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.

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  1. P

    Derive the wave equation for fields E, B from the potentials

    I'm studying for my electrodynamics exam and one of the past exam questions is: From the scalar and vector potentials, derive the homogenous wave equations for E and B fields in vacuum. I did derive the wave equation for the B field by simply taking the curl of the homogenous wave equation for...
  2. J

    Finding Density as a Function of Space and Time for 1D Wave Equation Problem

    Homework Statement Hello- I'm having trouble understanding a problem: Consider a sealed 1D pipe of length L. At t=0, v=0 everywhere and the pressure is given by: P=P_0 +δP and δP = (p-bar)x/L P_0 and (p-bar) are both constants. and I'm supposed to find density (ϱ) as a function of x and t...
  3. Marcus95

    Number of Different resonances in a closed Box

    Homework Statement Show that the possible resonance frequencies in a 3D box with side a are constant multiples of ##(l^2+m^2+n^2)^{1/2}##, where l, m and n are integers. Assume that the box with sides a is filled with a gas in which the speed of sound is constant. Hence show that the number of...
  4. Nikhil Rajagopalan

    I Exploring the Symmetry of Initial Conditions in Progressive Wave Equations

    For the wave traveling towards left, the equations is Asin(kx + ωt). How does the same mathematical equation explain the possibility of two initial conditions. In the case of the wave traveling towards right, Asin(kx - ωt) and Asin(ωt - kx) gives two initial conditions Asin(kx) and - Asin(kx) on...
  5. J

    Can somebody help me understand this BVP question?

    Homework Statement So I don't really understand what the professor means by "show why the displacements y(x,t) should satisfy this boundary value problem" in problem 1. Doesn't that basically boil down to deriving the wave equation? At least in problem 2 he says what he wants us to show...
  6. J

    Help understanding a vibrating string question

    Homework Statement So I don't really understand what the professor means by "show why the displacements y(x,t) should satisfy this boundary value problem" in problem 1. Doesn't that basically boil down to deriving the wave equation? At least in problem 2 he says what he wants us to show...
  7. A

    Looking for speed using the wave equation

    Homework Statement phi(x,t) = A e *[-a(bx+ct)*2] I'm trying to find the speed of the equation Homework Equations f(x+vt) +vt which means it is in the negative x-direction f(x)= e^-ax^2 plugging in x'=x+vt A e *[-a(bx+ct)*2] where a= constant A= amplitude...
  8. Nikhil Rajagopalan

    Deriving the progressive mechanical wave equation

    Is it correct to state that a progressive wave, originates when a simple harmonic motion is imparted continuously to adjacent particles from one direction to another moving with a velocity v. Using this idea, substituting (t - x/v) instead of t is the simple harmonic motion function...
  9. F

    Why is the wave equation different from the heat equation

    I have been thinking about this. For a wave equation, the acceleration of a point on a drumhead is proportional to the height of its neighbors $$U_{tt}=\alpha^2\nabla^2U$$ The heat equation, change in concentration or temperature is equal to the average of its neighbors...
  10. harini07

    A question about wave motion and beat frequency

    Homework Statement 3 tuning forks of frequencies 200, 203, 207 Hz are sounded together.find out the beat frequency. Homework Equations Beat frequency= n1-n2 (n=frequency). The Attempt at a Solution I know that beat frequency is the difference in the frequencies of two superposing notes. But...
  11. Rick16

    I What is the Direct Solution of the Wave Equation?

    Text books often give an expression like Asin(kx-ωt) as a solution of the wave equation, but they don’t show how to arrive at this solution. Other textbooks, which go through the complete solution process of the wave equation, determine the coefficients using Fourier series. My goal was to get...
  12. G

    I Wave Equation & Wave Displacement Invariance: Modern Physics

    This question concerns a section from the book Modern Physics by James Rohlf. http://srv3.imgonline.com.ua/result_img/imgonline-com-ua-twotoone-Bs4zgy7pruqG.png He shows that the form of the Wave equation for light remains invariant under a Lorentz boost (4.42)...
  13. T

    Python Testing code to solve 2nd order wave equation

    As an exercise, I am trying to solve the 2nd-order wave equation: $$ \frac {\partial ^2 E}{\partial t^2} = c^2 \frac {\partial ^2 E}{\partial z^2} $$ Over a domain of (in SI units): ## z = [0,L=5]##m, ##t = [0,t_{max} = 5]##s, ##c = 1## m/s and boundary/initial conditions: ## E(z[0]=0,t) =...
  14. B

    I Schrödinger's Equation Infinite Potential Well

    Given the equation ##\frac{d^2 \psi (x)}{{dt}^2}+\frac{2m}{{\hbar}^2}(E-V(x))=0## the general solution is: $$\psi (x)=A_1 e^{ix \sqrt{\frac{2m}{{\hbar}^2}(E-V(x))}} +A_2 e^{-ix \sqrt{\frac{2m}{{\hbar}^2}(E-V(x))}}$$ If we have an infinite potential well: ## V(x)=\begin{cases} \infty \quad x\ge...
  15. M

    A Wave equation in cylindrical coordinates - different expression?

