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Questions tagged [wave-equation]

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

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Pseudo-spectra of a dynamical system

This question is about the pseudo spectra of a dynamic system. What does a following system of two second order equations say in terms of pseudo-spectral analysis? Here is a system where $x(t)$ and $y(...
Edisher's user avatar
-1 votes
1 answer
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Good rigorous reference for wave phenomena [closed]

I'm interested in revisiting the physics of waves that I learned in college (diffraction, refraction, dispersion, interference) from a rigorous point of view, but I haven't found a PDE book which ...
Gordon Craig's user avatar
1 vote
1 answer
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Wave Equation general solution on 1D

I'm pretty sure the answer is simple but I need some guidance. I'm reading about the wave equation. And when we have it in the following form. $$ \frac{\partial^2 u(x, t)}{\partial t^2} = \frac{\...
silgon's user avatar
  • 161
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1 answer
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Two-dimensional wave equation and Bessel function [closed]

Show that the solution $w(x_1,t)$ to the initial-value problem $\begin{equation}\begin{cases} w_{tt}-c^{2}w_{x_{1}x_{1}}=c^{2}\lambda^{2}w\\ w(x_{1},0)=0, w_{t}(x_{1},0)=g(x_1)=\psi(x_1) \end{cases}\...
logarithm's user avatar
  • 530
1 vote
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Spherical Wave Equation With a Source and Time-dependent Velocity

In my research I am studying a type of soliton known as Q-balls. The standard equation of motion obtained for the spatial part of a Q-ball is (where $f = f(r)$) $$\frac{d^2f}{dr^2} + \frac{2}{r} \frac{...
Daniel Waters's user avatar
3 votes
1 answer
46 views

Deriving the Green's function for the D'Alembert operator from the Helmholtz equation with treatment of $\frac{\partial_t}{c}$ as a scalar

The 3-dimensional Green's function for the Helmholtz operator $$(\Delta_x + \omega^2)G(x,x') = \delta(x-x')$$ is given by $$G(x,x') = G(x-x') = -\frac{e^{\pm i \omega \|x-x'\| }}{4\pi \|x-x'\|}.$$ In ...
theta_phi's user avatar
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Solving the wave equation of a tensor $h_{\mu\nu} = (1/2) (e_\mu e_\nu + e_\nu e_\mu)$

It is known that the solution to the wave equation for a tensor $$ \square h_{\mu\nu} = 0 $$ is $$ h_{\mu\nu}(\vec{x}, t) = \int \frac{d^3k}{(2\pi)^3} \sum_{\lambda=+,\times} \left( \epsilon_{\mu\nu}^{...
Anon21's user avatar
  • 2,589
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21 views

Plotting Chladni pattern

Hi fellow mathematicians, I'm trying to simulate a 2d Chladni pattern in Matlab and Maple. Using Fourier series, I get: Solution for rectangular membrane However, comparing the result to previous 9 ...
TKO_N1's user avatar
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Solving differential equation with mixed partials [duplicate]

Lets take differential equation $$u_{tt}+2u_{tx}-3u_{xx}=0$$ where $-\infty < x < \infty$, $-\infty < t < \infty$ and the boundary conditions are $u(0,x) = \phi(x)$, $\;u_{t}(0,x) = \psi(x)...
Malum Phobos's user avatar
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Solution of Cauchy problem by Kirchhoff’s formula

while studying the Cauchy problem \begin{array}{l} {u_{tt}} - {\nabla ^2}u = 0,x \in {\mathbb{R}^3}\\ u\left( {x,0} \right) = 0\\ {u_t}\left( {x,0} \right) = f\left( {x} \right) \end{array} the ...
Jiabin Liu's user avatar
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Why is this Kirchhoff's formula for wave equation be different from the one on other books?

I'm reading this lectures, on page 15, the author gives the kirchhoff's formula as Theorem 4.11 (Kirchhoff's formula). Let $\phi$ be a solution to the equation $\square \phi=F$ with $\phi, F \in C^{\...
YuerCauchy's user avatar
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How to solve this 3-dimensional wave equation?

