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.

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  1. O

    Deriving Linear Wave Equation: Step-by-Step Guide

    Hi can anyone show me how to derive the linear wave equation mathematically or show me a link? I googled but unfortunately I am unable to find out anything about it. Wikipedia showed a derivation via Hooke's law, but I am not really interested since it is not a general derivation. My text...
  2. C

    Solved: Paraxial Wave Equation - Constant Phase Surfaces

    paraxial wave equation - Solved Homework Statement When a laser beam traveling is traveling in one direction, we can make the paraxial approximation. Question: Find an expression for the surfaces with constant phase in the beam. Homework Equations From a previous part of the...
  3. C

    Solving Wave Equation with Paraxial Approximation

    Homework Statement Homework Equations http://books.google.co.uk/books?id=4NXHYg70qqIC&pg=PA85&lpg=PA85&dq=paraxial+approximation+wave+equation&source=web&ots=6PbKKzSEz6&sig=bspXdKfxc-IiMV6AmoifMSJTHuk&hl=en&sa=X&oi=book_result&resnum=10&ct=result The Attempt at a Solution I...
  4. M

    Maxwell's Equations & Wave Equation: An Overview

    Let respectively b = (b1, b2, b3) and e = (e1, e2, e3) denote the magnetic and electric field in some medium. They are governed by Maxwell’s equations which look as follows: (0.1) \partialte = curl b (0.2) \partialtb = − curl e (0.3) div e = 0 (0.4) div b = 0. Show that each bi and each ei...
  5. L

    Half wave equation is this right?

    Hey, I've been asked to find the minimum thickness for a slab of crystalline sapphire. The equation for the half wave plate is : d(ne-no) = (m+1/2)*lambda I found the minimum by using the fact that m=0 It then asks what the other wavelengths are which will allow the plate to act as a...
  6. K

    Understanding the Wave Equation and Separation of Variables for Particle States

    A particle is in a state described by (\frac{mk}{\pi^2 \hbar^{2}})^{1/8}exp(- \frac{1}{2 \hbar} \sqrt{mk}x^{2})exp(-if(t)) When applying separation of variables here, my book ignores the first fraction and sets g(x) = exp(- \frac{1}{2 \hbar} \sqrt{mk}x^{2}) h(t) = exp(-if(t)) But...
  7. S

    Closed-form solutions to the wave equation

    Hi all, I'm interested to find a solution to the wave equation corresponding to Gaussian initial conditions \psi(0,x) = e^{-x^2/2} A solution which satisfies these initial conditions is (up to some constant factor) \psi(t,x) = \int \frac{d^3k}{(2\pi)^3} e^{-k^2/2 + i(k \cdot x -...
  8. redtree

    Integral of plane wave equation

    What is the integral for the following equation (e^{}ikx)/x
  9. W

    Solving inhomogenous wave equation

    I have made an attempt on this one, but I'm not quite sure that I have done it correctly so far..? I am now heading a (for me:)) massive partial integration, and therefore I think it's better to ask before I start. Homework Statement Find the solution u(x,t) of the inhomogenous wave...
  10. Somefantastik

    Wave Equation, stuck on a partial calculation

    Hey everybody, My professor started our PDE I class in Chapter six, so I am having a hard time with the really basic stuff to get the theory down. One of my questions to answer is to verify a solution by using direct substitution. u(x,t) \ = \ \frac{1}{2}\left[\phi(x+t) \ + \ \phi(x-t)...
  11. N

    Is the Wave Equation at Infinity Always Zero?