    Hello to everybody, I am solving some examples related to wave equation of shear horizontal wave in cylindrical coordinates (J.L Rose: Ultrasolic Waves in Solid Media, chapter 6), which is expressed as follows: ∇2u=1/cT2⋅∂2u/∂t2 The Laplace operator in cylindrical coordinates can can be...
  16. A

    I Exploring the 1D Wave Equation to Describe dΨ(x)/dt

    Hello! So if I want to describe dΨ(x)/dt can I use 1D wave equation?
  17. Quadrat

    Solving Wavefunction Problems: Tips and Examples

    Homework Statement [/B] I found a couple of assignements for a physics course I will take later this year- so I started looking into them a bit in advance. It concerns wavefunctions. I'm a bit rusty on my trigonometric identities So I would love if someone could try to help me solve these two...
  18. Wise Owl

    A Solution to the wave equation in Rindler coordinates

    I have been reading these notes on Rindler coordinates for an accelerated observer. In Rindler coordinates, the hyperbolic motion of the observer is expressed through the coordinate transformation $$t=a^{-1}e^{a{{\xi}}}\sinh a{\eta}\\ {}x=a^{-1}e^{a{{\xi}}}\cosh a{\eta}.$$On a space-time...
  19. T

    Wave Equation for a Progressive Wave

    Homework Statement In some places, the wave equation is y=Asin(wt-kx) and in other places they have y = Asin(kx-wt) and they treat them as if they are equal. How are they equal? They also had y=-Asin(wt-kx). What is the difference between all 3? Homework Equations As above The Attempt at a...
  20. T

    Partial Derivatives and the Linear Wave Equation

    Homework Statement I'm reading through the derivations of the linear wave equation. I'm following everything, except the passage I highlighted in yellow in the below attachment: Homework Equations I'm not understanding why partials must be used because "we evaluate this tangent at a...
  21. mertcan

    A Derive General Metric Wave Equation

    hi, I know the wave equation in terms of minkowski metric, and we use ordinary derivative in that equation. I also know this wave equation in terms of general metric form using covariant derivative, but I do not know the derivation of it. Could you spell out the steps of derivation?? How do we...
  22. G

    I Do non-monochromatic "waves" exist in dispersive media?

    Hi. Is the superposition of two different monochromatic waves in a dispersive medium still a wave (i.e. a solution of a wave equation) if the phase velocity is not the same? Since the wave equation contains the phase velocity, the two individual waves are solutions of different wave equations...
  23. W

    I Proving that a function is a solution to the wave equation

    How could you that y(x,t)=ƒ(x/a + t/b), where a and b are constants is a solution to the wave equation for all functions ,f ? many thanks.
  24. Z

    Satisfying the wave equation with 2 different speeds

    Hello, I have a wave of the form y = Asin(x-vt) + Asin(x+2vt) which I substituted into the wave equation to find out if it satisfies it. It didn't because of the speed of the left traveling wave being equal to 2v. What I got was: A[-sin(x-vt)-sin(x-2vt)] = 1/v2 * A[-v2sin(x-vt) -...
  25. F

    Eigenvalues of a string with fixed ends and a mass in the middle

    Hi there. First of all, sorry for my bad english. I ´m trying to solve next exercise, from Vibrations and Waves in Continuos Mechanical Systems (Hagedorn, DasGupta): Determine the eigenfrequencies and mode-shapes of transverse vibration of a taut string with fixed ends and a discrete mass in the...
  26. N

    Inverse Fourier Tranform of Transmission Lines Wave Equation

    Homework Statement From the derivation of v(x,t) and i(x,t) I am stuck on how the inverse Fourier transform of e^(-jwx/u) was calculated. I am trying to understand how the PDE was fully solved here: http://fourier.eng.hmc.edu/e84/lectures/transmission_line/node1.htmlHomework Equations Not...
  27. RJLiberator

