Mathematics Stack Exchange
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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.

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Wave equation and fourier transform

This is a famous problem, but I am stuck at the very last step. Given the wave equation $ y_{xx} = \frac{1}{v^2} y_{tt}$, I need to use Fourier transform to solve for $y(x,t)$ given the initial condition $y(x,0) = f(x)$ for some known $f(x)$ with FT…
drg
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How to generate a 3D sea wave like pattern at time t?

I'm using the following video background in one of my apps, and am interested in how I can programmatically generate position of a dot on a screen, as in this animation. The way I can describe this animation is as "waves" or "sea", with x,y and t…
Alex Stone
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How to calculate the PWM (Pulse Width Modulation) of a Pure Sine Wave of a specified Hz rate?

I am trying to calculate the pulse waves for the PWM (Pulse Width Modulation) of a Pure Sine Wave with 60 Hz or 50 Hz. Each pulse wave duration calculated, either to power on or power off, must be a whole number of 3 μs (microseconds) or greater.…
Mark Main
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Is there a formula relating wavelength to amplitude for a sinusoidal curve of fixed line length

I have a piece of string of fixed length 'x'. It is possible to lay it down in a set of sinusoidal waves of varying amplitude. The resultant wavelength will be a function of the amplitude. The maximum amplitude will be x/4, the minimum will be…
G Swain
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What is the formula for a square wave?

After writing this, I realised it's rather long. Please delete if too convoluted. Many many years ago I studied electronics and in a class we used excel to plot a sine wave. Simple. Get a sample rate (24) and make a column of 0 to 360 in 15°…
BBking
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About the wave equation

does the equation $ u_{tt} + \Delta u = f$ makes any sense? The usual wave equation is $ u_{tt} - c^2\Delta u = f$ What would happen if we changed the sign to the Laplace operator? Physically speaking, would it still represent anything?
Matt
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Finding all solutions of the wave equation using $u(x,t) = X(x)T(t)$

Assuming that function $u(x,t)$ in the wave equation $\partial_t^2u = c^2\nabla^2u$, can be factorized to $u(x,t) = X(x)T(t)$, this yields a viable strategy for solving the wave equation, and explicit solutions can be found for $X$ and $T$, and an…
Frank Vel
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Sine wave with postive and negative parts in orthogonal planes

Consider a sine wave in Cartesian coordinate system, with positive part in XY-plane and the negative part in XZ-plane: The two planes here are orthogonal, but they can have an angle $\theta$ , in general. How this wave is related to an ordinary…
Osh
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A problem about wave equation

The problem is from Evans PDE 2.5 problem 18 How to solve the equations? Assume u solves the initial-value problem $u_{tt}-\Delta u=0$ in $R^n \times (0,\infty)$ $u=0$, $u_t=h$ on $R^n\times \{t=0\}$ Show that $v:=u_t$ solves $v_{tt}-\Delta v=0$ in…
user140744
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Propagating wave spherical spreading (or geometric spreading) question

The intensity (power per unit area) of a spherical wave falls off as $1/4\pi r^2$. My question: Is that equation correct? Does this mean the wave amplitude falls off as $1/2\pi^{1/2} r$ ? I understand the $1/r$ and the $1/2$ but have not seen the…
user45664
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Finding wave solution for $\frac{\partial^2 f(x,t)}{\partial t^2} - g(t) \frac{\partial f(x,t)}{\partial t} - \frac{\partial^2 f(x,t)}{\partial x^2}$

I want to solve \begin{align} \frac{\partial^2 f(x,t)}{\partial t^2} - g(t) \frac{\partial f(x,t)}{\partial t} - \frac{\partial^2 f(x,t)}{\partial x^2} =0 \end{align} I am trying to find the general solution to this equation can you give me some…
phy_math
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Proving the Uniqueness of a Wave Equation with Energy Methods

The equations are \begin{equation*} \begin{cases} u_{tt} = (c^2(x)u_x)_x,&\quad 00\\ u(0,t) - u_x(0,t) = 0, &\quad t>0\\ u(L,t) + u_x(L,t) = 0,&\quad t>0\\ u(x,0) = f(x) , u_t(x,0) = g(x),&\quad 0
IReallyNeedHelp
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Get explicit formula for wave equation

Consider the following linear wave equation. $$ u_t+cu_x-\gamma u_{xx}+\delta u_{xxx}=0 $$ If we know the following initial data, $$ u(x,0)= 3\cos^2(x)+\sin(x) $$ how to get an explicit solution? I know that the general solution is: $$ v(x,t)=A\exp(…
jakeoung
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Harmonics on vibrating string

If we gently touch a string at a rational fraction of its length before plucking it, we get a pure harmonic. I've always wondered why that is the case. Wikipedia has a picture illustrating it, but I don't understand why the other higher harmonics…
Felipe Jacob
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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 for all times $t >> 0$. Determine the…
Luthier415Hz
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