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. 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...
  2. 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...
  3. 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} -...
  4. G

    A Seeking a derivation of Schrödinger's wave equation

    I am interested in the derivation of Schrödinger’s wave equation from the Klein Gordon equation. I have looked in Penfold’s ‘The Road to Reality’, the open University’s Quantum Mechanics books, Feynman’s lectures, the internet, but not found what I want. Everyone seems to take it as a given...
  5. 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...
  6. John Greger

    Expansion of the wave equation for a stationary wave

    Homework Statement A generic state represented by the wave function ##\psi (\vec(x)## can be expanded in the eigenstates with defined angular momentum. Write such an expansion for a plane wave traveling along the z direction with momentum ##p = \hbar k## in terms of unknown coefficients ##c ( k...
  7. It's me

    Show that the radiation field is transverse

    Homework Statement Show that the radiation field is transverse, ##\vec{\nabla}\cdot\vec{A}=0## and obeys the wave equation ##\nabla^2\vec{A}-\frac{1}{c^2}\partial_t^2\vec{A}=0##. You should start from the expansion of the quantum Electromagnetic field. Homework Equations ##H=\frac{1}{2}\int...
  8. B

    Resonant Lengths in open air column question

    Homework Statement An organ pipe 1.2m long and open at both ends produces a note with the fundamental frequency. If the speed of sound in air is 345 m/s, what is the fundamental frequency? Homework Equations Wave equation (f = v/lambda) The Attempt at a Solution My textbook solves the problem...
  9. Cathr

    Electromagnetic wave equation - phase and amplitude

    There are some things that confuse me about electromagnetic waves, and I haven't found good answers anywhere. Consider the following equation: E=E0 e i(wt-kx) (here E and E0 are vectors, I couldn't find the right symbols). The things that confuse me are the following: 1° We say that the power...
  10. evinda

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

    Hello! (Wave) I want to show for the initial value problem of the wave equation $$u_{tt}=u_{xx}+f(x,t), x \in \mathbb{R}, 0<t<\infty$$ that if the data (i.e. the initial data and the non-homogeneous term $f$) have compact support, then, at each time, the solution has also compact support. I...
  11. Jaden159

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

    h is plank constant and v is frequency. I was using this to derive the TDSE. But I ran into problem because to substitute k^2 in E=h^2/8mpi^2 * k^2, I can use single derivative of psi squared or double derivative, both of which tend to give the correct answer. So, is my assumption of energy...
  12. F

    I Why does the wave equation describe certain phenomena

    I have recently started learning about waves. We didn't really formally describe what a wave is, but instead started by looking at a concrete example namely harmonic sinusoidal waves in 1d. We then introduced the wave equation in 1d and showed that the sinusoidal waves indeed satisfy this...
  13. evinda

    MHB Initial value problem of the wave equation

    Hello! (Wave) I want to prove that if for the initial value problem of the wave equation $$u_{tt}=u_{xx}+f(x,t), x \in \mathbb{R}, 0<t<\infty$$ the data (i.e. the initial data and the non-homogeneous $f$) have compact support, then, at each time, the solution has compact support. I have...
  14. T

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

    Homework Statement I am having a issue understanding this question I have solve the PDE below, but I can't understand where or how you the characteristic frequency, what more confusing is that I don’t know if that lambda is just a constant or a wavelength. Homework EquationsThe Attempt at a...
  15. evinda

    MHB Compact Support of Wave Equation IVP Solutions

    Hello! (Wave) I want to prove that if the initial data of the initial value problem for the wave equation have compact support, then at each time the solution of the equation has also compact support. Doesn't the fact that a function has compact support mean that the function is zero outside...
  16. Mutatis

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

    Find the wave equation U(x,t) of a vibrating string with linear density d, tension p, initial velocity zero, weight L and initial displacement U0(x) = a1*sin(2*pi*x/L)+a2*sin(4*pi*x/L). Guys, please help me with this task. I did the following procedure: The U(x,t) solution must me a sum of...
  17. D

    I Exploring Wave Equation Solutions for E-fields

    Hi. I'm following the "derivation" in some lecture notes which shows that the Electric and Magnetic fields are perpendicular to each other and to the direction of propagation. There are 2 points I don't understand A solution to the wave equation for E-fields is given as E = E0 exp i(ωt-kz). It...
  18. 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...
  19. D

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

    Hi. I am working through " A Student's guide to waves " by Fleisch. In deriving the wave equation for a longitudinal wave it uses dψ = (∂ψ/∂x) dx where ψ is the displacement but ψ is a function of x and t ; so shouldn't this equation be dψ = (∂ψ/∂x)...
  20. SemM

    I Question about the solution to the Harmonic wave equation

    Hi, I have been looking in various text about how to find an admissible solution to the Schrödinger eqn in one dim. in the harmonic oscillator model. As in MQM, the solutions to this are said to be ##Ae^{ikx}+Be^{-ikx}##, which are then said to be not admissible. The book then goes straigtht to...
  21. V

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

    Homework Statement I am actually following the derivation of the wave equation from Maxwell equations. And I do not understand one step, because in the task for the derivation I get a slightly different result (maybe they are equivalent, but I am not sure). Homework Equations In the attached...
  22. Clara Chung

    I 2 sign questions related to the wave equation

    In the picture about, I gave 1. a picture of a segment of string for reference, 2. a description of the driving force and 3. a description about the potential energy per unit length of a point in the wave. I have two questions here. 1. Why does the driving mechanism produce a force to balance...
  23. Clara Chung

    I I don't understand the derivation of the wave equation

    If there is a net force along the y-axis, i.e. T sin(θ2) - T sin(θ1) Why is it equals to ma, where a is the acceleration of the piece of string along the y-axis? Shouldn't there be a torque so the piece of string rotates? Sorry for sounding stupid.
  24. S

    A Why Does the Wave Equation Solution Differ from Expected Traveling Waves?

