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Questions tagged [wave-equation]

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

The wave equation is a linear second order PDE that describe sound waves, light waves and water waves. It is defined by

\begin{equation*} \frac{\partial ^2 u}{\partial t^2}=c^2\frac{\partial^2 u}{\partial x^2} \end{equation*}

and can be derived from the mathematical model of a string vibrating in a two-dimensional plane where each elements are pulled in opposite directions.

1232 questions
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2D Wave equation with periodic BC-s

I am considering the problem in the picture[2]. [2]: https://i.stack.imgur.com/krlS4.jpg' I have so far got \begin{equation}\label{2.2.2.1} u(x,y,t) = X(x) \cdot Y(y) \cdot T(t) \end{equation} such that $$ T''(t) \cdot X(x) \cdot Y(y) = c^2…
Sivert Laukli
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Maximum derivative of sin(x)

I'm interested in the maximum flow $(Q\max)$ requirements of a hydraulic system. The system moves an actuator in a sinusoidal pattern from position 0 to $\pm A$ where $A$ is the volume of oil in the actuator. As I understand it, to find the flow…
miplums
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Limitations of D'Alembert solution of wave equation

A simple problem for a $1$D-wave equation $$u_{tt} = u_{xx}$$ with conditions \begin{align} u(0,t) &= 0; \\ u(\pi,t) &= 0; \\ u(x,0) &= f(x) = 1 - \cos(2x); \\ u_t &= 0; \end{align} can be solved using D'Alembert identity, the solution of this…
SSB
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Using d'Alemberts in a limited interval

I am a bit confused using d'Alembert's formula for solving the one-dimensional wave equation, and more precisely when it involves the Dirac-function. Some information before my question: Let's say we have the most basic wave equation in a limited…
ladvad
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function to generate the following signal

I don't know If this is the right place to ask but if not just tell me and I'll ask on another StackExchange forum! I'm currently working on a tiny game project and I'm trying to generate a basic day/night cycle. To do so, I would like to create the…
creekorful
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General solution to homogeneous 1D wave equation

I know that the general solution to the homogeneous 1D wave equation is f(x-vt) or f(x+vt). But how is this actually obtained? Was it just an educated guess, and if so how can we be sure it is the general solution?
ngc1300
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Wave equation in cylindrical coordinates

I am trying to find the displacement $u(r,t)$ in a circular membrane of radius $5$, that is clamped down along the circumference. If the initial displacement is $f(r)$ and the membrane is released from rest. $$a^2\left(\frac{d^2u}{dr^2}\right) +…
jjl93
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Calculation of an integral over the region.

Given the operator $\Lambda=(\partial_t^2-c_1^2\partial_r^2)(\partial_t^2-c_2\partial_r^2)$ prove that the spherical mean: $$M_u(x,r,t)=\displaystyle\frac{3}{4\pi r^3}\int_{B(x,r)}u dS(r)$$ Satisfies the equation $\Lambda(M_u)=0$. I have…
EQJ
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How to solve the following integral that gives the solution of the wave equation bi dimensional.

I ask to solve the following PDE: $$u_{tt}-\triangle u=0$$ $$u(x,y,0)=x$$ $$u_t(x,y,0)=0$$ I know that the general solution to this equation is $$u(x,t)=\displaystyle\frac{2}{3\pi t^2}\int_{B(x,t)}\displaystyle…
EQJ
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Graphical method for the wave equation

Can anyone teach me the "graphical method"? I think the parts of this question are based on looking at the graph. By graph I mean the two V-shaped lines that display domain of dependence and region of influence.
user225396
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find the formal solution of wave equation

$$\begin{align} &\frac{\partial^2u}{\partial t^2}+\frac{\partial u}{\partial t}+u = \alpha^2\frac{\partial^2u}{\partial x^2},\qquad && 00\\ &u(0,t) = u(L,t) = 0, && t>0\\[2ex] &u(x,0) = f(x), &&0
Togepi
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D'alembert solution for semi infintite string

Find the solution of the wave equation using d'Alembert solution. $u(0,t)=0$, $u(x,0)=0$ $u_t(x,0)=\frac{x^2}{1+x^3}, \, x\geq0$ $u_t(x,0)=0, \, x<0$ For a semi infinite string we have the solution $u(x,t)=\frac{1}{2}\left(…
James
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Wave Equation BC's

So, I'm trying to solve the wave equation with the Fourier transform, and I'm struggling to figure out how to apply the BC's. Here's the problem I considered: $$\frac{d^2u}{dt^2}=c^2\frac{d^2u}{x^2}$$ $$u(x,0)=g(x)$$ $$\frac{du}{dt}=0$$ at t =…
Incognito
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Sin wave formula simplification and explanation?

I have been trying to implement this Wave equation into java: A = amplitude of wave L = wave length w = spatial angular frequency s = speed wt = temporal angular frequency d = direction FI = initiatory phase $$Y(x，y，t)=A\cdot\cos(w \cdot(x，y)+…
user125740
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Simple Version of Gerstner Wave Formula?

I am attempting to implement these formulas into a wave simulator for java. But i am trouble understand a lot of parts of the equation: http://http.developer.nvidia.com/GPUGems/gpugems_ch01.html For example in equation 1: It returns $Wi(x,y,t)$ as…
user125740
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