    PDE: Nontrivial solution to the wave equation

    Homework Statement Consider the wave equation: u_{tt} - c^2u_{xx} = f(x,t), \hspace{1cm} for \hspace{1cm} 0 < x < l \\ u(0,t) = 0 = u(l,t) \\ u(x,0) = g(x), u_t(x,0) = f(x) \\ Find a nontrivial solution. Homework EquationsThe Attempt at a Solution Here's what I did, but I have little...
  28. It's me

    Velocity of propagation of an EM field in vacuum

    Homework Statement In a region of empty space, the magnetic field is described by ##\vec{B} = B_0e^{ax}\sin{(ky-\omega t)} \hat{z}##. Find the speed of propagation ##\vec{v}## of this field. Homework Equations ##\Delta \vec{B} = \frac{1}{v^2}\frac{d^2\vec{B}}{dt^2}## , ##k=\frac{\omega }{...
  29. I

    Showing f is a solution to quantum oscillator SWE

    Homework Statement For a 1-dimensional simple harmonic oscillator, the Hamiltonian operator is of the form: H = -ħ2/2m ∂xx + 1/2 mω2x2 and Hψn = Enψn = (n+1/2)ħωψn where ψn is the wave function of the nth state. defining a new function fn to be: fn = xψn + ħ/mω ∂xψn show that fn is a...
  30. nmsurobert

    Deriving the Wave Equation from Laplacian and Partial Derivatives

    Homework Statement derive the following wave equation ∇2H = 1/c2 (∂2H/∂t2) Homework EquationsThe Attempt at a Solution I'm not sure how to derive it. I suppose I can break it into a whole bunch of partial derivatives because of the del squared operator and then just lump the three partial...
  31. D

    How do i know what a function describe a wave?

    Homework Statement Show that the funtions y(x,t) and g(x,t) satisfy the differential equation of a wave unimensional. What function is a wave?Homework Equations y(x,t)=x² +v²t² ; d(x,t)= 2Acos(kx)cos(wt) frac{d²y}{dt²}=2v² frac{d²y}{dx²}=2 The Attempt at a Solution frac{d²y}{dt²}=2v²...
  32. RJLiberator

    PDE: Wave Equation with Neumann conditions

    Homework Statement Consider the homogeneous Neumann conditions for the wave equation: U_tt = c^2*U_xx, for 0 < x < l U_x(0,t) = 0 = U_x(l, t) U(x,0) = f(x), U_t(x,0) = g(x) Using the separation of variables, find a nontrivial solution of (1). Homework Equations Separation of variables The...
  33. N

    D'Alembert solution of wave equation on semi infinite domain

    Homework Statement Wave equation: ytt=yxx Initial conditions: Y(x,0) =f(x) = x (0 ≤ x < 1) 2.5(5-x) (1 ≤ x < 3) 0 (Otherwise) and yt(x,0) = 0 Boundary condition: y(0,t) =0 Semi infinite domain: 0 ≤ x < infinity Homework Equations d'Alembert solution...
  34. M

    Prove a satisfaction with the wave equation

    Homework Statement i want to prove that the functions u(r,t)=(1/r)f(r-v*t) and u(r,t)=(1/r)f(r+v*t) satisfy the wave equation in spherical coordinates, i have tried a lot to solve it but in each time i would face a problem. Homework Equations wave equation : grad^2(u)=(1/v)*(partial ^2...
  35. T

    Whats so amazing about the wave equation?

    From my understanding, all the wave equation says is the transverse acceleration of a particle on wave is equal to the the curvature of the of the wave at that point-particle times the propagation speed squared. Am i missing something?
  36. S

    How Does the Wave Equation Derive and What Solutions Exist?

    $$\frac{\partial^2}{\partial t^2}u(x,t)=c^2\Delta u(\vec{x},t)\qquad \vec{x}\in \mathbb{R}^n$$ is known as the wave equation. It seems not very trivial, so is there any derivations or inspirations of it? To solve this equation, we have to know the initial value and boundary conditions...
  37. G

    Wave Equation for Circular Waves

    What's the solution to the wave equation for circular waves on a two-dimensional membrane? The waves have a constant wavelength throughout. For spherical waves, you have to multiply the amplitude by 1/r. I tried 1/√r for circular waves but it didn't work. :blushing:
  38. S