Consider $\square u=0$ in $\mathbb{R}_{+} \times \mathbb{R}^3$, with $u(0, x)=\mathbb{1}_{B_R}$ and $\partial_t u(0, x)=0$. I want to find the solution $u(t, x)$ explicitly.here $\square u=u_{tt}-\...
YuerCauchy's user avatar
1 vote
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My solution to the fourier transform of multi-dimensional wave equation

Our definition of Fourier transform:$\hat{f}(\xi)=\int_{\mathbb{R}^n} f(x) e^{-i x \cdot \xi} d x$ now consider the wave operator $$ \square:=-\partial_t^2+\partial_{x^1}^2+\cdots+\partial_{x^d}^2=-\...
YuerCauchy's user avatar
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Elliptic problem of Euler Lagrangian form

Let $f, g \in C_c^\infty(\mathbb{R})$ be given, $c$ a positive constant, and suppose that $w$ solves the following initial value problem for the wave equation on $\mathbb{R}$: \begin{cases} c^{-2} w_{...
Russell Hua's user avatar
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Wave equation with periodic boundary values - D.Alembert's solution vs separation of variables.

The D.Alembert's solution to the initial value problem $$ \frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}, \quad u(0, x) = f(x), \quad \frac{\partial u}{\partial t}(0, x) = ...
Tomer's user avatar
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1 answer
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A problem in solving ODE generated by wave equation.

The problem is given below: Find the solution to the equation \begin{equation} u_{tt}-c^2 u_{xx}=k^2 u \end{equation} and the solution has the form $u=f(x^2-c^2 t^2)=f(s)$. And expand it to a power ...
Leven Wong's user avatar
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Two-Dimensional Wave Equation and Nodal Lines

I'm struggling with this question regarding the nodal lines of a 2D wave equation on a rectangular membrane. So I'm given the following equation: $u_{tt}-c^2(u_{xx}+u_{yy})=0, (x,y,t)\in R\times[0,\...
cillianlynch_'s user avatar
0 votes
1 answer
84 views

d'Alembert's solution to the wave equation via Fourier Transforms

I am trying to solve the wave equation $$v_{tt} = v_{xx} \text{ on } (x,t) = (-\infty, \infty) \times (0,\infty)$$ with initial conditions $$v(x,0) = f(x), \quad v_t (x,0) = g(x)$$ I have shown that ...
idk31909310's user avatar
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While solving Wave equation using FDM, how to solve and related eigenvalues

$\ t>0, x\in (0,\pi)$ \begin{cases} u_{tt}=u_{xx}\\ u(t,0) = u(t,\pi)=0\\ u(0,x)= exp(-32(x-(\pi/2))^2\\ u_t(0,x)=0 \end{cases} We assume that sufficient smoothness in $u$ and using \begin{cases} ...
JAEMTO's user avatar
  • 695
1 vote
1 answer
59 views

1D Wave Equation with Resonant Boundary Condition

The question is essentially how to find one particular solution of the following boundary-value problem: $$ \frac{\partial^2 y}{\partial x^2} = \frac{1}{c^2} \frac{\partial^2 y}{\partial t^2},\; y(0, ...
이희원's user avatar
  • 479
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61 views

Deriving the plane wave scattering from a conducting half plate

Consider the two-dimensional problem of a plane wave of angular frequency $\omega$ propagating in the direction $\hat{\textbf{k}}=cos(\phi^{i})\hat{\textbf{x}}+sin(\phi^{i})\hat{\textbf{y}}$ impinging ...
Chris's user avatar
  • 469
2 votes
0 answers
22 views

Any reference about these topics (waves on the sphere)?

At the end of the section 1.9.2 in the book A Panoramic View of Riemannian Geometry, it has the following text about waves on the sphere: "We will not say much now about spherical harmonics ... ...
yo-yos's user avatar
  • 63
1 vote
2 answers
41 views

A question about reflection method of the wave equation

For the Wave Equation: $ \left\{\begin{matrix} u_{tt}-c^2u_{xx}=0 & x>0, t>0\\ u(x,0)=f(x) & x>0 & \hspace{0.5cm} (1) \\ u_{t}(x,0)=g(x) & x>0 & \hspace{0.5cm} (2) ...
Yamato's user avatar
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1 answer
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Finding solution to the wave equation by the characteristics method

Given the wave equation in one spatial dimension, $\phi_{tt} -c^2\phi_{xx} = 0,$ setting $u:=\frac{\partial \phi}{\partial t}, v := \frac{\partial \phi}{\partial x},$ we get the matrix representation ...
user996159's user avatar
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What is the conservation form of this Euler equation?

We have a Euler equation: $$ \frac{\partial}{\partial t} L_1 + \frac{\partial}{\partial x} L_2 - L_3 = 0 $$ where $$ L_1 = \frac{\partial}{\partial \varphi_t} L, \qquad L_2 = \frac{\partial}{\partial ...
Gaelthorn's user avatar
  • 113
1 vote
1 answer
112 views

How to find the characteristic form of coupled PDEs?

