# Questions tagged [wave-equation]

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

912 questions
Filter by
Sorted by
Tagged with
30 views

### Wave Equation: what do Characteristic Curves mean?

This is my question: Compute the characteristic curves of the following wave equation $$\frac{\partial^{2} u}{\partial t^{2}}-a^{2} \frac{\partial^{2} u}{\partial x^{2}}=0$$ and draw them ...
5 views

### Can impulses offset in time be added in wave propogation?

I'm following this post to understand convolution. The post uses the example of stones being dropped on water to illustrate the uses of convolution. The post uses the idea of a fundamental solution i....
8 views

### harmonics and wave equation

I would like to ask if there how I could see from solution (by the method of separation) of 1-D wave equation that e.g. a string creates harmonics and not just overtones as e.g. a drumhead. What shall ...
20 views

### Solve Wave Equation Initial-Boundary-Value-Problem

I am trying to solve the following problem and this is my working so far. I'm struggling to get to the general solution for $X(x)$ as I'm not sure of the $\lambda$ value. Please could someone point ...
23 views

13 views

### spherical Bessel - Legendre relation, incident wave

I was reading the partial wave expansion for incident and scattered wave. I cannot understand two things: 1. Why in this process it indicates that the relation indicated in the following picture is ...
16 views

### Simplification of Three Integrals Partial Differential Equations [closed]

I am using Kirchoff's formula to solve wave equation in three dimensions, and I have the following three integrals: $\int_{\partial B(x_0,t)}t*|y-x_0|^2dS(y)$ $\int_{\partial B(x_0,t)}|y-x_0|^6dS(y)$...
38 views

### Mathematical references that deal with Kirchhoff's integral theorem?

I've seen Kirchhoff's integral theorem applied to the wave equation and similarly to the Helmholtz equation in optics books, but I am interested in gaining a mathematician's point of view on this ...
9 views

### non-homogeneous wave equation variable change

in the case of the non-homogenous wave equation with the general form of: Utt-c^2*Uxx=h(x,t) if we define m = x-ct and n = x+ct, then how we conclude the following equation? -(4*c^2)Umn=h(m,n) ...
152 views

### Help solving shallow water equations initial value problem?

The question is as followed: "Consider the initial value problem (IVP) for the linearised shallow water equations (1) $\frac{\partial h}{\partial t} + H_0 \frac{\partial u}{\partial x} = 0$ ; (2) ...
32 views

### I do not understand what this PDE problem is looking for?!

This is an exercise in Qing book. the given hint is not helping me with what the problem is looking for and how to begin. I would appreciate any help. ps. I am studying uniqueness of solutions of ...
10 views

### Wave equation: function 2L-periodic.

I have to prove that any function that is even with respect to x=0 and x=L is necessarily 2L-periodic. We are studying wave equations but I don´t know how to prove it. Can you give me a clue?
50 views

### Transforming PDE $u_{tt}+ (u^2)_{xx} + u_{xxxx} = 0$ into travelling wave ODE

I was wondering on how can we transform a pde to an ode using $$\partial_{tt}u+ \partial_{xx}u^2+\partial_{xxxx} u= 0$$ $$u=f(w)=f(x-vt)$$ how can I transform it into the ode $$f'' +v^2f+f^2=aw+b$$...
14 views

### Wave equation, kirchhof furmula

Given that $u(x,t)\in C_{x,t}^{2,2}(R^3\times(0,\infty))$ solution of $\frac{\partial^2 u}{\partial t^2}$-$\Delta u$=0 ,t>0 $x\in R^n$ $u(x,0)=1$ $u_t(x,0)=0$ for all : $r_0\leq x\leq r_1$ Prove u(0;...
11 views

39 views

### Ferris Wheel Trig Problem

Your ride on a clockwise Ferris wheel begins at the top of the ride, and your height is described by the function $$h(t)=4\cos{\left(\frac{\pi}{18}t\right)}+50,$$ where $h$ is in feet, and $t$ is in ...
61 views

16 views

### Area of Spherical Cap to solve a PDE

there StackExchange, In order to solve a PDE I need to solve for the area of a spherical caps and what I get doesn't reduce to anything nice or that makes sense in the context of the IVP of the PDE ...
29 views

34 views

### Monotone increasing solutions of the wave equation

If I have the following PDE; $${{\partial_t^2 u(t,x)}} - {\partial_x^2 u(t,x)} = 0 \\ u(0,x) = g(x) \\ {\partial_t u(0,x)} = h(x)$$ Is is true if $g$ and $h$ are monotone increasing then $u(x,t)$ ...
20 views

### d'Alembert solution PDE

Consider a wave described by the one-dimensional wave equation ∂ttu = ∂xxu with zero initial velocity and initial configuration given by φ(x) := {$2 − 2|x|$ if $|x| ≤ 1$ and $0$ Otherwise ...
39 views

### Linearized operator around a solution by conserved quantities

Let $$u_{tt}-u_{xx}= u-u^3 ,\: (t,x)\in \mathbb{R}\times \mathbb{R}.$$ I know that this equation admits wave traveling solution of the form $u(x,t)=\varphi(x-ct)$, where $c\in \mathbb{R}$ is a ...
27 views

### Why does a sine wave speed up and slow down?

I was playing with the resource linked below, which allows you to see how the value of sine changes as the angle changes. I understand that the hypotenuse side must move faster than the adjacent side ...
106 views

### How to find the conserved quantities of $\phi^4$ model?

Consider \begin{equation}\label{1} \partial^2_t\phi-\partial^2_x\phi=\phi -\phi^3,\: \ (x,t) \in \Bbb{R}\times \Bbb{R} \tag{1} \end{equation} the $\phi^4$ model, often used in quantum field theory and ...
12 views

### 1D Wave on the Half-Plane with piecewise defined conditions

I'm studying for an exam and I'm completely lost for one particular problem. We are asked to solve by the method of characteristics \begin{align*} u_{tt}-4u_{xx}& = 0 \qquad x>0,\, t>0\\ u(x,...
12 views

Recently, I was attempting to solve the following boundary-value problem for the wave equation with a student that I tutor: $$\frac{\partial^2 u}{\partial x^2}=\frac{1}{4}\frac{\partial^2 u}{\partial ... 1answer 15 views ### Neumann functions of integer order - are they undefined by definition? I have the following definition for Neumann functions (in terms of Bessel functions)$$ Y_n(z) = \frac{J_n(z) \cos(n\pi) - J_{-n}(z)}{\sin(n \pi)}. $$Now my problem is concerned only with n \in \... 0answers 20 views ### Solving PDE using separate variables method Given u_{tt} - u_{xx} = 0 where 0 < x <3 with boundary conditions u(0,t) = u(3,t) = 0 To solve it, separate the variables$$u(x,t) = T(t)X(x)$$and get$$\frac{T''}{T} = \frac{X''}{X} = - ...
I am currently studying a nonlinear wave equation taking the form $$v_{tt} - v_{rr} + v + \frac{1}{r^6}v^7 = 0$$ Some might recongnize this from W. Strauss' book on PDEs. In this example, he goes on ...