    Homework Statement Hi all. The wave equation at plus/minus infinity is zero: \left. {\left| {\psi (x,t)} \right| } \right|_{ - \infty }^\infty= 0 Does this also mean that: \left. {\left| {\psi (x,t)} \right|^2} \right|_{ - \infty }^\infty=0 ?
  12. M

    How to Normalize the Basic Wave Equation

    This is a fairly simple question, but the first such question I have done. Inorder to check my work I was hoping somone could show me how to normalize the following. \Psi(x,t) = Ae^{-a[(mx^{2}/\hbar)+it] where m is the particles mass And also that the expectation values of x and x2 would...
  13. Peeter

    Inhomogenous electrodynamics wave equation

    I was playing around with some manipulations of maxwell's equations and seeing if I could work out the wave equation for light. I get: (\nabla^2 -{\partial_{ct}}^2) \mathbf{B} = -\mu_0 \nabla \times \mathbf{J} (\nabla^2 -{\partial_{ct}}^2) \mathbf{E} = \nabla \rho/\epsilon_0 + \mu_0...
  14. P

    Wave Equation in Space with Angle "p

    What is the equation of wave traveling in space making an angle "p" to the horizontal with wave length 'a',frequency 'n',time period 'T'. Please anyone can help me
  15. N

    EM waves - wave equation derivation

    Hi, Something has been bothering me about deriving the wave equation for a plane EM wave. We were showed this derivation in class and had to reproduce it but something is not making sense to me... The derivation is as follows: Suppose you have a plane EM wave (in a vaccuum) traveling in the...
  16. G

    Solving a Wave Equation with 0 < \lambda < 1

    Homework Statement if 0<\lambda<1 and f(x) = x for 0<x<\lambda\pi and f(x) = (\lambda/(1-\lambda))(\pi-x) for \lambda\pi<x<\pi show that f(x)= 2/(\pi(1-\lambda))\Sigma(sin( n\lambda\pi)sin(nx)(/n^{}2 Homework Equations The Attempt at a Solution am i right in saying that...
  17. O

    How Does the Electromagnetic Wave Equation Represent Electric Fields?

    Hi I have a presentation tomorrow and have to explain a few wave equations. I am using a book to walk me through them but there is one point I don't understand: At one point the book states: Because k=(angular frequency)/c, we will represent the waves of the electric field as: e^i...
  18. M

    How is the wave equation derived?

    Hi All, Question: "How is the wave equation derived? This is the question. Here is my answer. I am trying to ensure that it is correct. "To derive wave equation, we apply Newton's law to an elastic string, concluding that small amplitude transverse vibrations of the string obey the...
  19. L

    Wave equation - vector notation

    wave equation -- vector notation what does the solution to the SE look like if expressed in vector notation? say if we just used phi as a function of x and t.
  20. P

    Wave equation in curved spacetime

    Does anyone know how to derive the wave equation in curved spacetime? (-g)^{-1\over 2}\partial_\mu((-g)^{1\over 2}g^{\mu \nu}\partial_\nu \phi) = 0 A reference, or an outline of the derivation would be very helpful. Thanks.
  21. N

    General question about wave equation

    Homework Statement How do you find the phase angle of a wave equation given in the form y(x,t) = Acos(kx - wt) thanks Homework Equations The Attempt at a Solution
  22. C

    Seperation of variables in the 2 dimensional wave equation

    [SOLVED] Seperation of variables in the 2 dimensional wave equation I'd like to apologize right away for the terrible formatting. I was trying to make it pretty and easy to read but I guess I'm just not used the system yet and I had one problem after another. As you'll see at one point the...
  23. J

    Substituting the plane wave solution into the wave equation

    what do you have to do to solve this
  24. N

    Solving a Wave Equation with Electric Fields of Different Frequencies

    Homework Statement I need to solve the following wave equation: [\nabla^2 + \frac{\omega_a^2}{c^2}\epsilon]\mathbf{E_a} = -\frac{4\pi\omega_a^2}{c^2}\mathbf{P}^{(3)} Homework Equations \mathbf{P}^{(3)}=\chi^{(3)}:\mathbf{E_1E_1E_2^*} E_1 and E_2 are two electric fields with...
  25. ~christina~

    How Do You Derive the Linear Wave Equation from Basic Physics Principles?

    Homework Statement As a wave passes through any element of a stretched string under tension T, the element moves perpendicularly to the wave's direction of travel. By applying the laws of physics to the motion of the element, a general differential equation, called the linear wave equation...
  26. D

    Does the N-Dimensional Wave Equation Apply Beyond 1-2 Dimensions?

