Wave equation Definition and 543 Threads

Homework Statement Find the solution u, via the Fourier sine/cosine transform, given: u_{tt}-c^{2}u_{xx}=0 IC: u(x,0) = u_{t}(x,0)=0 BC: u(x,t) bounded as x\rightarrow \infty , u_{x}(0,t) = g(t) 2. The attempt at a solution Taking the Fourier transform of the PDE, IC and BC...

Hello! When considering the acoustics wave equation \frac{\partial^{2}P}{\partial t^{2}} = c^{2} \nabla^{2} P I don't really understand why you can say that the applicability of this equation varies for different sound pressure levels. I don't see why this shouldn't hold for all...

Hello. If I have this equation: And this general solution: Would it then be wrong to write the above solution with only positive values of n? In my textbook they often write the result from a superposition with only positive values of n, becasue the negative values of n already...

Homework Statement Consider the following system of equations: \frac{\partial \vec H}{\partial t} -i \vec \nabla \times \vec H =0 where \vec H is a vector field. 1)Show that \vec Y =\partial _t \vec H satisfies the wave equation. 2)Demonstrate that if \vec \nabla \cdot \vec H=0 initially...

Hi, I want to solve the forced wave equation u_{tt}-c^2u_{xx} = f''(x)g(t) (primes denote derivatives wrt x). The forcing I am interested in is f(x,t)= e^{-t/T} (\alpha_o+\alpha_1 Tanh(-\frac{(x-x_o)}{L}) . I also am imposing causality, i.e. u =0 for t<0 . In the case...

I'm doing a project on a vibrating guitar string and I have completed all the simulation and experimental work, but I do not fully understand the theory behind it. I need to derive the 1 dimensional case of the wave equation, as the 1 dimensional case is considered to be the most convenient...

If I understood well my professor, he showed that "playing" mathematically with Maxwell's equation \frac{\partial \vec E}{\partial t} = c \vec \nabla \times \vec B can lead to the result that \frac{\partial \vec E}{\partial t} satisfies the wave equation (only in vacuum). So what does this...

Homework Statement Solve the IVP for the wave equation: Utt-Uxx=0 for t>0 U=0 for t=0 Ut=[dirac(x+1)-dirac(x-1)] for t=0 2. The attempt at a solution By D' Almbert's solution: 1/2 integral [dirac(x+1)-dirac(x-1)] dx from (x-t) to (x+t) I apologize for not using Latex- my...

In solving the driven oscillator without damping, I need to solve the integral { exp[-iw(t-t')] / (w)^2 - (w_0)^2 } .dw where w_0 is the natural frequency. I know the poles lie in the lower half plane, yet I cannot see why. If (t - t') < 0, the integral is zero. I am not exactly sure...

So, I do not think I did this properly, but if f(-x)=-f(x), then u(-x,0)=-u(x,0), and if g(-x)=-g(x), then ut(-x,0)=-ut(x,0). According to D`Alambert`s formula, u(x,t)=[f(x+t)+f(x-t)]/2 + 0.5∫g(s)ds (from x-t to x+t) so, u(0,t)=[f(t)+f(-t)]/2 + 0.5∫g(s)ds (from -t to t) f is odd, and so is...

Homework Statement Solve the boundary value problem (1)-(3) with a=b=1, c=1/Π f(x)=sin(3 \pi x) sin(\pi y),g(x)=0 (1)\frac{\partial^{2}u}{\partial t^{2}}=c^{2}\left(\frac{\partial^{2}u}{\partial x^{2}}+\frac{\partial^{2}u}{\partial y^{2}}\right) 0 < x < a, 0< y <b, t > 0 (2)...

Homework Statement I have a homework problem that says that any function of the below form is a solution to the homogeneous wave equation. Any function of this form is a solution to the following equation: I would be able to solve it if the function was defined, but I'm not...

Homework Statement Derive the general nontrivial relation between \phi and \psi which will produce a solution to u_{tt}-u_{xx}=0 in the xt-plane satisfying u(x,0)=\Phi(x) and u_t(x,0)=\Psi(x) for -\infty\leq x \leq \infty and such that u consists solely of a wave traveling to the left along...

Homework Statement Consider the partial differential equation u_{xx}-3u_{xt}-4u_{tt}=0 (a) Find the general solution of the partial differential equation in the xt-plane, if possible. (b) Find the solution of the partial differential equation that satisfies u(x,0)=x^3 and...

Hi all I was wondering if I could solve the schrodinger's equation to see the limiting velocity for a proton to tunnel through the coulomb gap in order for the first equation in the fusion reaction to occur Thanks a lot

On my notes, the lecturer left out some of the formulae as blanks which we were supposed to fill in as we went a long but I'm missing a few of them. The 1st one is: [PLAIN]http://img213.imageshack.us/img213/6627/screenshotdh.png I'm stuck here, I can't figure out what equation he's...

