Wave equation Definition and 595 Threads

Homework Statement Please find question attached. The context is a lecture course on seismic theory for exploration geophysics. Homework Equations The substitution for the elasticity tensor made in my solution is given in the lecture notes. Please find my attempt at the solution attached...

In the equation regarding an array of masses connected by springs in wikipedia the step from $$\frac {u(x+2h,t)-2u(x+h,t)+u(x,t)} { h^2}$$ To $$\frac {\partial ^2 u(x,t)}{\partial x^2}$$ By making ##h \to 0## is making me wonder how is it rigorously demonstrated. I mean: $$\frac {\partial ^2...

Homework Statement http://web.phys.ntnu.no/~ingves/Teaching/TFY4240/Exam/Exam_Dec_2008_tfy4240.pdf problem 2a) Homework EquationsThe Attempt at a Solution Hi. In problem 2a I was supposed to find a wave equation, however while digging around in maxwell's equations, I found this result...

Hi, I am trying to plot a function subjected to a nonlinear wave equation. One of the method I found for solving the nonlinear schrodinger equation is the split step Fourier method. However I noticed that this method only works for a specific form of PDE where the equation has an analytic...

Hi PF! SO we have defined energy per unit mass as $$E(t) = \int_0^L \frac{1}{2} u_t^2 + \frac{c^2}{2} u_x^2 dx$$. We are given a vibrating string that exhibits ##u_x(0,t) = 0## and ##u(L,t)=0##. I am trying to figure out what is happening with total energy, ##E(t)##. My work is $$\int_0^L...

Suppose that we have the four-vector potential of the electromagnetic field, [texA^i[/tex] The wave equation is given by $$(\frac {1}{c^2} \frac {\partial^2}{\partial t^2}-\nabla^2) A^i=0$$ Now the solution, for a purely spatial potential vector, is given by $$\mathbf{A}(t...

Hello, I hope somebody can help me with this. 1. Homework Statement I am supposed to show that if there is a function \phi(x,t) which is real, satisfies a linear wave equation and which satisfies \phi(x,0)=0 for x<0 then the Fourier Transform \tilde{\phi}(k) of \phi(x,0) is in the lower...

Homework Statement Reading the very first chapter of Weinberger's First Course in PDEs, I stumbled over the derivation of the tensile force in the horizontal direction. The question was posted already in this thread: https://www.physicsforums.com/threads/one-dimensional-wave-equation.531397/...

Hello question is: As you see when we do del operator on A vector filed in below example it removes exponential form at the end.why does it remove exponential form finally?

I am studying the sound wave equation deducted by Feynman in his lectures. In section 47-3: P0 + Pe = f(d0 + de) = f(d0) + de f'(d0) Where f'(d0) stands for the derivative of f(d) evaluated at d=d0. Also, de is very small. I do not understand the second step of the equality. Can anyone help...

Homework Statement Show that the solution \textbf{E}=E(y,z)\textbf{n}\cos(\omega t-k_xx) substituted into the wave equation yields \frac{\partial^2 E(y,z)}{\partial y^2}+\frac{\partial^2 E(y,z)}{\partial z^2}=-k^2E(y,z) where k^2=\frac{\omega^2}{c^2}-k_x^2 Homework Equations See above. The...

We have a string of length \(\ell\) with fixed end points. At \(x(a)\), we have a mass. We can break the string up into two sections \(a + b = \ell\); that is, a is the distance up to the mass and b afterwards. The string is under tension \(T\). My question is why is the DE then \[...

I've been stuck trying to figure out what's going on in a particular section of my notes for the last couple days. The biggest issue is the lecturer has just not explained where the example has come from and what it represents. I thought I would post the relevant section here and see if anyone...

Is it possible to solve these partial differential equations directly, relating to Antenna Theory; ∇^2 E - μ_0 ε_0 \frac{∂^2E}{∂t^2} = -μ_0 \frac{∂J}{∂t}. AND ∇^2 B - μ_0 ε_0 \frac{∂^2B}{∂t^2} = -μ_0 ∇ x J. I don't like the idea of having to make up fields that don't exist in order to make...

The wave function solution psi is a function of time and position. Hence the integral of its square over all x will, in general, give a function of time. To normalize this, we must multiply with the inverse of the function. Therefore it seems that the normalization constant does not remain...

Homework Statement I'm working on using the wave equation to prove that EM waves are light. Homework Equations Here's what I'm working with: E = Em sin(kx-wt) B = Bm sin(kx-wt) ∂E/∂x = -∂B/∂t -∂B/∂x = μ0ε0 ∂E/∂t and the wave equation: ∂2y/∂x2 = 1/v^2(∂2y/∂t2) The Attempt...

