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. deuteron

    I Physical Meaning of the Imaginary Part of a Wave Function

    We know the wave function: $$ \frac {\partial^2\psi}{\partial t^2}=\frac {\partial^2\psi}{\partial x^2}v^2,$$ where the function ##\psi(x,t)=A\ e^{i(kx-\omega t)}## satisfies the wave function and is used to describe plane waves, which can be written as: $$ \psi(x,t)=A\ [\cos(kx-\omega...
  2. F

    I Non-wave solution to wave equation and virtual particles

    Hello everyone. The 1D wave equation is written: $$ \left( \partial_t^2/c^2 - \partial_x^2 \right) \Psi = 0$$ An electromagnetic wave or matter wave, like free electron (unnormalized here), can be written with the following wave function ##\Psi_m## of energy ## \hbar k c ##: $$ \Psi_m \propto...
  3. jjson775

    I Don’t understand the general form of the Sinusoidal Wave Equation

    I am a retired engineer, 81 years old, self studying modern physics using Young and Freedman University Physics. I am familiar with the wave equation y(x,t) = A cos (kx - wt) where A = amplitude, k = wave number and w (omega) = angular frequency. in the chapter introducing quantum mechanics...
  4. C

    Showing that this equation is a solution to the linear wave equation

    For this problem, Where equation 16.27 is the wave equation. The solution is I don't understand how they got the second partial derivative of ##y## with respect to ##x## circled in red. I thought it would be ##1## since ##v## and ##t## are constants Many thanks!
  5. M

    A Wave energy dissipated due to geometry

    Hey Condition 1: A 2D infinite plane and there is a circular hole in the middle. When t=0, an impulsive loading, P=f(t), is applied to the boundary of the circle(outward), so the wave will start at the boundary of the circle and propagate in the plane Condition 2: A 3D infinite plane and there...
  6. S

    Determine eigenvalue-problem for steel pole

    If we assume that ##\psi## has a Fourier transform ##\hat{\psi}##, so that ##\psi(x,t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}\hat{\psi}(x,\omega)e^{i\omega t}\mathrm{d}\omega##, then the wave equation reduces to ##-\rho\omega^2\hat{\psi}(x,\omega)=E\frac{\partial^2 \hat{\psi}(x,\omega)}{\partial...
  7. MevsEinstein

    B Particle Function? Particle Equation?

    The following is the wave equation from Electrodynamics: $$\frac{\partial^2 \Psi}{\partial t^2} = c^2\frac{\partial^2 \Psi}{\partial x^2}$$ Where ##\Psi## is the wave function. But because of Heisenberg's Uncertainty, physicists had to come up with another equation (the Schrodinger equation)...
  8. person123

    I Shallow Water Wave Equation Simulation: Reverse Shoaling?

    I'm creating a simulation of the shallow water wave equation in MATLAB. I'm using the equations: $$\frac{\partial v}{\partial t}=-g\frac{\partial \eta}{\partial x}$$ $$\frac{\partial h}{\partial t}=-h\frac{\partial v}{\partial x}$$ Iteratively updating the velocity from neighboring heights...
  9. bob012345

    I FEM Method for the Wave Equation

    I am trying to understand how to apply the finite element method for a simple 1D wave equation with four elements for learning purposes. $$\frac{d^2 T(x)}{dx^2} = -f(x)$$ I am stuck because the structure of the equations set up in Numerical Methods for Engineers by Chapra and Canale as shown...
  10. Afo

    Solving a Sound Wave Equation in Physics 1: Halliday, Resnick, and Krane

    Homework Statement:: This is from 5 ed, Physics 1Halliday, Resnick, and Krane. page 428 about sound waves I have highlighted the equation that I don't understand. How did the author get it? I understand how they get from the middle side to the RHS of the equation, but I don't understand how...
  11. chwala

    Solving the wave equation with change of variables approach

    I am refreshing on the pde's, and i am trying to understand how the textbook was addressing change of variables, i find it a bit confusing. I will share the textbook approach, then later share my own understanding on change of variables approach. Here is the textbook approach; My approach on...
  12. Poetria

    Critical points of a multivariable function (wave equation)

    I would like to check my understanding of this problem. There are the following possibilities: a. Isolated points where the gradient is 0. b. The level curves of height 0 c. The level curves of height 1. d. The level curves of height -1. e. None of the above. I would choose a, c, d. Where...
  13. redtree

