Wave equation Definition and 595 Threads

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. T

    Numerical solution to the second order wave equation

    Homework Statement Consider the second order wave equation u_{tt} = 4u_{xx} There are initial and boundary conditions attached, but I'm less concerned with those for the moment. I think I can figure those out if I can figure out where to get started. Rewrite this as a system of first order...
  2. C

    How Do You Approach Solving a Forced Wave Equation with Sinusoidal Terms?

    Homework Statement If a system satisfies the equation \nu^2 {\partial^2 \psi\over \partial x^2}={\partial^2 \psi\over \partial t^2}+a{\partial \psi\over \partial t}-b\sin\left({\pi x \over L}\right)\cos\left({\pi \nu t\over L}\right) subjected to conditions: \psi(0,t)=\psi(L,t)={\partial...
  3. J

    Matlab, wave equation, and a flick of a string

    Homework Statement You know how if you flick one end of a garden hose you can watch the wave travel down, and then back to you? I wrote this MATLAB code to solve the associated PDE via the Fourier method and the resulting animation looks good for a few time steps, but then the solution...
  4. R

    Intuitively d'Alembert's solution to 1D wave equation

    D'Alembert's solution to the wave equation is u(x,t) = \frac{1}{2}(\phi(x+ct) + \phi(x-ct)) + \frac{1}{2c}\int_{x-ct}^{x+ct} \psi(\xi)d\xi where \phi(x) = u(x,0) and \psi(x) = u_t (x,0). I'm trying to understand this intuitively. The first term I get: a function like f = 0 (x/=0), = a (x=0)...
  5. R

    D'Alembert Problem for 1-D wave equation

    [b]1. For the 1-D wave equation, the d’Alembert solution is u(t, x) = f (x + ct) + g(x − ct) where f , g are each a function of 1 variable. Suppose c = 1 and we know f (x) = x^2 and g(x) = cos 2x for x > 0. Find u(t, x) for al l t, x ≥ 0 if you are also given the BC: u ≡ 1 at x = 0...
  6. M

    Maxwell equations and wave equation in a medium

    Homework Statement Consider an isotropic medium with constant conductivity \sigma. There is no free charge present, that is, \rho = 0. a)What are the appropriate Maxwell equations for this medium? b)Derive the damped wave equation for the electric field in the medium. Assume Ohm's...
  7. C

    Create a wave equation with the following properties.

    Homework Statement Write down an equation to describre a wave \psi(x,t) with all of the following properties a) It is traveling in the negative x direction b) It has a phase velocity of 2000ms-1 c) It has a frequency of 100kHz d)It has an amplitude of 3 units e) \psi(0,0)= 2 units...
  8. Telemachus

    How Is Tension Calculated in the One Dimensional Wave Equation?

    Hi there. I was trying to understand this deduction of the one dimensional wave equation developed at the beggining of the book A first course in partial differential equations of H.F. Weinberger. You can see it right here...
  9. M

    Use Fourier transform to solve PDE damped wave equation

    This question is also posted at http://www.mathhelpforum.com/math-help/f59/use-fourier-transform-solve-pde-damped-wave-equation-188173.html Use Fourier transforms to solve the PDE \displaystyle \frac{\partial^2 \phi}{\partial t^2} + \beta \frac{\partial \phi}{\partial t} = c^2...
  10. Z

    Wave equation ( partial differential equations)

    Consider a string of length 5 which is fixed at its ends at x = 0 and x = 5. The speed of waves along the string is v = 2 and the displacement of points on a string is defined by the function f(x,t). At the initial time the string is pulled into the shape of a triangle, defined by f(x,0) =...
  11. H

    Normalization factor in wave equation

    (Note: although arising in QM, this is essentially a calculus question) Ѱ (x) = A sin (n╥x/a) 1 = ∫ l Ѱ (x) l^2 dx with limits of integration a to 0 1 = ∫ A^2 sin^2 (n╥x/a) dx with limits of integration a to 0 Indefinite integral ∫ sin^2 x dx = x/2 - sin2x/4 I know this integral...
  12. Saitama

    What's in the schrodinger wave equation?