Infinite string at rest for t<0, has instantaneous transverse blow at t=0 which gives initial velocity of V \delta ( x - x_{0} ) for a constant V. Derive the position of string for later time. I thought that this would be y_{tt} = c^{2} y_{xx} with y_{t} (x, 0) = V \delta ( x - x_{0} ) ...

For plane wave travel in +ve z direction in a charge free medium, the wave equation is: \frac{\partial^2 \widetilde{E}}{\partial z^2} -\gamma^2 \widetilde E = 0 Where \gamma^2 = - k_c^2 ,\;\; k_c= \omega \sqrt {\mu \epsilon_c} \hbox { and } \epsilon_c = \epsilon_0 \epsilon_r...

Homework Statement Two vibrating sources emit waves in the same elastic medium. The first source has a frequency of 25 Hz, while the 2nd source's frequency is 75 Hz. Waves from the first source have a wavelength of 6.0 m. They reflect from a barrier back into the original medium, with an...

A scientist on a ship observes that a particular sequence of waves can be described by the function y(x,t) =(0.800 m)⋅ sin[(0.628 m−1 )⋅ {x − (1.20 m/s)t}]. (a) At what speed do these waves travel? (b) What is the wavelength? (c) What is the period of these waves? Can anyone tell me what...

Problem: show that the series \sum(1/n^2)*sin(nx)*exp(-ny) converges to a continuous function u(x,y), Then show that U satisfies Uxx + Uyy = 0 Attempt: By the M-test, I know it converges, but I have to find the function it converges to. I tried to simplify the sum by using an identity...

u''tt=a^2*u''xx + t*x 0<x<l; t>0 u(0,t)=u(l,t)=0 u(x,0)=u't(x,0)=0 http://eqworld.ipmnet.ru/en/solutions/lpde/lpde202.pdf ^^Here i found how to solve this problem using Green's function, however i am told to solve this using the method of separation of variables. But i cannot find any theory...

Recently I was going through the derivation of wave equation I want to discuss it to get my concepts fully clear by deriving and comparing the two major type of eqtns i came across. I found two equations 1) When initial positon is x' and t=0 a) y=f(x-vt) for +ve direction b)...

Jackson electrodynamics 3rd. p244 I understood that G=\frac{e^{ikR}}{R} is a spetial solution for ( \nabla ^2 + k^2 )G =0 (R>0) . but,why G=\frac{e^{ikR}}/{R} satisfy ( \nabla ^2 + k^2 )G =-4\pi \delta (\mathbf{R}) ? How to normalize the Green function? ( \nabla ^2 + k^2...

Hi, everyone, I have a question about the acoustic wave equations in two different forms (see the attached). I think the simpler form is more general (in terms of density variation) than the complex one, although the latter looks more general at first sight. But my advisor thinks it's the...

Homework Statement Hello I am asked to find the solution to the following equation no infinite series solutions allowed. We are given that there is a string of length 4 with the following... ytt=yxx With y(0,t) = 0 y(4,t) = 0 y(x,0) = 0 yt(x,0) = x from [0,2] and (4-x) from [2,4]. Homework...

Homework Statement The question is here: http://ocw.mit.edu/courses/mathematics/18-303-linear-partial-differential-equations-fall-2006/assignments/probwave1solns.pdf It's a long question and I figured attaching the link here would be better. I need help with the question on page 4. when...

I need to use partial derivatives to prove that u(x,t)=f(x+at)+g(x-at) is a solution to: u_{tt}=a^{2}u_{xx} I'm stuck on how I'm supposed to approach the problem. I'm lost as to what order I should do the derivations in. I tried making a tree diagram, and I came out like this. The arrow...

Homework Statement The Green function for the three dimensional wave equation is defined by, \left ( \nabla^2 - \frac{1}{c^2}\frac{\partial^2}{\partial t^2} \right ) G(\vec r, t) = \delta(\vec r) \delta(t) The solution is, G(\vec r, t) = -\frac{1}{4 \pi r} \delta\left ( t - \frac{r}{c}...

Homework Statement This question is closely related to physics but it's in a maths assignment paper i have so here it is: By taking curls of the following equations: \nabla \times \bf{E} = -\frac{1}{c}\frac{\partial\bf{B}}{\partial t} \nabla \times \bf{B} =...

Homework Statement The problem is to solve \phi_{yy}-c^2 \phi_{xx} = 0 \phi_y (x,0) = f'(x), x>0 \phi_x (0,y) = \phi(0,y) = 0, y>0 or y<0 Homework Equations The solution, before applying boundary conditions is obviously \phi(x,y)=F(x+c y)+G(x-cy) The Attempt at a Solution I start...

Hi, Has there to your knowledge been developed any wave equation for for water waves?

In deriving the governing equation for a vibrating string, there are several assumptions that are made. One of the assumptions that I had a hard time understanding was the following. Once the string is split into n particles, the force of tension on each particle from the particles in the...

Homework Statement struggling with a problem and hoping someone could help me out. the problem reads, Let F and G be arbitrary differentiable functions of one variable. Show that u(x,t) = f(x+ct) + G(x-ct) is a solution to the wave equation, provided that F and G are sufficiently smooth...