    Wave equation indicates the probability density

    First, we know for every wave function $$p(x)=\psi(x)^*\psi(x)$$ indicates the probability density of a particle appearing at the point x. So if we calculate $$P=\int _M p \text{d}x$$ this gives the probability of the particle appearing in the range M. On the other side, I was thinking about...
  39. M

    Derivation of wave equation using tension of a string

    Homework Statement I'm currently following the textbook Advanced Engineering Mathematics by Erwin Kreyszig. I'm learning the derivation of the Wave equation using the method shown in the book, but when I reached the final part of the derivation, the working just confuses me. (1/Δx)[ (u/dx)|...
  40. M

    Finding the General Solution of a Wave Equation with Fourier Transformations

    Hi, following the attached paper I try to find the general solution of the following wave equation: \frac{1}{a^2} \frac{\partial^2 \phi}{\partial t^2} + \frac{2M}{a}\frac{\partial^2 \phi}{\partial x \partial t} + \overline{\beta}^2 \frac{\partial^2 \phi}{\partial x^2} =...
  41. L

    What Is the Analytic Solution to the Wave Equation Problem?

    Homework Statement [/B] Plot the solution at times t = 0, t = 0.25 and t = 0.5. Based on your knowledge of the general solution of the wave equation, the behaviour you see here, and the initial data profile, what is the analytic solution u(x, t) to this problem? (In deriving the analytic...
  42. W

    Vector Wave Equation: Uses & Benefits

    hi, my question is , when do we need to have vector wave equation. So far in Maxwell equation you can find scalar as well as vector wave equation, I figure out when we are looking for the scattering we need vector wave equation. Second isn't simple to work out scalar potential and then by its...
  43. H

    How to check a numerical simulation is energy conservative?

    I am modelling a 1D fluid wave propagation problem and I needed to know how I can check that my results are energy conservative. please reply, urgent. thank you.
  44. F

    Is y(x,t) a Solution to the Wave Equation?

    Homework Statement Is the function y(x,t) = Ae(−β2x2−2βxt−t2) + Be(−x2+2αxt−α2t2) a solution to the wave equation ∂2y / ∂t2 = v2 (∂2y / ∂x2) Homework Equations ∂2y / ∂t2 = v2 (∂2y / ∂x2) The Attempt at a Solution I have found the solution through finding the partial derivatives (∂2y / ∂t2...
  45. DrPapper

    Plane Wave Equation Propagation and Oscillation Directions

    Homework Statement Write down the equation for a plane wave traveling in perpendicular to the plane x+y+z=constant traveling in the direction of increasing x, y, and z. Homework Equations From the given information how do I determine the unit vector that goes next to E(0)? How do I determine...
  46. G

    What is this version of the wave equation?

    I came across this expression for the wave equation: \nabla^2E + \mu\sigma\frac{\partial{E}}{\partial{t}} - \frac{n^2}{c^2}\frac{\partial{E}}{\partial{t^2}} = 0 My question is what kind of medium is it for/where did it come from?
  47. W

    How do I plot ψ(x,t) as a function of x at time t=a/v?

    Homework Statement A 1D wave function ψ(x,t) satisfies these initial conditions: ψ(x,0) = 0 for all x ∂ψ/∂t (x,0) is v for -a≤x≤a 0 otherwise Plot ψ(x,t) as a function of x at time t=a/v. Homework EquationsThe Attempt at a Solution I know the 1D wave equation is given by...
  48. W

    Plot graph of 1D wave equation (using d'Alembert's formula)

    Homework Statement [/B] Don't know if this goes here or in the advanced bit, thought I'd try here first! I know the general solution of a 1D wave equation is given by d'Alembert's formula ##u(x,t) = 0.5[u(x+vt,0) + u(x-vt,0)] + \frac{1}{2v} \int_{x-vt}^{x+vt} \frac{\partial u}{\partial...
  49. N

    Making a program to model the Double Slit Experiment?

    Hey guys the Idea of simulating the DSE with a program caught my interest but I just had a few questions regarding the DSE Is there a function that models the probability of finding an electron at a certain point ONCE It's BEEN FIRED FROM AN ELECTRON GUN? like an amplitude function squared or...
  50. D

    Proving Non-linear Wave Equation for Riemann Tensor

    Hello, I am working through Hughston and Tod "An introduction to General Relativity" and have gotten stuck on their exercise [7.7] which asks to prove the following non- linear wave equation for the Riemann tensor in an empty space: ∇e∇eRabcd = 2Raedf Rbecf − 2Raecf Rbedf − Rabef Rcdef I have...
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