I'm reading a textbook named linear and nonlinear waves. In chap.14, a method was used to solve the coupled equations which can reduce PDEs to ODEs. But I don't know how to find the needed ...
Gaelthorn's user avatar
  • 113
1 vote
0 answers
36 views

Stable timestep criterion for variable density acoustic wave equation

As part of my PhD, I am implementing the isotropic variable density acoustic wave equation numerically with finite-difference and I have a question regarding its stable timestep criterion. The ...
Anon's user avatar
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1 vote
0 answers
29 views

How to use Stokes expansion method for solving nonlinear Klein-Gordon equation?

A simple nonlinear Klein-Gordon equation can be written as: $$ \varphi_{tt} - \varphi_{xx} + V'(\varphi) = 0 $$ If we take $$ \varphi = \Psi(\theta), \qquad \theta = kx - \omega t $$ then the equation ...
Gaelthorn's user avatar
  • 113
1 vote
0 answers
45 views

Is the Fourier series solution to the plucked string problem a weak solution to the wave equation?

I am looking at the Fourier series method applied to the plucked string problem: \begin{align} u_{tt} = u_{xx} & , \qquad 0 < x < \pi, t>0 ; \nonumber \\ u(0,t) = u(\pi,t) & , \qquad ...
ivan44's user avatar
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4 votes
0 answers
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Attenuation estimation of the solution of the two-dimensional wave equation Cauchy problem

This is the equation given, $$\begin{array}{l} u_{tt}=a^{2}\left(u_{x x}+u_{y y}\right), \\ \left\{\begin{array}{l} \left.u\right|_{t=0}=\varphi(x, y), \\ \left.u_{t}\right|_{t=0}=\psi(x, y) . \end{...
Zydragon's user avatar
1 vote
0 answers
38 views

Integrate $u_{\xi\eta}$ with respect to $\xi$.

I'm studying about the non-homogeneous wave equation $u_{tt}(x,t)-c^2u_{xx}(x,t)=F(x,t)$. After the change of variables $\xi=x+ct,~\eta=x-ct$, this takes the form $$u_{\xi\eta}\left(\dfrac{\xi+\eta}{2}...
Fabrizio G's user avatar
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2 votes
0 answers
98 views

$L^p$-estimates for one dimensional wave-equation with lower order pertubation

Suppose $b\in L^\infty(\mathbb{R})$ and $u\colon \mathbb{R}\times \mathbb{R}\to \mathbb{R}$, where $(t,x)\mapsto u(t,x)$ is a solution to $$ \partial_t^2 u = \partial_x^2 u +b(x)\partial_x u,\quad (x,...
ym94's user avatar
  • 873
1 vote
0 answers
29 views

Given the wave equation and $u_{yw}=0$, conclude $u(x,t)=\phi(x-\alpha t)+\psi(x+\alpha t)$.

I know the wave equation is $\alpha^2u_{xx}=u_{tt}$. I am given the result from a previous problem, $u_{yw}(x,t)=0$ where we let $y=x-\alpha t$ and $w=x+\alpha t$. For this, $\phi$ and $\psi$ are ...
PeakyBlaze7788's user avatar
1 vote
0 answers
27 views

Given the wave equation, show $u_{yw}=0$

Consider the wave equation: $\alpha^2 u_{xx}=u_{tt}$. I am told to let $y=x-\alpha t$ and $w=x+\alpha t$ and use chain rule but I'm not confident in my attempt. I got $x=\frac{y+w}{2}$ and $t=\frac{w-...
PeakyBlaze7788's user avatar
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0 answers
8 views

How would I approach a question on using d'Alembert's solution to find the solution of the initial boundary problem?

the question i am stuck on I have obtained the following answers: c = 2 f(x) = sin(3x)cos(6t) g(x) = 28sin(7x)(cos(14*t)) however these don't appear to make sense with the boundary conditions. What is ...
user avatar
4 votes
1 answer
22 views

How to find the time period of the sum of two waves with different frequency

I'm dealing with a few sin waves that represent music. An example is the following f(x)=sin($\frac{1.47x}{440}$)+sin(x) How would I represent the time period of this superposed wave? Also if the ...
WhoShotJJ's user avatar
0 votes
0 answers
28 views

Sound level dropoff from sources of different shapes

I'm looking into sound propagation for an audio system in a game engine and in the book "game engine architecture" I found that there could be different types of sound sources and the ...
WhoLeb's user avatar
  • 1
1 vote
1 answer
66 views