Hello exalted ones. I am working on a set of differential equations for my research and there is one that is becoming mortal. I am solving a mechanical system whose behavior eq. is that of a one dimensional wave PDE. Namely: u_{tt}=a^{2}u_{xx} For which I would derive two parametrized...

Homework Statement Suppose an element of a string, called \[\triangle x\] with T being the tension. The net force acting on the element in the vertical direction is \[\sum F_{y} = Tsin(\theta _{B}) - Tsin(\theta _{A}) = T(sin\theta _{B} - sin\theta _{A})\] I know what small-approximation...

Hi! I've just finished learning the basics of MATLAB from an internet tutorial. I know a the basics of how to represent and manipulate vectors,matrices,graphs and plots on MATLAB. Now,my H.O.D wants me to make a programme that will simulate the Schrodinger wave equation on MATLAB...and I...

I'll need some help and clarification about solving this equation. After some non-dimensionalization, I can arrive at the following wave equation with a moving point source. The initial conditions are zero. \Delta P - \frac{\partial^2 P}{\partial \tau^2} = - A \cos(\tau) \delta^3(\vec{r} -...

I've been trying to non-dimensionalize a wave equation with a moving point source, but the peculiar properties of the delta function have confused me. How does one non-dimensionalize an equation with a delta function? For example, the equation I'm looking at is something like the one below...

Hello, I've been working for a while with the following wave equation PDE: \[ \frac{{\partial ^2 p}}{{\partial x^2 }} + \frac{{\partial ^2 p}}{{\partial y^2 }} = A\frac{{\partial ^2 p}}{{\partial t^2 }} + B\frac{{\partial p}}{{\partial t}} \] In preparation for the application of a...

Are there general boundary conditions for the wave equation PDE at infinity? If there is, could someone suggest a book/monograph that deals with these boundary conditions? More specifically, if we have the following wave equation: \[ \nabla ^2 p = A\frac{{\partial ^2 p}}{{\partial t^2...

Hi. I think the wave equation for a flexible cable including gravity should look like this \frac{\partial^2}{\partial x^2}f(x,t)-\frac{1}{c^2}\frac{\partial^2}{\partial t^2}f(x,t)=g It this true? (g is the gravitational constant) Now if I put the boundary conditions f(x=0,t)=0 , f(x=1,t)=0...

I am trying to write a solver for a 1D wave equation in MATLAB, and I have run into interesting problem that I just can't find a way out of. I start with the wave equation, and then discretize it, to arrive at the following, U{n+1}(j)=a*(U{n}(j+1)-2*U{n}(j)+U{n}(j-1))+2*U{n}(j)-U{n-1}(j)...

Hi, I am looking at electron beam going through a plasma. I am modelling it using two regions, the electron beam and external to the electron beam. I am using the potential formulation of electrodynamics and I am modelling a rigid electron beam and assuming cylindrical symmetry for...

Homework Statement I'm given that the motion of an infinite string is described by the wave equation: (let D be partial d) D^2 y /Dx^2 - p/T D^2/Dt^2 = 0 I'm asked for what value of c is Ae^[-(x-ct)^2] a solution (where A is constant) Then I am asked to show that the potential...

Homework Statement [PLAIN]http://img33.imageshack.us/img33/8236/waveeq.jpg The Attempt at a Solution We calculate second differential with respect to x, and t, substitute into the wave equation. We then equate the coefficients: [A''(x) + (w/v)^2A(x)]sin(wt)=0 We know from...

Hallo Every one, Homework Statement y(x,t)=sin(x)cos(ct)+(1/c)cos(x)sin(ct) Boundary Condition: y(0,t)=y(2pi,t)=(1/c)sin(ct) fot t>0 Initial Condition : y(x,0)=sin(x),( partial y / Partial t ) (x,0) = cos(x) for 0<x<2pi show that y(x,t)=sin(x)cos(ct)+(1/c)cos(x)sin(ct)...

Homework Statement Show that u(r,t)=\frac{f(r-vt)}{r} is a solution to the tridimensional wave equation. Show that it corresponds to a spherical perturbation centered at the origin and going away from it with velocity v. Assume that f is twice differentiable.Homework Equations The wave...

Homework Statement Show that the function u(x,y,z,t)=f(\alpha x + \beta y + \gamma z \mp vt) where \alpha ^2 + \beta ^2 + \gamma ^2 =1 satisfies the tridimensional wave equation if one assume that f is differentiable twice.Homework Equations \frac{\partial ^2 u}{\partial t ^2}-c^2 \triangle...

Homework Statement I have the damped wave equation; u_{tt} = 4 u_{xx} -2 u_{t} which is to be solved on region 0 < x < 2 with boundary conditions; u(0,t) = 2, u(2,t) = 1. i must; 1) find steady state solution u_{steady}(x) and apply boundary conditions. 2) find \theta(x,t)...