Homework Statement set \phi = f(x-t)+g(x+t) a) prove that \phisatisfies the wave equation : \frac{\partial^2 \phi}{\partial t^2} = \frac{\partial^2 \phi}{\partial x^2} b) sketch the graph of \phi against t and x if f(x)=x^2 and g(x)=0The Attempt at a Solution part a, I have already gotten the...

Homework Statement Assume that the wavelength of acoustic waves in an organ pipe is long relative to the width of the pipe so that the acoustic waves are one-dimensional (they travel only lengthwise in the pipe). Therefore, the equation governing the pressure in the wave is: ∂2p/∂t2-c2*∂2p/∂x2...

Homework Statement A mobile phone signal with a frequency of 1945Mhz is being broadcast from a transmitter with a peak output of 3kW. A: What part of the EM spectrum is the signal. Classify it in terms of its orientation of oscillation and propagation. B: Write a general equation for the...

Homework Statement utt=a2uxx Initial conditions: 1)When t=0,u=H,1<x<2 and u=0,x\notin(1<x<2) 2)When t=0,ut=H,3<x<3 and u=0,x\notin(3<x<4) The Attempt at a Solution So I transformed the first initial condition \hat{u}=1/\sqrt{2*\pi} \int Exp[-i*\lambda*x)*H dx=...

Hi, This is a worked example in the text I'm independently studying. I hope this isn't too much to ask, but I am stupidly having trouble understanding how one step leads to the other, so was hoping someone could give me a little more of an in-depth idea of the derivation. Thanks. Homework...

Hi, considering the scalar wave equation $$ { \partial^2 u \over \partial t^2 } = c^2 \nabla^2 u $$ (where ∇^2 is the (spatial) Laplacian and where c is a fixed constant) how can I derive the potential and kinetic energy for a given state u and u' ? Thanks and cheers

I am very familiar with the equation: $$f(t)=Asin(ωt+ϕ)$$ Used to describe the instantaneous value f(t) of a wave with amplitude A, frequency ω, and phase shift ϕ at time t. This equation is very intuitive to understand: As t increases the value within the sin operator will increase from ϕ...

Hello, I am looking at the wave equation for the casimir effect and I was hoping if some could tell me what type of what equation is it and what techinque is used to derive it. The wave equation can be found here: http://en.wikipedia.org/wiki/Casimir_effect Thank you

1.This equation in the link below refers to the small angle approximation regarding deriving the wave equation from Newtons laws from small amplitude waves in a single string with fixed tension. 2.http://imgur.com/NGSwzcl 3. I'm a bit rusty on the maths and have no idea how these...

Solve U_xx=U_tt with c=1. Dirchlet boundary conditions U(x,0)=1 for 5<x<7 U(x,0)=0 for everywhere else U_t(x,0)=0 I know that by taken an odd extension I can get rid of the boundary condition and then solve the initial value problem using the d'alembert solution and only care for x>0...

Hello guys, I would like to ask some questions regarding my coursework, which is about 2nd ODE and multivariable calculus. Since we have the one-dimensional wave equation and values for the string stretched between x=0 and L=2: 0≤x≤L, t≥0 The string is fixed at both ends so we have ...

Homework Statement Prove by direct substitution that any twice differentiable function of (t-R\sqrt{με}) or of (t+R\sqrt{με}) is a solution of the homogeneous wave equation. Homework Equations Homogeneous wave equation = ∂2U/ ∂R2 - με ∂2U/∂t2 = 0 The Attempt at a Solution Could you...

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*cos（w *（x，y）+ wt*t + FI; I...

Homework Statement Our teacher asked us to find a useful use of the wave equation, in which we must explain how the wave equation is used in the application we chose, in which will be a 2 minutes at max speech. Homework Equations Wave equation v=f*λ The Attempt at a Solution I...

Homework Statement Prove that y(x,t)=De^{-(Bx-Ct)^{2}} obeys the wave equation Homework Equations The wave equation: \frac{d^{2}y(x,t)}{dx^{2}}=\frac{1}{v^{2}}\frac{d^{2}y(x,t)}{dt^{2}} The Attempt at a Solution 1: y(x,t)=De^{-u^{2}}; \frac{du}{dx}=B; \frac{du}{dt}=-C 2...

Hi, Apologises if I have submitted this issue into the wrong Math forum. However, I was wondering if anybody could help me with 2 steps in a derivation of an equation. Simply by way of background, the derivation is linked to formation of a superposition wave subject to a Doppler effect [1]...

Hello, How does the change of variables ## \alpha = x + at , \quad \beta = x - at ## change the differential equation $$ a^2 \frac{ \partial ^2 y}{ \partial x^2 } = \frac{ \partial ^2 y} {\partial t ^2} $$ to $$ \frac{ \partial ^2 y}{\partial \alpha \partial \beta } = 0$$ ? I'm having a...