    I have just completed Atomic Structure from my textbook. In that a Schrodinger Wave Equation is mentioned and after that it is written that it is not in the scope of this book to solve this equation. I want to know what is so hard in the schrodinger wave equation that it is not of my level?
  13. M

    How Does Polarization Arise in the Solution of the Wave Equation?

    Homework Statement The wave equation is \nabla^2 \mathbf{A}(\mathbf{r},t) = \frac{1}{c^2} \frac{\partial^2 \mathbf{A}(\mathbf{r},t)}{\partial t^2} I want to get a solution for the vector potential A. Homework Equations we can use the Fourier transformation...
  14. L

    What's the different between wave equation and Schrodinger's eq?

    Homework Statement As I know wave equation has d^2/dt^2,but Schrodinger's equation has only d/dt (Time-dependent). Why these eq has different thing(d/dt, d^2/dt^2)? I assume if Schrodinger's equation has d^2/dt^2(not d/dt), eigenfunction of Schrodinger's equation is not stable along with...
  15. D

    The Wave Equation: Schrodinger, Physical Interpretation, Scope & Limitations

    How did Schrodinger derive the wave equation? Does it have a physical interpretation? And, what are its scope of applicability and limitations?
  16. S

    From wave equation to maxwell equation

    in electromagnetic books, we see by the aid of vector calculus, we can reach to wave equation from Maxwell 's equations. is it possible to reach to Maxwell 's equations from wave equations? in the other word, in electromagnetic books we get Maxwell 's equations as phenomenological...
  17. I

    1+1 Radial wave equation- numerical. BC near origin

    Homework Statement Well, I'm not sure if this is a correct subforum to post my problem, but to me it does seem to me as an academic problem. One I can not solve, apparently. Well, anyway. I'm solving the 1+1 radial wave equation using finite difference. I shifted my grid, so that the origin...
  18. I

    Numerical FDM - wave equation - boundary conditions question

    Hello everyone and greetings from my internship! It's weekend and I'm struggling with my numerical solution of a 1+1 wave equation. Now, since I'm eventually going to simulate a black hole ( :D ) I need a one-side open grid - using advection equation as my boundary condition on the end of my...
  19. A

    Show EM Wave equation invariant under a Lorentz Transformation

    Homework Statement Show that the electromagnetic wave equation \frac{\partial^{2}\phi}{\partial x^{2}} + \frac{\partial^{2}\phi}{\partial y^{2}} + \frac{\partial^{2}\phi}{\partial z^{2}} - \frac{1}{c^2}\frac{\partial^{2} \phi}{\partial t^2} is invariant under a Lorentz transformation...
  20. M

    Solving the Wave Equation with Method of Characteristics

    Hello, My question is about method of characteristics used in solving wave equation. I've found a book on dynamics of structures, and what I cannot understand is a part when it is talked about method of characteristic. Can somebody try to read the shoert article attached below and see if...
  21. T

    Can the Forced Wave Equation Be Solved Numerically?

    Hi, I want to solve the following wave equation: u_{tt} - c^2 u_{xx} = f(x,t)u What is the best way to do it? I don't think I can use Duhamel's principle since I have a u in the forcing. Doing a change of variables of the form w=x+ct, v=x-ct Seems to make things worse. Any ideas...
  22. H

    Coupled pendulums and wave equation.

    Homework Statement (A) [PLAIN]http://remote.physik.tu-berlin.de/farm/uploads/pics/Gekoppeltes_Pendel_01.png What happens when you swing pendulum P1? (B) How does the position of the spring affect the outcome? (C) If the length of the string of one pendulum was longer than the...
  23. B

    Solving the Damped Wave Equation: A Study of u(x,t) and its Derivatives

    For a traveling wave u(x,t) = u(x-ct) How is the relation below hold? u_{x}u_{xt}=-u_tu_{xx} I don't understand why there is (-) sign . Thanks in advance ! PS. Here is the URL of the book I am having trouble with https://www.amazon.com/dp/0198528523/?tag=pfamazon01-20...
  24. A

    How Do You Simplify a Wave Equation in a Non-Uniform Pipe?

