\cos x ~\text{for}~x\in \mathbb{R}\\&u_t(x,0)=\sin x~\text{for}~x\in ...
0
votes
0answers
11 views
Exactly what is the cause for the zero reactance seen in the impedance at the center feed point of a resonant half wave dipole?
Lots of explanations i've read on the internet written by ham radio operators and in some text books state that a half wave dipole is resonant when the inductive and capacitive reactances cancel out. ...
0
votes
1answer
23 views
Wave packets and duration of wave packet
Let $T=\frac{1}{30}$ be the time it takes for the wave to complete one cycle. That is, T is the time period. Then the duration of the wave packet is $\frac{10}{T}$.
Now my question is, how does one ...
1
vote
0answers
22 views
How to solve the 2D wave equation given the Helmholtz solutions containing bessel functions?
I am struggling conceptually with what this question is asking me to do. The question is:
In polar coordinates the Helmholtz equation has solutions of the form:
$$u_m=(AJ_m(\sqrt{\lambda}+(BY_m(\sqrt{\...