    B Difference between a continuously differentiable function and a wave

    What is the difference between an absolutely continuously differentiable function and a wave? Are all absolutely continuously differentiable equations waves?
  14. baby_1

    Attenuation and Phase constant values in wave equation

    Regarding wave equation we are faced with this form $$\nabla^2 \vec{E}=j\omega \mu \sigma \vec{E}-j\omega \mu\varepsilon \vec{E}=\gamma ^2\vec{E}$$ where $$\gamma ^2=j\omega \mu \sigma -j\omega \mu\varepsilon $$ $$\gamma =\alpha +j\beta $$ where alpha and beta are attenuation and phase...
  15. U

    I Why is Scalar Massless Wave Equation Conformally Invariant?

    It can be shown mathematically that the scalar massless wave equation is conformally invariant. However, doing so is rather tedious and muted in terms of physical understanding. As such, is there a physically intuitive explanation as to why the scalar massless wave equation is conformally invariant?
  16. Arman777

    Wave equation for Schwarzschild metric

    I am trying to find the $$\nabla_{\mu}\nabla^{\mu} \Phi$$ for $$ds^2 = (1 - \frac{2M}{r})dt^2 + (1 - \frac{2M}{r})^{-1}dr^2 + r^2d\Omega^2$$ I have did some calculations by using $$\nabla_{\mu}\nabla^{\mu}\Phi = \frac{1}{\sqrt{-g}}\partial_{\mu}(\sqrt{-g}g^{\mu \nu}\partial_{\nu}\Phi)$$...
  17. RealKiller69

    Discrete 2D wave equation

    I am having problems writing the initials conditions.
  18. Haorong Wu

    I Weak Gravitational Field & Wave Eq. - Analyzing Effects on Massless Scalar Field

    A massless scalar field in a curved spacetime propagates as $$(-g)^{-1/2}\partial_\mu(-g)^{1/2}g^{\mu\nu}\partial_\nu \psi=0 .$$ Suppose the gravitational field is weak, and ##g_{\mu\nu}=\eta_{\mu\nu}+\epsilon \gamma_{\mu\nu}## where ##\epsilon## is the perturbation parameter. And let the field...
  19. B

    Solving the wave equation for standing wave normal modes

    ## \frac {\partial^2 \psi} {\partial t^2} = v^2 \frac {\partial^2 \psi} {\partial x^2} ## has solution ## \psi (x, t) = \sum_{m=0}^\infty A_m \sin(k_mx + \alpha_m)sin(\omegat + \beta_m) ## The boundary conditions I can discern $$ \psi (0, t) = 0 $$ $$ \frac {\partial \psi} {\partial x} (L, t)...
  20. L

    I Connecting wave equation with a non-constant force only in bounds

    Hi hi, I'm confused about how to mix this two concepts, actually the wave equation: ##\frac {\partial^2 u} {\partial t^2} = v_x^2 \frac {\partial^2 u} {\partial x^2} + v_y^2\frac {\partial^2 u} {\partial y^2} + force## The equation will apply the rule all over the space, but I have the next...
  21. person123

    I Computing the Speed of Traveling Waves in 2 Dimensions

    I have a 2-dimensionsal smooth function ##f(x,y,t)##. There may be multiple traveling waves across the domain. None of them are precisely traveling waves (the shape of the wave changes as it travels). Here is how one of these waves would look in 1-dimension: I want to find the speed of these...
  22. J

    A Nonlinear Wave Equation (Nonlinear Helmholtz)

    I am trying to solve a PDE (which I believe can be approximated as an ODE). I have tried to solve it using 4th Order Runge-Kutta in MATLAB, but have struggled with convergence, even at an extremely high number of steps (N=100,000,000). The PDE is: \frac{\partial^2 E(z)}{\partial z^2} +...
  23. O

    A Inhomogeneous wave equation: RHS orthogonal to homogeneous solutions

    Hi, I've been reading Brillouin's 'Wave Propagation in Periodic Media'. About the following equation $$\nabla^2u_1+\frac{\omega^2_0}{V_0}u_1 = R(r)$$ Brillouin states that "it is well known that such an equation possesses a finite solution only if the right-hand term is orthogonal to all...
  24. E

    B Wave equation with a slight alteration

    I want to find the particular solution to the differential equation$$g(L-x) \frac{\partial^2 y}{\partial x^2} = \frac{\partial^2 y}{\partial t^2}$$with the boundary condition ##y(0) = 0## for all ##t##. If the coefficient of ##\frac{\partial^2 y}{\partial x^2}## were constant then it could be...
  25. T

