What is Wave equation: Definition and 594 Discussions

The wave equation is an important second-order linear partial differential equation for the description of waves—as they occur in classical physics—such as mechanical waves (e.g. water waves, sound waves and seismic waves) or light waves. It arises in fields like acoustics, electromagnetics, and fluid dynamics. Due to the fact that the second order wave equation describes the superposition of an incoming and outgoing wave (i.e. rather a standing wave field) it is also called "Two-way wave equation" (in contrast, the 1st order One-way wave equation describes a single wave with predefined wave propagation direction and is much easier to solve due to the 1st order derivatives).
Historically, the problem of a vibrating string such as that of a musical instrument was studied by Jean le Rond d'Alembert, Leonhard Euler, Daniel Bernoulli, and Joseph-Louis Lagrange. In 1746, d’Alembert discovered the one-dimensional wave equation, and within ten years Euler discovered the three-dimensional wave equation.

View More On Wikipedia.org
Recent contents Top users
  1. F

    Series converges to a function that satisfies the wave equation

    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...
  2. A

    Inhomogeneous Wave Equation: How to Solve using Separation of Variables?

    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...
  3. E

    What is the derivation of the wave equation for traveling waves?

    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)...
  4. N

    Jackson p244,Green function for wave equation

    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...
  5. H

    A question about acoustic wave equation

    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...
  6. C

    Wave Equation with Continuous Piecewise Initial Velocity

    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...
  7. R

    PDEs- D'Alembert Solution of the Wave Equation

    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...
  8. L

    Using Partial Derivatives To Prove Solution To Wave Equation

    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...
  9. kreil

    Green's Function in the wave equation

    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}...
  10. K

    "Showing E&B Obey Wave Equation w/ Maxwell's Curl

    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} =...
  11. B

    What are the Boundary Conditions for Solving a Wave Equation?

    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...
  12. T

    Wave equation for water waves?

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

    About Duhamel Principle in Wave Equation

    \begin{array}{l} \frac{{\partial ^2 u}}{{\partial t^2 }} - a^2 \frac{{\partial ^2 u}}{{\partial x^2 }} = f(x,t) \\ t = 0:u = 0,\frac{{\partial u}}{{\partial t}} = 0 \\ \end{array}...(*) \begin{array}{l} \frac{{\partial ^2 W}}{{\partial t^2 }} - a^2 \frac{{\partial ^2 W}}{{\partial x^2...
  14. N

    Fundamental solution to the radial wave equation

    i was wondering if anyone could point me towards any resources (including books, papers, and/or notes) that discusses and explicitly derives the fundamental solution to the radial wave equation. i have evans' PDE book, but it isn't contained there, and I've been having trouble searching for it...
  15. S

    What Value of b Makes the Complex Function Satisfy the 3D Wave Equation?

    Homework Statement Given a complex function \tilde{f}(\mathbf{r},t)=Aexp(i(b(x+y)-wt+d)) I am asked to determine for what value of b this satisfies the three-dimensional wave equation. x and y are components of r.Homework Equations 3d wave equation: \nabla^2 f=\frac{1}{v^2} \frac{d^2...
  16. M

    Explaining Wave Equation Solution for Vibrating Strings

    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...
  17. M

    How Does u(x,t) = F(x+ct) + G(x-ct) Satisfy the Wave Equation?

    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...
  18. V

    Variable coefficient Wave Equation

    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...
  19. J

    Help with understanding the linear wave equation

    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...
  20. T

    Semi-infinite wave equation

    Homework Statement solve the wave equation (dx^2 -dt^2)\Phi = 0 on the semi-infinite line x<=0 with boundary conditions \Phi at x=0 = 0 and initial conditions \Phi at t=0 = tanh(x) Homework Equations solution of the wave equation is of the form \Phi = f(x-t) + g(x+t). \Phi at t=0...
  21. U

    MATLAB Simulating Schrodinger wave equation using MATLAB.

    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...
  22. L

    Linear wave equation with moving point source

    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} -...
  23. L

    Non-dimensionalization of a wave equation with point source

    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...
  24. N

    Splitting wave equation into x-component and y-component

    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...
  25. N

    Wave equation boundary conditions at infinity

    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...
  26. D

    What is the issue with the wave equation for a flexible cable including gravity?

    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...
  27. M

    Nonhomogeneous wave equation with vanishing initial conditions

    Homework Statement Let u(x,t) be the solution of the following initial value problem for the nonhomogeneous wave equation, u_{x_1x_1}+u_{x_2x_2}+u_{x_3x_3}-u_{tt}=f(x_1,x_2,x_3,t) u(x,0)=0 and u_t(x,0)=0 x\in\Re^3 , t>0 Use Duhamel's Principle and Kirchoff's formula to show that...
  28. E

    1D wave equation open boundary

    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)...
  29. H

    Inhomogeneous electromagnetic wave equation

    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...
  30. B

    Solving the Wave Equation for c and Examining Potential/Kinetic Energy Equality

    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...
  31. L

    How Do You Solve the Wave Equation Using Coefficient Equations?

    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...
  32. M

    Wave equation with initial and boundary conditions.

