Questions tagged [wave-equation]
For questions related to solutions and analysis of the wave equation.
1,206
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Does the Wave Equation (and other PDEs) define an ODE if I took the behavior in 1D of just one point in space? How I find the ODE from the PDE eqn.?
Does the Wave Equation (and other PDEs) define an ODE if I took the behavior in 1D of just one point in space? How I find the ODE from the PDE eqn.?
Intro______________
Since is a conceptual question,...
- 1,043
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Finding volume element in a wave function
Im working on wave function. I dont know how to find this volume element from the figure eventhough some explanation for factors under the figure . Any help? [ In that paper, the author used $(s,t,u)$ ...
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Wave equation on 2d semi-infinite plane
I'm trying to understand what a correct procedure is for solving the following wave equation on a semi-infinite plane $-\infty < x < \infty$, $0 \le z < \infty$:
$$\nabla^2 \Psi(x,z,t) - \...
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Using method of characteristics to solve the p.d.e. $u_t + u_x = \sin(x-t)$ given the initial condition $u(x,0) = \sin(x)$
I am taking a numerical method P.D.E. class and want to check my numerical solution against the analytic solution but I am going wrong somewhere. I am using the method of characteristics to solve
$$...
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Smooth interpolation between two sinusoidal functions, with constraints
I am trying to create a function $f(t)$ that smoothly interpolates between $\sin(tg)$ and $\sin(t(g+1))$, where $g$ is a small integer. Additionally, I want the function to satisfy the constraint $f(t)...
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Wave equation with initial conditions question
Small transverse disturbances $u(x, t)$ of a uniform ideal string of length $\pi$, fixed at its ends and stretched straight under constant tension, obey the one-dimensional wave equation $$u_{tt}=c^...
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How many boundary conditions do I need to use this finite difference spatial approximation to solve the wave equation?
In the book, on page $233$, I found that the wave equation:
$$ u_{tt}=u_{xx} \tag 1$$
can be discretized using the following finite difference approximation:
$${ U_{i+1,j}-2U_{i,j}+U_{i-1,j} \over \...
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Natural symeetry of the wave equation?
What does "natural" symmetry of the wave equation even mean?
Given a one dimensional wave equation $$\frac{1}{c^2}\frac{\partial ^2 \phi}{\partial t^2}-\frac{\partial ^2 \phi}{\partial x^2}=...
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Find the energy of a PDE, and show it is conserved
In the domain $(0,1) \times [0,T], T>0$, we consider the boundary value problem
$$V_{tt} + \eta V = (\xi V_x - \beta V_{xxx})_x,\,\,\,\,\,V(0,t)=0, V(1,t)=0, V_x(0,t)=0, V_x(1,t)=0,$$
where $\eta, \...
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Does a weak solution have weak derivatives?
Suppose
$$ Lu = (\partial_t^2 - g^{ij}\partial_i \partial_j + b ^i \partial_i + c)u$$
has an elliptic spatial part (i.e. $g^{ij}$ with $i, j = 1,...,n$ is uniformly bounded as a linear operator from ...
- 217
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Eikonal equation reflection to solve for boundaries.
Given a point source of radially expanding wave (2D) I need to change its wavefront shape (partially) to planar by reflection.
A simple parabolic reflector will not work since the circular wave ...
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use separation of variables to solve differential wave equation with force term and steady state solution
Consider the problem of a string held at both ends and acted on by gravity. Take the coordinate x to be along the string with the ends of the string at x = 0 and x = L. Take t as time and the variable ...
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Deriving the 2D Wave Equation for a Vibrating Membrane using Newton's Law of Motion and Taylor Series
This is my first time on this site so i hope i followed the guidelines well enough. I am working on deriving the two-dimensional wave equation for a vibrating membrane in my partial differential ...
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How can I use the Von Neumann stability analysis on a system of finite difference equations?
I am trying to numerically solve the following system of partial differential equations:
$$ k^2u_{xx} - u_{tt} = f \tag 1$$
$$ c^2f_{xx} - f_{tt} = 0 \tag 2$$
I am using the method of finite ...
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Uniqueness of solution to the wave equation
Consider the Cauchy problem to the free wave equation
$\begin{align}
u_{tt}-\Delta u=0,\newline
u(0,x)=\phi(x), \newline
u_t(0,x)=\psi(x)
\end{align} \quad \text{with } t \geq 0, x \in \mathbb{R}^n$
...
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Inequality for Solution of 3-D Wave Equation.
My Problem is about the following question found at Bounding the solution of a wave equation in 3 dimensions:
: Let $u: \mathbb{R}^3 \times (0,\infty) \rightarrow \mathbb{R}$
be a solution of the ...
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General solution of the wave equation in 3D
In the context of a physics calculation, I have a wave equation with the following form, where $v$ is a constant and $f$ depends on $\vec{x}=(x,y,z)$ and on the time $t$:
$$\bigg(\dfrac{\partial^2}{\...
