Articles of wave equation

Solution of 1d wave equation by Laplace transform

This is a homework problem that I can almost finish. I just can’t invert the Laplace transform at the end. $$u_{xx}=u_{tt}, u(t=0)=u_t(t=0)=0, u(x=0)=\sin\omega t, u(x=2)=0.$$ Taking the Laplace transform with respect to $t$ and using the zero initial conditions, I obtain $U” = s^2U$, where $s$ is the transform variable and $U$ is the transform […]

Compact support

From PDE Evans, 2nd edition, page 204 Example 9 (Wave equation from the heat equation). Next we employ some Laplace transform ideas to provide a new derivation of the solution for the wave equation, based–surprisingly–upon the initial-value problem. $\quad$ Suppose $u$ is a bounded, smooth solution of the initial-value problem: \begin{cases} u_{tt}-\Delta u = 0 […]

An exercise from Stein's Fourier analysis about wave equation

Show that the solution of the equation $\frac { \partial^2u}{\partial t^2}= \frac { \partial^2u}{\partial x_1^2}+ \frac { \partial^2u}{\partial x_2^2}+ \frac { \partial^2u}{\partial x_3^2}$ Subject to $u (x,0)=f (x)$ and $\frac {\partial u}{\partial t}(x,0)=g (x )$ , is given by $u (x,t)=\frac {1}{|S (x,t)|}\int _{S (x,t)}[tg (y)+f (y)+\nabla f (y ). (y-x )]d\sigma ( y)$. Where […]

Does the wave equation require an initial function for one of its derivative?

Is it possible to find an explicit solution to the wave equation: $$ \partial_t^2u-c^2 \partial_x^2 u=0 \\ u(x,0)=f(x), \ u(cx,x)=g(x) $$ or do we need information about a derivative of $u$ as well?

General solution to wave equation of half-line with nonhomogeneous Neumann boundary

Problem Use the general solution to solve the signalling problem with homogeneous wave equation on the half line, homogeneous IC and nonhomogeneous Neumann boundary conditions. Where c>0 is a constant, and h is continuous function. The solution u should be continuous. $u_{tt} – c^2 u_{xx} = 0, \quad\quad \quad \quad 0 < x < \infty,\quad […]

Why are the limits of the integral $0$ and $s$, and not $-\infty$ and $+\infty$??

The D’Alembert solution of the wave equation $$u_{tt}-c^2u_{xx}=0, x \in \mathbb{R}, t>0$$ $$u(x,0)=\phi(x), x \in \mathbb{R}$$ $$u_t(x,0)=\psi(x), x \in \mathbb{R}$$ is the following: $$u(x,t)=\frac{1}{2}\begin{bmatrix} \phi(x-ct)+\phi(x+ct) \end{bmatrix}+\frac{1}{2c} \int_{x-ct}^{x+ct} \psi(\tau)d \tau$$ I am looking at the proof of this solution, which is the following: The general solution of the wave equation is given from: $\displaystyle{u(x,t)=f(x+ct)+g(x-ct)} \ \ […]

water wave and fluids dispersion relation

Small-amplitude water waves travel on the free surface $y = \eta(x, t)$ of an incompressible inviscid fluid of uniform depth $h$. Derive the linearised boundary conditions $$\text{ at }y=0\quad \frac{\partial\varphi}{\partial t}=\frac{\partial\eta}{\partial t},\quad \frac{\partial\varphi}{\partial t}+g\eta=0 $$ and write down the boundary condition satisfied by the velocity potential $\varphi$ at the rigid boundary $y = −h$. Show […]

Why don't elliptic PDE's have a time coordinate?

Usually second-order linear PDE’s are classified as elliptic, parabolic, or hyperbolic (or ultrahyperbolic) depending on the eigenvalues of the coefficient matrix. The three cases correspond to the three most famous second-order PDE’s: Elliptic – Laplace’s equation $\nabla^2 u = 0$. Parabolic – the heat equation $u_t = \nabla^2 u$. Hyperbolic – the wave equation $u_{tt} […]

Solving Wave Equations with different Boundary Conditions

Right now I’m studying the wave equation and how to solve it with different boundary conditions (i.e. $u(x,0);u(0,t);u_t(x,0);u_x(x,0);u(x,x);u_t(x,x)…$) I know how to solve it when the boundary conditions are $u(x,0)=f(x)$ and $u_t(x,0)=g(x)$ with d’Alambert’s formula. But I don’t know how to solve it in any other case. Does anyone know of any book, notes, etc. […]

Range of Influence of the Wave Equation?

Suppose $u$ is a solution of the two-dimensional wave equation $$u_{tt}-c^2 \Delta u=f(x)$$ with initial values $u(0),u_t(0)$ that have support on the disc $x_1^2+x_2^2 \le 1$. Up to what time can you be sure that $u=0$ at the point $(x_1,x_2)=(2,3)$? My attempt: I think it depends on the point farthest from $(2,3)$ in the disc, […]