PDE of second order

Can someone tell me what type of equation is this and how would one go about solving it. \begin{align} &u_{xx}-u_{tt}=u+\sin \left(\frac{\pi x}{4}\right),\quad x\in (0,2),\, t>0, \\ &u(0,t)=...
Vuk Stojiljkovic's user avatar
1 vote
0 answers
33 views

D'Alembert Solution - 1D Wave equation, integration step

I am working through D'Alembert's solution to the 1D wave equation using the substitution of canonical coordinates. I have an initial condition of: $$u_{t}(x,0) = g(x) $$ with a general solution ...
Alexander Savadelis's user avatar
2 votes
0 answers
62 views

Obtain an analytic solution to a hyperbolic PDE

I have been dealing with the following PDE, where $f:\mathbb{R}^2\rightarrow\mathbb{C}$: $$ \frac{\partial^2 f}{\partial x^2}-A \frac{\partial^2 f}{\partial y^2}-Be^{4x}f+Ce^{6x}g(y)f=0 , $$ where $A$,...
physics-its's user avatar
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0 answers
46 views

General solution to finite interval simple wave equation

I have a wave equation with fixed interval with the following initial and boundary conditions. $$u_{tt} = u_{xx}$$ $$u(x, 0) = \phi(x) = x(1-x)$$ $$u_t(x, 0) = x^3 - x$$ $$u(0, t) = u(1, t) = 0$$ ...
Occhima's user avatar
  • 203
0 votes
0 answers
65 views

How to find eigenvectors for this spectral decomposition?

I want to gain more intuition on the spectral theorem, so I came up with an exercise to test it in a concrete case. I thought considering the second order ODE: $$A\frac{d^2}{dx^2}f(x)+Bf(x)=\lambda f(...
Simón Flavio Ibañez's user avatar
9 votes
2 answers
2k views

Terry Tao didn't list the "dozens and dozens of wave equations out there" in his talk, but is there such a list somewhere?

In the new Simons Foundation video Terence Tao - From Rotating Needles to Stability of Waves: Local Smoothing... (November 29, 2023) after 01:15, Tao says: In mathematics we describe waves by partial ...
uhoh's user avatar
  • 1,893
0 votes
0 answers
16 views

Appropriate definition of the energy functional for one-dimensional wave equation with Robin BC

$$\def\d{\mathop{}\!\mathrm{d}}$$ In the domain $Q_T = (0,l)\times (0,T)$ we consider the following problem $$\begin{align} & u_{tt} - a^2 u_{xx} + b(x,t)u_x + b_0(x,t)u_t + c(x,t)u = f(x,t), \...
Stephen's user avatar
  • 786
0 votes
1 answer
36 views

Possibly ambiguous boundary conditions for wave equation

Consider the wave equation $$u_{tt}=u_{xx}$$ with boundary conditions $$u\left(x,t\right)=X\left(x\right)T\left(t\right)$$ $$u\left(\pm a,t\right)=0$$ $$u\left(x,0\right)=\max\left(0,1-\left(x/b\right)...
Gabsmacked's user avatar
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0 answers
69 views

One dimensional wave equation - getting slightly different result

Consider the one dimensional wave equation $$ u_{tt}=c^2u_{xx} $$ where the initial conditions are $$ \begin{cases} u(x,0) = Ax\\ u_x(0,t)=u_x(L,t)=0. \end{cases} $$ With separation of variables $u(x,...
Gabsmacked's user avatar
1 vote
1 answer
35 views

Sum of waves with equal amplitudes and frequencies but in different phases

I have a problem where I need to sum four cos wave functions with equal amplitudes and frequencies, but different in phases. Presumably it should be solved using complex numbers: $f(t) = \cos(\omega ...
Gustamons's user avatar
-1 votes
1 answer
82 views

A variant of the wave equation

I was working on the following generalization of the wave equation; $u_{tt}-c^2u_{xx}+m^2u=0$ My professor had suggested a technique similar to d'Alembert's principle would work but I see no useful ...
aritracb's user avatar
  • 745
0 votes
0 answers
29 views

Free space vector Helmholtz equation with spherically symmetric source

I have issues or confusions to solve the vector Helmholtz equation in free space for a spherically symmetric source. Let the equation: $$ \left( \widehat{\Delta} + k^2 \right) \boldsymbol{a} = f(r) \...
Fefetltl's user avatar
  • 191
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0 answers
56 views

The real CFL condition for cylindrical laplacian

I've been exploring the CFL (Courant-Friedrichs-Lewy) condition in polar coordinates and have observed that previous inquiries haven't yielded a satisfactory answer. I've come across this paper which ...
Manuel Borra's user avatar

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