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Show that a solution of the wave equation is time independent
I am trying to solve this problem:
Let $u$ solve the wave equation $u_{tt}-\Delta{u}=0$ in $\mathbb{R}^n\times[0,\infty)$ and $h:\mathbb{R}^n\rightarrow\mathbb{R}$ be a harmonic with $h(x)=u(x,0)$ and ...
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Fourier transform of homogeneous wave equation
I have the following PDE $u_{tt} - \Delta_xu = 0, x \in \mathbb{R}_n, t>0$. I suppose $\Delta_{(x_1,...,x_n)}u \equiv u_{x_1x_1} + ... + u_{x_nx_n}$, because it wasn't defined in the textbook. The ...
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Proving the constancy of the speed of light using Maxwell's equations
I've seen several places on the internet say that the constancy of the speed of light can be proven using Maxwell's equations. All of the derivations of the speed of light from Maxwell's equations I'...
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Is this the correct derivation of the CFL condition and how can I interpret the error?
Consider the non-dimensionalised form of the wave equation:
$$ u_{xx} = u_{tt} \tag 1$$
Let $e^n_j$ be the difference between the numerical grid point and the function value at that point:
$$ e^n_j = ...
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Calculate integral $2\int_{0}^{1}e^{-400(x-0.3)^2}\sin(h\pi x) dx$ where $h \in \mathbb{N}$
I'm solving the wave equation
$$
\begin{cases}
\dfrac{\partial^2 u}{\partial t^2} - v^2 \dfrac{\partial^2 u}{\partial x^2} =0, &(x,t) \in (0;1) \times \mathbb{R}^+_{*},\\
u(t,0) =u(t,1) =0, &...
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Finite Speed of Propagation for $u_{tt} - \Delta u + qu = 0$
I am looking to show that there is some type of finite speed of propagation property for PDEs of the form
$$
u_{tt} -\Delta u + q u = 0
$$
Where we can assume that $q$ is as smooth or integrable ...
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IBVP wave equation with a Robin condition, using d'Alambert's formula
We are given $$\frac{\partial ^2u}{\partial x^2}=\frac{\partial ^2 u}{\partial t^2},x>0,t>0,$$with the initial conditions
\begin{align*}
u(x,0)&=2x+1\\
\frac{\partial u}{\partial t}(x,0)&...
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What is the exact minimal condition for the emergence of stationary intensity patterns in a chaotic wave field?
Consider the fact that a superposition of two wave functions with different frequencies $\omega_A$ and $\omega_B$ ($\omega_A \neq \omega_B$),
$$\begin{align}
\Psi(\vec x, t) &= \Psi_A(\vec x, t) + ...
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Condition on structure for lower order terms for local existence to quasilinear waves
In Jonathan Luk's lecture notes, section 6, in order to get the local theory for a wave equation
$$a^{\alpha\beta}(\phi,\partial \phi) \partial_\alpha \partial_\beta \phi=F(\phi,\partial \phi) $$
he ...
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Where did I make a mistake while trying to derive the CFL condition using the matrix norm method
Using the central difference derivative approximation, the finite difference approximation of the following wave equation:
$$ c^2 u_{xx} = u_{tt} \tag 1$$
can be written as:
$$ U_{i+1, j}= kU_{i, j+1} ...
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A tough wave-equation problem
A string is at rest and in a straight line. At t = 0 it
is subjected to a constant force distribution perpendicularly from
above and along the entire string. This force distribution remains constant ...
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Can you explain why $du/\sqrt{c-2u} = d\varepsilon$ using hyperbolic substitution integral becomes $A\operatorname{sech}^2(t)$?
Here I have a 3rd order ODE wave equation
$$
-cu' + 6uu' + u''' = 0
$$
where $u(\varepsilon) = u$ and $\varepsilon = x - ct$; a wave (assume $\varepsilon$ is one variable; hence $u' = du/d\varepsilon$)...
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Analytical methods for solving PDE's with time-varying boundary conditions
I'm trying to find solutions for the wave equation when the boundary condition changes over time. I've seen some specific examples of the heat equation online, but I haven't been able to find any ...
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Analytical Equation of the gaussian 1D wave equation with periodic Boundary condition
I am trying to validate the 1D analytical wave equation with a numerical solution with periodic boundary conditions. I have implemented the periodic boundary condition for the numerically calculated ...
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Solution to wave equation using odd extension of initial conditions
I'm trying to solve this problem from Haberman's book.
I solved the PDE and got
$$
u(x,t) = \sum_{n=1}^{\infty} A_n \sin{\frac{n\pi}{L}x} \cos{\frac{n\pi c}{L}t}
$$
I used the trigonometric identity ...
- 55
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Analytic solutions for damped semi-linear wave equation
I am looking for the analytic solution to the initial value problem of a semi-linear wave equation of the form
\begin{align}
u_{tt} - \Delta u + au_{t} + f(x,u) &= 0 \qquad \text{on } \Omega\times[...
