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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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Laplace operator. Wave equation.

Why does Laplace operator affect only x-coordinates but not time variable in wave equation. Does t variable in U(X, t) is different from x coordinates? I think they are equal.
Russiancold
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Solution to wave equation case three

I know how to find the solution to the wave equation. When I solve it, there are three cases, when A (constant) is equal to zero, positive and negative. It is this third case, when the constant is negative which we use to continue solving the wave…
user536981
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General solution of the wave equation

everyone!! Could you please help me with a question? Using separation of variables one can derive the general solution of the wave equation as can be seen in the following picture: General solution There is a solution to the following problem, which…
euler132
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Can I obtain an analytical solution for the wave equation with a non-zero neumann BC?

For the 1-D 2nd order wave eqn: $$\frac{d^2u}{dt^2}=c\frac{d^2u}{dx^2}$$ subject to the boundary conditions: $$u(0,t)=0$$ $$\frac{du}{dx}(L,t)=C$$ where $C$ is some finite non-zero constant independent of time, is there an analytical solution to…
David
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Finding highest value of curve

I have the question "The instantaneous values of two alternating voltages are given by v1 = 5sinwt and v2 = 8sin(wt - pi/6). By plotting v1 and v2 on the same axes, using the same scale, over one cycle, obtain expressions for v1 + v2." Here is the…
Dan
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Obtaining sinusoidal expression

I have the question " The instantaneous values of two alternating currents are given by i1 = 20sinwt amperes and i2 = 10sin(wt + pi/3) amperes. By plotting i1 and i2 on the same axes, using the same scale, over one cycle, and adding ordinates at…
Dan
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The derivationof sine to cosine

I am new to mathematics somehow, and I know that: $\frac{d(\sin x)}{dx}= \cos x$ But I can't understand this: $\frac{d (a \sin(\omega.t))}{dt} = a.\omega\cos(\omega.t)$ where $a$ is the peak of the wave and $\omega$ is $2\pi f$ (where $f$ is…
Asmaa
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Two-dimensional plane wave: Why not plotting it with a third axis?

Here I found a summary about two-dimensional plane waves. I have a maybe naive question. The right-hand side of the first figure shows a plot of $\Re(\exp(i\theta))$ as a function of $(x,y)$ for fixed $t$ and the lines are are lines of constant…
mathfemi
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in the equation y = 4*sin(z*x) where z is a constant , why in certain values of z super-position is observed?

when a graph of the equation y= 4* sin(x) is drawn the graph looks like result of a graphical calculator when given this equation it looks like a , normal harmonic wave but when the equation y= 4 sin(90*x) is drawn on a graph the wave looks like a…
Shashwat
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Wave equation even solution

$u_{tt} -c^2u_{xx}=F(x,t) $ , $ x>0$ $u(x,0)=f(x)$ , $ x\ge 0$ $u_t(0,t)=g(x) $ , $ x\ge0$ $u_x(0,t)=0 $ , $ t\ge0$ I did an even expansion to solve it for $-\infty
Sijaan Hallak
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How to break down a complex wave to basic waves

Let's say we have a wave that is made up of 3 sine waves all at different frequencies. It would be easy to figure out what those 3 original sine waves were by either basic observation, or trial and error. My question in regards to this though, is…
Michael King
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Derive the amplitude of superimposed harmonic waves in trigonometric form

We have two waves $\psi_1(x) = a_1\mathrm{cos}(kx-\omega t + \phi_1)$ $\psi_2(x) = a_2\mathrm{cos}(kx-\omega t + \phi_2)$ We know that the superimposed wave can be written as $\psi(x) = \psi_1(x) + \psi_2(x)$ How can I derive from this that the…
hm8
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Deriving basic form of sine wave

I'm trying to derive the basic form of a sine wave: $$y = A \cdot \sin(\omega t + \theta)$$ I'm guessing I could probably first derive the cosine wave as follows and then add a phase of $-\frac{\pi}{2}$. $$y = Re(z) = Re(A \cdot \cos(\omega t) +…
Tarius
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Property of wave equation's solutions derivatives

Let $\square u$ be $u_{tt}-c^2u_{xx}$ where $u$ is a function to times differentiable. Suppose that $\square u=0$, $\square v=0$ for $a0). Prove that $$…
EQJ
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The Wave Equation and Linear Combinations

Say you are trying to solve the wave equation, $$\frac{\partial^2y}{\partial t^2}=c^2\frac{\partial^2y}{\partial x^2}$$ If the boundary is finite, you find that you need an infinite sum of solutions to fit some initial condition. $y = \sum y_n$.…
DWade64
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