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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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What are acoustic modes?

In the context below "In Pidduck [8, 9] it was suggested that the water compressibility be taken into account to resolve the instantaneity paradox in the well-known Cauchy-Poisson problem. Pidduck described the water flow as isothermal and weakly…
Caslu
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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 the correct way to approach a problem like…
John
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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 wavefronts would be different, here's a picture that's in the…
WhoLeb
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Possibly ambiguous boundary conditions for wave equation

Consider the wave equation $$u_{tt}=u_{xx}$$ with boundary conditions $$u\left(x,t\right)=X\left(x\right)T\left(t\right)$$ $$u\left(\pm a,t\right)=0$$ $$u\left(x,0\right)=\max\left(0,1-\left(x/b\right)^2\right)$$ $$u_t\left(x,0\right)=0$$ $$-a\le…
Gabsmacked
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Wave Equation when initial displacement $f(x)=\sin{(mx)}, m=1,2,...$

I am studying 1D Wave Equation $u_{tt}=c^2 u_{xx}$ with B.C. $u(0,t)=0,u(L,t)=0$ and I.C. $u(x,0)=f(x), \frac{\partial u}{\partial t}|_{t=0}=g(x) $. The question is what is $u(x,t)$ when $L=\pi, c=1,g(x)=0 \ and \ f(x)=0.01\sin{3x}$. Since f(x) in…
KenN
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Average values of the waveform

What is the formula to find the average value of this waveform? What is the relation between the peak to peak value and average value of this waveform? Note that it's not an alternating wave. It's positive but changing wave.
Alex
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Finding volume element in a wave function

Im working on wave function. I dont know how to find this volume element from the figure eventhough some explanation for factors under the figure . Any help? [ In that paper, the author used $(s,t,u)$ as coordinates, where $s=r_1+r_2, t=r_2-r_1,…
Chinnapparaj R
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How to solve this wave equation in a triangle?

I want to solve the following problem. We have a triangle with side lengths $a,b,c$. We have a potential inside the triangle satisfying the 2D wave equation with source at $(x_0,y_0)$ which is $\partial^2_x \phi(x,y)+ \partial^2_y \phi(x,y) =…
zooby
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Amplitude of wave question

Suppose the following wave has amplitude 1. $y(x,z,t)=Ab\cos(px)e^{iwt+ikz}$ Find a relationship between $A$ and $b$. Then rewrite $y$ in terms of that relationship. My thoughts. The amplitude is the largest value $y$ can take. So…
user643073
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Creating a sine wave with wavelength that increases by 2^(n+1) each half-period

The roots of the sine wave should start at 1, then: 2, 4, 8, 16, 32, etc. I have tried playing around with formulas such as: y = sin(sqrt(x)) And I believe this is the key however I am not sure how to progress from here. Perhaps I am wrong though.
BFI01
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Sove the wave equation

Previous exercise Express f(t)=\begin{cases} 0,& 0\leqslant t<\pi\\ -\sin t,& \pi\leqslant t\leqslant 2\pi. \end{cases} as a full-range Fourier series. \begin{align} a_n &= \frac 2{2\pi} \int_\pi^{2\pi}-\sin t\cos\left(2\pi t\frac n{2\pi}…
Iqish
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Formula to convert sine waves into other waveforms?

I finally found a way to get a python program to produce sound. It involves using a sine wave. The problem is, I want to make chiptune music, so sine waves just won't do. I need square waves, triangular waves, 'sawtooth' waves, and a fourth that's…
user708316
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What is the point of absolute value squared in wave function's probability density?

The wave function is defined as $$ \Psi(x,t) $$ To get the probability, they squared it with a modulus bracket $$ |\Psi(x,t)|^2 $$ Because amplitude can also be -ve but the probability cannot be. My question is, What is the actual point of both mod…
weegee
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Solving wave equation with two spatial dependancies

I have an elastic wave equation $$\rho \frac{\partial^{2}}{\partial t^{2}} u(x_n) = \frac{\partial}{\partial x_n}u(x_n).$$ If I want to solve when $u = u(x_1,x_3)$, can I solve the wave equation for $u(x_1)$ and $u(x_3)$, and combine the solutions…
rami_salazar
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List of solutions to the wave equation in three dimensions

Wikipedia describes plane waves and $e^{i(\omega t\pm kr)}$ as well as a general integral formula as solutions to the wave equation. As far as I understand, all solutions can be constructed from these by finite and infinite sums. But: are there…
Harald
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