    Wave Equation (urgent) Sounds waves in a pipe of varying cross-section are described by the wave equation v2 d/dx .(1/A.dAu/dx) = d2u /dt2 Where A = 0.2 +0.3x simplify the equation My attempt at a solution Sub in A: v2 d/dx ( 1/(0.2+0.3x) . d(0.2+0.3x)u/dx) =d2u/dt2 Not to...
  25. X

    Solution for Vibrating String Problem: Wave Equation Problem Explained

    This is the problem, it says to solve the solution to the vibrating string problem. \frac{\partial^2 u}{\partial t^2}=\frac{\partial^2 u}{\partial x^2} u(0,t)=u(1,t)=0,t>0 u(x,0)=x(1-x),0<x<1 \frac{\partial u}{\partial t}(x,0)=sin(7\pi x),0<x<1 The solution form I obtained (without showing my...
  26. K

    What is the solution for the wave equation using u = cos(kx-wt)?

    For a real stretched string, the wave equation is (partial deriv)^2 (u)/partial deriv t^2 = (T/mu) (partial deriv)^2 (u)/partial deriv x^2 - B/mu(y) where T is the tension in the string, mu is its mass per unit length and B is its "spring constant". Show that the wave given by u =...
  27. G

    Why do the pulses move like that?

    Hello everybody! I have a really silly question concerning wave equation: consider the problem \left\{ \begin{matrix} u_{tt} &=& u_{xx} & x \in \mathbb{R}\\ u(x,0) &=& 0 & \\ u_t(x,0) &=& x(1-x)\chi_{\left[0,1\right]}(x)& \end{matrix} \right. the solution is given by d'Alembert's...
  28. M

    Wave equation derivation Why is the angle assumed to be small?

    http://www.math.ubc.ca/~feldman/apps/wave.pdf is the link from where I understood how to derive the wave equation. But why is theta assumed to be small? As I understand it, theta is the angle that the string segement we're considering makes with the horizontal. Even a simple sine wave seems to...
  29. C

    Fourier sine transform for Wave Equation

    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...
  30. R

    Understanding the Applicability of the Acoustics Wave Equation

    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...
  31. S

    Can the Wave Equation Solution Use Only Positive Values of n?

    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...
  32. fluidistic

    Equation related to the wave equation

    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...
  33. N

    Solving Forced Wave Equation with Causal Boundary Conditions

    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...
  34. D

    Deriving 1D Wave Equation for Vibrating Guitar String

    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...
  35. fluidistic

    Solving Maxwell's Equations: Wave Equation in Vacuum

    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...
  36. F

    How to solve the wave equation with Dirac delta function initial conditions?

    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...
  37. M

    Green's Functions, Wave Equation

    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...
  38. M

    Wave Equation with initial conditions, boundary condtions

    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...
  39. E

    PDE - Two Dimensional Wave Equation

    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)...
  40. J

    Need help proving that a function is a solution to the homogeneous wave equation

    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...
  41. R

    Wave Equation traveling to the left

    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...
  42. R

    Check Homework on Partial Differential Wave Equation

    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...
  43. N

    Can you have fourier transform + boundary condition? (solving wave equation)

    Homework Statement "Solve for t > 0 the one-dimensional wave equation \frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2} with x > 0, with the use of Fourier transformation. The boundary condition in x = 0 is u(0,t) = 0. Assume that the initial values u(x,0) and...
  44. R

    Coulomb's gap and Wave Equation

    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
  45. Z

    Inhomogeneous wave equation solution?

    inhomogeneous Klein-Gordon equation solution? Homework Statement Psi_xx - Psi_tt - 4Psi = exp(exp(3it))*dirac_delta(x) DE valid for all x,t (no boundary conditions specified). Homework Equations Solve for Psi. If the DE is singular, then nontrivial solutions are okay. The Attempt at a...
  46. H

    Wavefunction solution to the Schrödinger Wave Equation for a H atom

    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...
  47. J

    How Does an Instantaneous Transverse Blow Affect a String's Position Over Time?

    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} ) ...
  48. Y

    Question on wave equation of plane wave.

    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...
  49. V

    What is the range of angular frequencies for this waveform?

    I've attached the problem sheet with the given bottom line numerical answers. I'm struggling with question 3 part d and the solution sheet doesn't include the answer to this part of the question. I've completed the previous parts to question 3 already. Can someone please guide me through...
  50. V

    Help with Transverse Wave Equation

    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...
Back
Top