    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)...
  33. fluidistic

    Wave Equation Spherical Perturbations

    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...
  34. fluidistic

    Solve Wave Equation Problem: f(\alpha x + \beta y + \gamma z \mp vt)

    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...
  35. S

    Damped Wave equation, PDE.

    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)...
  36. Advent

    Wave equation and Galilean Transformation

    Hi! I was reading some notes on relativity (Special relativity) (http://teoria-de-la-relatividad.blogspot.com/2009/03/3-la-fisica-es-parada-de-cabeza.html) and it says that the classical wave equation is not Galilean Invariant. I tried to show it by myself, but I think there is some point that...
  37. Y

    What is the meaning of the wave equation .in English?

    What is the meaning of the wave equation...in English??! Everybody knows one dimensional wave equation \frac{\partial^2u}{\partial t^2} = c^2 \frac{\partial^2u}{\partial x^2} This together with verious boundary and initial condition give various solution of u(x,t). Also it can be transform...
  38. J

    Standing Wave Equation: Exploring Y = 2Asin(\omegat)cos(kx)

    My lecturer said that a standing wave is formed when two waves that travel in the opposite have the same frequency. He said that if the waves are y1 and y2, then the resulting wave y can be given as the sum: y = y1 + y2. y = Asin(\omegat - kx) + Asin(\omegat + kx). (1) Where the...
  39. I

    How to Approach Solving a 2D Damped Wave Equation?

    Hi to all! I need to solve following equation: \frac{\partial^2 u}{\partial t^2} + 2 \beta \frac{\partial u}{\partial t} -c^2\nabla^2u=0 It describes a damped wave on a x-y plane. 2\beta is damping factor and c is wave speed. I haven't had any luck finding a PDE class that looks...
  40. M

    Space and Time Invariance (Classical Wave Equation)

    Hey, I've come across a part in my notes which I can't figure out. Essentially it says: \frac{\partial^{2}y}{\partial t^{2}} = v^{2} . \frac{\partial^{2}y}{\partial x^{2}} is space and time invariant. Whereas: \frac{\partial y}{\partial t} = -v . \frac{\partial y}{\partial x} is not...
  41. B

    Wave equation with Neumann BDC

    The problem statement is: Solve the Neumann problem for the wave equation on the half line 0<x<infinity. Here is what I have U_{tt}=c^{2}U_{xx} Initial conditions U(x,0)=\phi(x) U_{t}(x,0)=\psi(x) Neumann BC U_{x}(0,t)=0 So I extend \phi(x) and \psi(x) evenly and get...
  42. B

    Quantum mechanics for wave equation solution

    1. Homework Statement consider the differential d²ψ(x)/dx²=k²ψ(x); for which values of a is the equation e^(a*x) is a solution to the above equation. 2. Homework Equations 3. The Attempt at a Solution I have been working on this problem but I do not know how relate the 2...
  43. B

    Differntial of wave equation solution

    Homework Statement consider the differential d²ψ(x)/dx²=k²ψ(x); for which values of a is the equation e^(a*x) is a solution to the above equation. Homework Equations The Attempt at a Solution I have been working on this problem but I do not know how relate the 2 equations, or if...
  44. K

    Water Waves - Universal Wave Equation

    Water Waves -- Universal Wave Equation Homework Statement Attached. Homework Equations v=fλ The Attempt at a Solution f= cycles/time f= 45/60 f= 0.75 Hz The trouble I am having is wondering if λ is 28 m or do i have to do something else to find λ?
  45. C

    Unraveling the Wave Equation: Exploring its Origins

    Hi, Why the wave equation is \frac{\partial^2 u}{\partial^2 x}=c^2\nabla^2 u? Where does it come from?Thanks.
  46. R

    When is it appropriate to use a time dependent Schrodinger's wave equation?

    In what conditions do we use time dependent and time independent Schrodinger's wave equations?
  47. B

    Solving Wave Equation with D'Alembert: Step by Step

    Homework Statement Ok so hope someone will be able to help... I've used the D'Alembert method to solve the wave equation and have got that the general form should be y(x,t) = f(x+ct) + g(x-ct) Now I am also told that the time dependence at x=0 is sinusoidal.. that is, y(x,0) =...
  48. T

    Wave Eqn: Show v(x,t)=u(x,t)-ue(x) Satisfies Wave Eqn

    how do i show that v(x,t)=u(x,t)-ue(x) satisfies the wave equation? =( i get that ue(x)=gx2/2c2 + ax + b where a and x are just constants but how does this satisfy the wave equation?
  49. K

    Wave Equation and Energy Conservation for a Stretched String

    Homework Statement The ends (x=0,x=L) of a stretched string are fixed, the string is loaded by a particle with mas M at the point p (0<p<L). 1. What are the conditions that the transverse displacement y must satisfy at x=0, x=p and x=L? 2. Show that the energy of the system is E(t) =...
  50. Y

    Question regarding D'Alembert solution for one dimension wave equation

    In am studying PDE and I have question about D'Alembert solution for one dimension wave equation. I am going to reference Wolfram: http://mathworld.wolfram.com/dAlembertsSolution.html 1) I want to verify the step of \frac{\partial y_0}{\partial t} of step (14) of the page...
Back
Top