Homework Statement A string of length l has a zero initial velocity and a displacement y_{0}(x) as shown. (This initial displacement might be caused by stopping the string at the center and plucking half of it). Find the displacement as a function of x and t. See the following link for...

Homework Statement Show that any function of the form ##z = f(x + at) + g(x - at)## is a solution to the wave equation ##\frac {\partial^2 z} {\partial t^2} = a^2 \frac {\partial^2 z} {\partial x^2}## [Hint: Let u = x + at, v = x - at] 2. The attempt at a solution My problem with this is...

Hi all, I have the question: Consider a flag blowing in the wind. Assume the transverse wave propagating along the flag is one dimensional. Solve the wave equation for the wave on the flag, assuming the displacement of the flag is zero at the flag pole and the other end of the flag is...

Homework Statement One end of a long horizontal string is attached to a wall, and the other end is passed over a pulley and attached to a mass M. The total mass of the string is M/100. A Gaussian wave pulse takes 0.12 s to travel from one end of the string to the other. Write down the...

The wave on the string could be described with wave equation. Wave equation has a factor v^2 = Tension/linear density. It has dimensions of speed, but from where exactly does it follow that this is actually speed of propagation of the wave?

Homework Statement The attempt at a solution I'm using the method of separation of variables by first defining the solution as u(x,t) =X(x)T(t) Putting this back into the PDE I get: T''X = x^{2}X''T + xX'T which is simplified to \frac{T''}{T} = \frac{x^{2}X'' + xX'}{X} = -\lambda^{2} The...

Solve ##u_{xx} - 3u_{xt} - 4u_{tt} = 0##, ##u(x,0) = x^{2}##, ##u_{t}(x,0) = e^{x}##. (Hint: Factor the operator as we did for the wave equation.) (From Partial Differential Equations An Introduction, 2nd edition by Walter A. Strauss; pg. 38) This is the first of a set of three exercises on...

Consider the classical wave equation in one dimension: \frac{\partial^2 \psi}{\partial x^2}=\frac{1}{v^2} \frac{\partial^2 \psi}{\partial t^2} It is a linear equation and so the set of its solutions forms a vector space and because this space is a function space,its dimensionality is...

Homework Statement A spring of mass m, stiffness s and length L is stretched to a length L + l. When longitudinal waves propagate along the spring the equation of motion of a length dx may be written pdx second partial derivative of n with respect to t = partial derivative of F with respect to...

If the solution to the electric part of the spherical wave equations is: E(r, t) = ( A/r)exp{i(k.r-ωt) What happens when t=0 and the waves originates at the origin, i.e. r=0 ... which I assume can't be right as you of course cannot divide by zero. Thanks!

If the solution to the electric part of the spherical wave equations is: E(r, t) = ( A/r)exp{i(k.r-ωt) What happens when t=0 and the waves originates at the origin, i.e. r=0 ... which I assume can't be right as you of course cannot divide by zero. Thanks!

If motion of an object obeys the wave equation, then it will display wave like behaviour. If you solve the wave equation, you get things like y = Asin \frac{2∏}{\lambda}(x - vt) which is a sinosodial wave. But from the second order differential equation v^{2}\frac{d^{2}y}{dx^{2}} =...

Homework Statement I must show that the one dimensional wave equation ##\frac{1}{c^2} \frac{\partial u}{\partial t^2}-\frac{\partial ^2 u}{\partial x^2}=0## is invariant under the Lorentz transformation ##t'=\gamma \left ( t-\frac{xv}{c^2} \right )## , ##x'=\gamma (x-vt)##Homework Equations...

Homework Statement The cross-section of a long string (string along the x axis) is not constant, but it changes wit the coordinate x sinusoidally. Explore how a wave, caused with a short stroke, spreads through the string. Homework Equations Relevant is the one-dimensional wave...

Homework Statement Solve using separation of variables utt = uxx+aux u(0,t)=u(1,t)=0 u(x,0)=f(x) ut=g(x) The Attempt at a Solution if not for the ux I'd set U=XT such that X''T=TX'' and using initial conditions get a solution. In my case I get T''X=T(aX'+X'') which is...

I am studying Coulomb and Lorentz gauge. Lorentz gauge help produce wave equation: \nabla^2 V-\mu_0\epsilon_0\frac{\partial^2V}{\partial t^2}=-\frac{\rho}{\epsilon_0},\;and\;\nabla^2 \vec A-\mu_0\epsilon_0\frac{\partial^2\vec A}{\partial t^2}=-\mu_0\vec J Where the 4 dimensional d'Alembertian...

Homework problem: For the wave equation: Utt-Uxx=0, t>0, xER u(x,0)= 1, |x|<1 0, |x|>1 sketch the solution u as a function of x at t= 1/2, 1, 2, and 3 I am able to use d'Alemberts and solve for u however the boundaries and the odd/even reflections are throwing me off and...