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. Robin04

    I Wave equation and the d'Alembert solution

    I have a few questions about the wave equation and the D'Alambert solution: 0) First of all, I'm a bit confused with the terminology. Wikipedia says that THE wave equation is a PDE of the form: ##\frac{\partial^2 u}{ \partial t^2 } = c^2 \nabla^2 u##, however there are other PDEs that have...
  2. Prez Cannady

    I Inhomogeneous Wave Eq. & Minkowski Spacetime Interval

    Answer is probably not, but is there some connection between the inhomogeneous wave equation with a constant term and the spacetime interval in Minkowski space? $$ 1) ~~ \nabla^2 u - \frac{1}{c^2} \frac{\partial^2 u}{\partial t^2} = \sum_{i=0}^2 \frac{\partial^2 u}{\partial x_i^2} -...
  3. G

    A Seeking a derivation of Schrödinger's wave equation

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  4. G

    Are superpositions of waves with different c still waves?

    Hi. As far as I know, superpositions of waves are normally considered to be waves too, even in dispersive media. But how can they still be solutions of a wave equation of the form $$\left(\frac{1}{c^2}\frac{\partial^2}{\partial t^2}-\Delta\right)u=0$$ if ##c## isn't the same for all of them...
  5. John Greger

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  6. It's me

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  7. B

    Resonant Lengths in open air column question

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  8. Cathr

    Electromagnetic wave equation - phase and amplitude

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  9. evinda

    MHB Is this the desired bounded set of the wave equation?

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  10. Jaden159

    A Is energy contained in matter wave equals hv like EM waves?

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  11. F

    I Why does the wave equation describe certain phenomena

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  12. evinda

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  13. T

    What is the characteristic frequency in a PDE modified wave equation?

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  14. evinda

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  15. Mutatis

    How can I find the wave equation u(x,t) of a string

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  16. D

    I Exploring Wave Equation Solutions for E-fields

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  17. harambe

    B The Wave Equation and Traveling Waves

    The wave equation in one space dimension can be written as follows: .A traveling wave which is confined to one plane in space and varies sinusoidally in both space and time can be expressed as combinations of What is the difference between these two wave equations?? And is traveling wave always...
  18. D

    I Should the Wave Equation for a Longitudinal Wave Include Time?

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  19. SemM

    I Question about the solution to the Harmonic wave equation

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  20. V

    Equivalence of Derivations in the Wave Equation from Maxwell's Equations

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  21. Clara Chung

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  22. Clara Chung

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  23. S

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  24. J

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  25. T

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  26. M

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  27. T

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  28. N

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  29. Johny911

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  30. B

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  31. F

    I Homogeneous Wave Equation and its Solutions

    Hello, There are many different wave equations that describe different wave-like phenomena. Being a differential equation, the WE is a pointwise relation and applies to the wavefield at spatial points. The equation is homogeneous when the source term is zero. That means that the solution...
  32. D

    I Alternate form of wave equation

    Apologies if this question is better posed in the mathematics section, it is for a quantum mechanics class so I decided to post it here: We are asked to verify that the following equation is a solution to the Schrodinger wave equation for a free particle: Psi(x,t) = Ae^i(kx-wt) - Ae^-i(kx+wt)...
  33. T

    Wave equation for an elastic rod

    1. Homework Statement The figure below shows a section of a thin, elastic rod of density ##\rho##, cross sectional area ##A##, and modulus of elasticity ##E##. By considering the net force acting on an element of the rod, derive the wave equation governing its longitudinal motion...
  34. G

    I How do I name a generalisation of the wave equation?

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  35. P

    Solving a wave equation with seperation of variables.

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  36. G

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  37. Eclair_de_XII

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    Homework Statement "The solution ##u(x,t)=f(x+ct)+g(x-ct)## solves the PDE, ##u_{tt}=c^2u_{xx}##. By graphing the solution ##u(x,t)=f(x+ct)## on the ##ux##-plane, please show that as ##t## increases, the graph shifts to the left at a velocity ##c##. Conversely, show that for ##u(x,t)=g(x-ct)##...
  38. B

    A Intuition on integral term in D'Alembert's formula

    If $$\phi(t,x)$$ is a solution to the one dimensional wave equation and if the initial conditions $$\phi(0,x) , \phi_t(0,x)$$ are given, D'Alembert's Formula gives $$\phi(t,x)= \frac 12[ \phi(0,x-ct)+ \phi(0,x+ct) ]+ \frac1{2c} \int_{x-ct}^{x+ct} \phi_t(0,y)dy . \tag{1}$$ which is...
  39. S

    I Difference between field and wave equation

    Hello! I am reading some introductory stuff on Klein-Gordon equation and I see that the author mentions sometimes that in a certain context the K-G equation "is a classical field equation, not a quantum mechanical field equation". I am not sure I understand. What is the difference between the...
  40. Pushoam

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    For using Galilean transformation, I have to assume that speed of light w.r.t. ether frame is c. W.r.t. ether frame, E = E0 eik(x-ct) W.r.t. S' frame which is moving with speed v along the direction of propagation of light, E' = E0 eik(x'-c't') Under Galilean transformation, x' = x-vt, t' = t...
  41. evinda

    MHB Solving the Wave Equation: Finding $u(x,t)$

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  42. snate

    I Troubleshooting a Derivation: Why μ*ε=1/c^2?

    I've been following the derivations in the following video up until that point. I don't quite understand why does it imply that μ*ε=1/c^2. Thanks.
  43. O

    Why plucking of string creates two pulses?

    When we pluck a string and a triangle is formed. Why does this triangle form into two opposite moving pulses? If we have reflective edges the two pulses will reflect, invert and superposition into the same triangle wave on the under side of the string. Let's say we have no dampening. I think...
  44. C

    Solutions of wave equation but not Maxwell equations

    Maxwell equation in absence of charges and currents are $$\nabla \cdot \bf{E} = 0 \\ \nabla\cdot B=0 \\ \nabla \times E=-\frac{\partial B}{\partial t} \\\nabla \times B=\mu \epsilon \frac{\partial E}{\partial t}$$ Wave equation is $$\nabla ^2 \bf{E}=\mu \epsilon \frac{\partial^2...
  45. asteeves_

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  46. F

    I D'Alembert's Solution – Domain of dependence

    When considering the Wave equation subject to initial conditions as follows… …then D'Alembert's solution is given by (where c is wave speed): I'd like to understand physically how this formula allows us to know the value of u (where u is the height of the wave, say) at some point (x0,t0)...
  47. P

    I Similarities / diffs between diffusion & wave propagation

    Hi, I'm a second year undergrad and we've covered the heat equation, \begin{equation} ∇^{2}\Psi = \frac{1}{c^{2}}\frac{\partial^2 \Psi}{\partial t^2} \end{equation} and the wave equation, \begin{equation} D∇^{2}u= \frac{\partial u}{\partial t} \end{equation} in our differential equations...
  48. L

    I Wave equation solution using Fourier Transform

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  49. Riverbirdy

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  50. P

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