    I Looked around the web for a while and had not found anything so I figured I'd ask you all about this. It's been awhile since I took a PDE course, but given your standard homogeneous /\u = 0 wave equation, does it scale above and beyond the typical 1-2 dimensional cases? If so, what are some...
  27. K

    How to find the speed and direction of propagation from the wave equation?

    Homework Statement how to find the speed and direction of propagation from the wave equation? Homework Equations y(x,t)=Aexp{B(x-ct)^2} The Attempt at a Solution
  28. D

    Wave Equation Solution: v = w/k | y(x,t) = Ae^(i(kx-wt))

    Homework Statement Show that, as long as v = w/k, the wave equation is solved by y(x,t) = Ae^(i(kx-wt)) v=velocity w=angular velocity k=wave number
  29. N

    Non-Reflective Boundary Conditions for the Wave Equation

    I wasn't completely sure where to put this (programming or Diff.E.'s), so if there's a better place, maybe the mentors could move it for me. I'm doing some numerical simulations involving the (2-D) wave equation, and was wondering if anyone could tell me (or give a reference to a paper which...
  30. S

    What is the velocity of the given wave based on the wave equation?

    [SOLVED] Satisfying wave equation Homework Statement Confirm that the following wave satisfies the wave equation and obtain an expression for the velocity of a wave Y=Asin(2x-5t)*e^(-2t) Homework Equations the wave equation is (d^2y/dt^2)=(V^2)*(d^2y/dx^2) The Attempt at a...
  31. M

    Determining uncertanity from the wave equation

    Hi! How does one find out dx and dp from the wave equation? Appreciate ur help:)
  32. D

    Electromagnetism Homogeneous Wave Equation

    I've managed to derive from Maxwell's equations the homogeneous electromagnetic wave equation with respect to the magnetic field. (The one that goes Del Squared of H minus (The second order partial derivative of B multiplied by the recipricol of C squared all equal to zero) Hopefully that...
  33. N

    Calculate time evolution of Schrodinger wave equation

    [SOLVED] Calculate time evolution of Schrodinger wave equation Homework Statement At time t=0 particle is in state: \psi\left(x\right)=\sqrt{2}A\phi_{1}(x)+\frac{A}{\sqrt{2}}\phi_{2}(x)+A\phi_{3}(x) where \phi_{n}(x) are eigenfunctions of 1-D infinite potential well. a) Normalize...
  34. R

    How the wave equation relates to Newton's Second Law of Motion

    Homework Statement Open Question 3.bmp Homework Equations The Attempt at a Solution Open Answer 3.bmp Any help with this would be greatly appreciated
  35. S

    Wave equation PDE, can't match initial conditions

    First post, hooray! Undergrad nuke engineer here, trying to figure out a really annoying PDE. My notation for U_xx = 2nd partial of U with respect to x, U_tt = 2nd partial of U with respect to t, etc. Homework Statement I'm working a nonhomogenous PDE with homogeneous initial and boundary...
  36. P

    Wave equation for sound waves

    i saw the 'proof' of the wave equation for a sound wave in a medium assuming the wave equation for a dissplacement wave. that is the equtaion s=s_{0} \sin(kx-wt) is supposed to hold for all points for a wave propagating in the x direction. then using this he found out the excess pressure at...
  37. J

    How Do You Calculate the Amplitude of a Standing Wave at a Specific Point?