    B Complete set of solutions to the wave equation

    I am solving the wave equation in z,t with separation of variables. As I understand it, Z(z) = acos(kz) + bsin(kz) is a complete solution for the z part. Likewise T(t) = ccos(ω t) + dsin(ωt) forms a complete solution for the t part. So what exactly is ZT = [acos(kz) + bsin(kz)][ccos(ωt) +...
  26. Athenian

    One-Dimensional Wave Equation & Steady-State Temperature Distribution

    To begin with, I can first let ##T(x,y) = X(x) Y(y)## to be the given solution. With this, I can then continue by writing: $$Y \frac{\partial^2 X}{\partial x^2} + X \frac{\partial^2 Y}{\partial y^2} = 0$$ $$\Longrightarrow \frac{1}{X} \frac{\partial ^2 X}{\partial x^2} + \frac{1}{Y}...
  27. jeremiahrose99

    I Solution to the 1D wave equation for a finite length plane wave tube

    Hi there! This is my first post here - glad to be involved with what seems like a great community! I'm trying to understand the acoustics of a finite plane-wave tube terminated by arbitrary impedances at both ends. So far all of the treatments I've managed seem only to address a different...
  28. Q

    Action of the time reversal operator on the QM wave equation

    Applying the time reversal operator to the plane wave equation: Ψ = exp [i (kx - Et)] T[Ψ ] = T{exp [i (kx - Et)]} = exp [i (kx + Et)] This looks straightforward as I have simply applied the 'relevant equation' however my doubt is in relation to the possible action of operator T on the i...
  29. M

    Wave Equation: d'Alembert solution -- semi-infinite string with a fixed end

    Hi, I was trying to get some practice with the wave equation and am struggling to solve the problem below. I am unsure of how to proceed in this situation. My attempt: So we are told that the string is held at rest, so we only need to think about the displacement conditions for the wave...
  30. Y

    Question about travelling wave equation

    There is a multiple choices question about traveling wave in my book. Based on the graphic, if T = 2s, the wave equation is ... My answer : ω = 2π/T = 2π/2 = π k = 2π/λ = 2π/4 = 0,5π → in my country, we use comma (,) for point (.) y = ±A sin (ωt - kx) y = -0,5 sin (πt - 0,5πx) y = -0,5 sin...
  31. M

    Engineering Power Loss Definition in a Damped Wave Equation (Skin Depth Problem}

    Hi, So the main question is: How to deal with power loss in E-M waves numerically when we are given power loss in dB's? The context is that we are dealing with the damped wave equation: \nabla ^ 2 \vec E = \mu \sigma \frac{\partial \vec E}{\partial t} + \mu \epsilon \frac{\partial ^ 2 \vec...
  32. jk22

    I Solving the Wave Equation via complex coordinates

    I'm looking for material about the following approach : If one suppose a function over complex numbers ##f(x+iy)## then ##\frac{df}{dz}=\frac{\partial f}{\partial x}\frac{1}{\frac{\partial z}{\partial x}}+\frac{\partial f}{\partial y}\frac{1}{\frac{\partial z}{\partial y}}=\frac{\partial...
  33. jk22

    I D'Alembert solution for the wave equation: question about the speeds

    The solution for the wave equation with initial conditions $$u(x,0)=f(x)$$ and $$u_t(x,0)=g(x)$$ Is given for example on wikipedia : $$u(x,t)=(f(x+ct)+f(x-ct)+1/c*\int_{x-ct}^{x+ct}g(s)ds)/2$$ So a vibrating string, since there is no conditions on ##g## (like ##\sqrt{1-g(x)^2/c^2}##), could...
  34. JD_PM

    Deriving the wave equation using small perturbations

    Note that the wave equation we want to derive was introduced by Alfven in his 1942 paper (please see bottom link to check it out), but he did not include details on how to derive it. That's what we want to do next. Alright, writing the above equations we assumed that: $$\mu = 1 \ \ \ ; \ \ \...
  35. JD_PM

    Understanding how to apply the method of images to the wave equation

    Exercise statement Find the general solution for the wave equation ftt=v2fzzftt=v2fzz in the straight open magnetic field tube. Assume that the bottom boundary condition is fixed: there is no perturbation of the magnetic field at or below the photosphere. Solve means deriving the d’Alembert...
  36. ContagiousKnowledge