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Solve a wave-equation problem
We have the following problem
\begin{cases}
u_{tt}=u_{xx}\\
u_x(0,t)=u_x(\pi,t)=0\\
u(x,0)=\cos x\\
u_t(x,0)=-\cos x
\end{cases}
The correct approach should be:
We have homogeneous Neumann conditions ...
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Using Fourier Sine Series to Solve a Dirichlet PDE Without Separation of Variables
Suppose we are given the following PDE:
$$u_{tt} - c^2u_{xx} + m^2u = 0$$
which is equipped with the following boundary conditions: $u(0,t) = u(L,t) = 0, u(x,0) = \Phi(x), u_t(x,0) = \Psi(x)$ where we ...
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How to solve partial differential equation of the form $\partial_x f\ (dx/dt)+\partial_t f =0$
In some of my clases, they teach me some function which describe infiltration in the soil. The equation is known as Kostiakov's equation. We denote the Kostiakov's equation as $I(t)$, which maps $I:\...
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Domain of dependence for wave equation on bounded domain
Consider a wave equation, say in $1+1$ dimensions for $\phi(x,t)$, on a bounded domain, say $x \in (0,L)$ and $t \in \mathbb{R}$, with initial values $\phi(x,0)=u(x)$ and $\partial_t \phi(x,0)=v(x)$ ...
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Solving Wave Equation with strange boundary conditions
I'd wish to solve
$$\boxed{\begin{align}{\partial_x}^2\,\psi(x,t) - \frac{1}{c^2}{\partial_t}^2\,\psi(x,t) = 0 \qquad (\star)\end{align}}$$
with somewhat strange boundary conditions:
$$\psi(x,0) = e^{\...
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Solving the 3D Nonhomogeneous Wave Equation
I am attempting to solve the following differential equation:
$$c^2 \nabla^2 u(\vec{r}, t)-\ddot{u}(\vec{r}, t)=f(\vec{r}, t)$$
... where the forcing function $f(\vec{r}, t)= R_f(\vec{r})T_f(t)$ is ...
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More Exacting A Wave
I have a strange wave and I am trying to make it an equation. These are the points I have plotted:
x
y
$0$
$8$
$\arcsin\left(\frac{\sqrt{40}}{20}\right)$
$\sqrt{40}$
$\arcsin\left(\frac{\sqrt{80}}{...
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Solution to PDE wave equation using d'Alembert's formula and given that $f(x)$ and $g(x)$ are both odd functions.
Show that if $f(x)$, $g(x)$ are both odd functions about $x_0$, then the solution of the wave equation, $u_{tt}=c^2u_{xx}$, is also odd about $x_0$. J. d'Alembert's solution is \begin{equation}u(x,t)=\...
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Initial value problem for PDE wave equation using D'Alembert's formula
$$\frac{\partial^2}{\partial t^2}u(x,t)=c^2 \frac{\partial^2}{\partial x^2}u(x,t) , \qquad -\infty<x<\infty, \qquad t>0,$$
\begin{equation}u(x,0)=e^{-x^2}\end{equation}
\begin{equation}u_t(x,...
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How would one model the propagation of a wave given an initial state but with no sources?
I'm not very fluent in this area of math and I'm just dipping my toes into solving the wave equation and simulating their propagation. I recently learned to use Green's functions to find the ...
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Wave equation solved by MOL - stability condition for 5-point central formula
I want to derive the stability condition for the wave equation (5-point central finite difference formula).
Wave equation 1D is defined as:
$$
\frac{\partial^2 u}{\partial t^2} = \alpha\frac{\partial^...
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How to find a PDE satisfying $\rho_{tt}=c^2(\rho_{rr}+2r^{-1}\rho_r)$ with $R=r\rho$?
I'm working on a course problem,
In a compressible and uniform fluid the equilibrium density and pressure are $\rho_0$ and $p_0$, respectively. Due to the passage of a compressible perturbation, ...
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Understanding the boundary condition of spherical waves in the flat spacetime
I am trying to understand one of the two boundary conditions one has to impose to find the solutions of the wave equation in the flat space-time inside a collapsing null shell. For the spherical wave, ...
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1
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Intuitive explanation of initial condition for wave equation
I've just started a course in Fourier analysis, and have some problem understanding the initial condition of wave equation, and would appreciate if someone would like to explain to me in the easiest ...
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1
answer
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1D Wave Equation with Coupled IC's and Non-Homogeneous BC's
Consider the following wave equation:
\begin{align}
&u_{tt} = u_{xx}, \\\\
&u(x,0) = \frac{1}{2 + \sin x} =: \psi(x), \\\\
&u_t(x,0) = -\frac{\cos x}{(2 + \sin x)^2} = \psi'(x), \\\\
&...
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Fundamental mode of Bessel functions (vibration in a circular membrane)
How come the fundamental mode of Bessel's equations in solutions of the PDE in the case of the vibrations of circular membrane must be angles independent? (thus making it the axisymmetric case).
So in ...
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Wave-equation for a string under a constant perpendicular force
A string is at rest and in a rectilinear form when at $t=0$ begins to
be subjected to a constant force distribution perpendicularly from
above and along the entire string. This force distribution ...
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