    URGENT. Wave Equation question I have a standing wave and it's various parameters. I need to work out the amplitude at a point 3 cm to the right of an antinode. I'm stumped as to how to approach it. A pointer in the right direction would be great!
  38. E

    Constructing a Tubular Bell Array: Solving the Wave Equation

    I have the next theoretical-practical problem. I have to build a tubular bell array (like that at symphonic orchrestas) with tubes (not rods) of aluminium or copper. The principal problem I have is I don't know how to state the wave equation for a tube (I have done it for a string). How I do it...
  39. K

    Wave on a rope - question concerns the maths of the wave equation

    Homework Statement [(w^2).b - Tk^2]/Qw = tan(kx - wt + P) This can't be solved for all (x,t) with constant values of w and k Can you explain why this is so please? ive used b to represent the mass per unit length, and T is the tension Homework Equations This is the answer to a...
  40. M

    Solving 1-D Wave Equation w/ Fixed Boundaries

    [SOLVED] 1-D Wave Eqn Alright, so this problem is giving me troubles, and I must just be missing the trick. The equation to solve is the one dimensional wave equation with isotropic, homogeneous, etc. (i.e. wave in a vacuum). Which means the PDE is \frac{\partial^2 u}{\partial t^2} = c^2...
  41. N

    Solutions to Schrodinger's Wave Equation

    Homework Statement Assume that \psi_{1}(x,t) and \psi_{2}(x,t) are solutions of the one-dimensional time-dependent Schrodinger's wave equations. (a) Show that \psi_{1} + \psi_{2} is a solution. (b) Is \psi_{1} \cdot \psi_{2} a solution of the Schrodinger's equation in general...
  42. J

    Mass term in wave equation

    I know how to write down solutions of wave equation \partial^2_t u(t,x) = \partial^2_x u(t,x) for given initial u(0,x) and \partial_t u(0,x) like this u(t,x) = \frac{1}{2}\Big( u(0,x+t) + u(0,x-t) + \int\limits^{x+t}_{x-t} \partial_t u(0,y) dy\Big), but what about \partial^2_t u(t,x) =...
  43. I

    Understanding the Mean Value Theorem for Differentiation

    can someone tell me why this is true \vec{F}=T(\frac{dy(x+\Delta x)}{dx}-\frac{dy(x)}{dx})\cong T(\frac{d^2y(x)}{dx^2})\Delta x and am i correct in understanding the notation in that that dy(x) simply means the same as dy when it is implied dy is a function of x?
  44. N

    Form of the solution of wave equation

    It is just a mere question...Can we write the solution of wave equation as f=g[(+-)ct(+-)x]?
  45. T

    What is the relationship between E and B in the wave equation?

    First of all I have to say that translating specific words from native language to english, is not easy. So I hope that you realize what is going on: What did I do wrong ? (Traveling waves from: http://en.wikipedia.org/wiki/Waves ).
  46. B

    Energy flow in the wave equation (PDE)

    Homework Statement I have a problem that I'm trying to make sense of. Note y_t is the partial derivative of y with respect to t and y_tt is the second order partial derivative of y with respect to t, etc. The complete problem statement is the following: Show that for the equation...
  47. A

    I don't get something about the speed wave equation

    a. As the wavelength of a wave in a uniform medium increases, its speed will _____. a. decrease b. increase c. remain the same The correct answer is c?? I thought that the change in wavelength always incrases the speed of the medium? What does it mean by uniform medium? Can anyone...
  48. L

    How can the invariance of the wave equation be shown without using tensors?

    The problem is, rather briefly: Show that the wave equation is INVARIANT The equation is given as: [the Laplacian of phi] - 1/(c^2)*[dee^2(phi)/dee(t^2)] dee being the partial derivative.. phi is a scalar of (x, y, z, t) Now, i want, and think i should be able, to solve this problem...
  49. S

    Solve 1-D Wave Equation: Superposition of Complex Exponentials

    Homework Statement Express the solution P(t, x1) = cos(!t − kx1) as the superposition of two complex exponentials. Show that each complex exponential is also a solution of the 1-D wave equation. Homework Equations just that THETA=P !=w whoops, made a type The Attempt at a...
  50. W

    Inhomogeneous Electromagnetic Wave Equation

    Homework Statement Consider a medium where \vec{J_f} = 0 and {\rho_f}=0, but there is a polarization \vec{P}(\vec{r},t). This polarization is a given function, and not simply proportional to the electric field. Starting from Maxwell's macroscopic equations, show that the electric field in...
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