    General solution of the spherical wave equation

    Since the spherical wave equation is linear, the general solution is a summation of all normal modes. To find the particular solution for a given value of i, we can try using the method of separation of variables. $$ ψ(r,t)=R(r)T(t)ψ(r,t)=R(r)T(t) $$ Plug this separable solution into the...
  37. J

    A Using the 2d wave equation or the Kirchhoff plate equation

    I would like to model the dynamics of a plate. Is it ok to use just the 2d wave equation if the plate will be under tension and fixed at the boundaries? I am a bit confused what the point of the Kirchhoff plate equation is in that case, is it for when the plate is self supporting? Many thanks
  38. B

    I Basic doubt on chain rule in DAlemberts soln to wave equation

    In D Alembert's soln to wave equation two new variables are defined ##\xi## = x - vt ##\eta## = x + vt x is therefore a function of ##\xi## , ##\eta## , v and t. For fixed phase speed, v and given instant of time, x is a function of ##\xi## and ##\eta##. Therefore partial derivative of x w.r.t...
  39. B

    I D'Alembert's solution to the wave equation

    Hi On page 81 of the book "A student's guide to waves by Fleisch and Kinneman a conclusion is made while differentiating D Alembert's solution to the wave equation. Will someone explain this please ? The details are in the attachment TIA
  40. dRic2

    I Green's function for the wave equation

    Hi, I'm reading "Wave Physics" by S. Nettel and in chapter 3 he introduces the Green's function for the 1-dimensional wave equation. Using the separation of variables method he restricts his attention to the spatial component only. Let ##u(x)## be the spatial solution to the wave equation and...
  41. W

    Derivation of the wave equation on a curved space-time

    I'm confused by this question, from minimal coupling shouldn't the answer simply be ## \nabla^a \nabla_a F_{bc} = 0 ##? Any help would be appreciated. EDIT: I should also point out ##F_{ab}## is the EM tensor.
  42. fluidistic

    A Schrödinger's equation: a diffusion or a wave equation?

    From many sources (Internet, Landau & Lifshitz, etc.), it is claimed that the Schrödinger's equation is a wave equation. However I do not understand why for the following reasons: It is Galilean invariant, unlike the wave equation which is Lorentz invariant. Note that the diffusion/heat...
  43. S

    MHB Solving wave equation using Fourier Transform

    I am having trouble with doing the inverse Fourier transform. Although I can find some solutions online, I don't really understand what was going on, especially the part that inverse Fourier transform of cosine function somehow becomes some dirac delta. I've been stuck on it for 2 hrs...
  44. AncientOne99

    Wave equation, wavelenght not given

    Homework Statement Wave problem Given data : f = 20 Hz y0 = 0.005 m t = 0.1 s x = 0.4 m1. The end portions of stretched string oscillates in the transverse direction with a frequency of 20 Hz and an amplitude of 0.005 m. Wave , which travels along the string, made in 0.1 second, 0.4 m long...
  45. F

    I Meaning of the Wave Equation

    If I’m not mistaken, a system can be described as a wave if it follows the wave equation. On Wikipedia, the general solution for the one-dimensional wave equation is written as u(x,t) = F(x - ct) + G(x + ct). I don’t see the connection between this solution and what I understand waves to be...
  46. C

    MHB Wave Equation Question Part 3

    Dear Everybody, I do not know how to begin with the following problem: you are asked to solve the wave equation subject to the boundary conditions ($u(0,t)=u(L,t)=0$), $u(x,0)=f(x)$ for $0\le x\le L$ and ${u}_{t}(x,0)=g(x)$ for $0\le x\le L$ . Hint: using the...
  47. C

    MHB Wave Equation Question Part 2

    Dear Everybody, I am confused about how to start with the following problem: using the solution from ex. 3: $u(x,t)=F(x+ct)+G(x-ct)$ "For data u(x,0)=0 and ${u}_{t}=\frac{x}{(x^2+1)^2}$ where x is from neg. infinity to pos. infinity." Thanks Cbarker1
  48. C

    MHB Solving the Wave Equation with Initial Data: Ex. 5 Solution and Derivation

    Dear Everyone, Hi. I do not how to begin for the following question: Ex. 5. Using the solution in Ex. 3, solve the wave equation with initial data $u(x,t)=\frac{1}{{x}^2+1}$ and $\pd{u}{t}(x,0)=0$ for $x\in(-\infty,\infty)$. The solution, (I have derived this solution in Ex. 4), that is...
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