    Hi everyone, I'm reading about the solution of the wave equation in free space on Stratton - Electromagnetic Theory and Snider - PDE and I got a little confused. The wave equation in 3D (plus time) is the following $$\frac{\partial^{2} \Psi (x,y,z,t)} {\partial t^{2}}=\nabla ^{2}\Psi...
  25. J

    B Wave equation, psi with dots and things like that....

    Hi everyone! I'm a psychologist form Brazil, so sorry for the bad English and for the lack of knowledge in math! I ve been trying to understand the Schrodinger equation and, as predicted, it's very hard! Please, help me with this: A sine wave function can be written as: F (x) = sin (x) And...
  26. T

    A What resources can help me solve the Spheroidal Wave Equation?

    Hi! I would like to Start from Maxwell's equations in order to solve the wave equation analytically for oblate and prolate spheroids. Could you suggest me any help?
  27. M

    Wave equation in inhomogeneous media -- Question about the formula

    https://i.hizliresim.com/ZO0bvG.jpg I don't understand that where ε0/ε is coming from? Can you explain this?
  28. T

    Wave equation, taut string hit with hammer

    Homework Statement A string of length L is fixed at both ends ##u(0,L)=u(L,t)=0## The string is struck in the middle with a hammer of width a, leading to an intial condtion ##u(x,0)=0## and $$U_t(x,0)=v_0 $$ for $$\frac{l}{2}-\frac{a}{2} \leq x \leq \frac{l}{2}+\frac{a}{2} $$ and...
  29. N

    About acoustics physics -- The Wave Equation and diminishing sound intensity

    Hello everyone! :-) Actually I'm starting to understand acoustics physics and I figured actually out about this equation: $$\frac{\partial^2\psi}{\partial t^2}=c^2 \nabla^2 \psi$$ which describes practically about pressure and propagation speed into space and time. I know also this equation...
  30. Johny911

    Wave equation, separation of variables and the Laplace transform

    Homework Statement Homework Equations If i solve the wave equation using separation of variable and laplace tranform. Will i get the same answer ? The Attempt at a Solution
  31. B

    Derive a wave equation for an n mass coupled system

    1. Derive the wave equation for longitudinal vibrations in an extended 1-D system of masses and springs. The average distance between masses is D [m], the spring constants are K [kg/s2 ], and the masses are M [kg]. b) Determine the wave speed c as a function of D, K, and M. Verify that it has...
  32. 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...
  33. 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)...
  34. 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...
  35. G

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

    I am interested in discussing those phenomena which can be defined at a point. The wave equation is the simplest example. Is it acceptable to use the term 'wave' to indicate any phenomenon that is defined at a point, and to call the equation that results a wave equation? To illustrate the...
  36. P

    Solving a wave equation with seperation of variables.

    Homework Statement I am trying to solve the given wave equation using separation of variables, u_{tt} - 4u_{xx} = 4 for 0 < x < 2 and t > 0 (BC) u(0,t) = 0 , u(2,t) = -2, for t>0 (IC) u(x,0)=x-x^2 , u_t(x,0)=0 for 0\leq x \leq2 Homework Equations We are told we will need to use, x =...
  37. G

    Half-life using Schrodinger's wave equation

    Hello guys! I've been learning how to estimate half life using Schrodinger's time-independent wave equation. In class, we divided the energy barrier into five smaller segments just like this webpage http://hyperphysics.phy-astr.gsu.edu/hbase/Nuclear/alpdet.html#c1 I was wondering if we could...
  38. Eclair_de_XII

    Understanding the wave equation

    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)##...
  39. 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...
  40. 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...
  41. Pushoam

    Variance of the EM wave equation under Galilean transformation

    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...
  42. evinda

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

    Hello! (Wave) Let $$u_{tt}-c^2 u_{xx}=0, x \in \mathbb{R}, t>0 \\ u(x,0)=0, u_t(x,0)=g(x)$$ where $g \in C^1(\mathbb{R})$ with $g(x)>0$ for $x \in (0,1)$, $g(x)=0$ for $x \geq 1$ and $g(x)=-g(-x)$ for $x \leq 0$. I want to find the sets of $\{ (x,t): x \in \mathbb{R}, t \geq 0 \}$ where $u=0...
  43. 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.
  44. 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...
  45. 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...
  46. asteeves_

    Calculating Amplitude in Simple Harmonic Motion

    I have encountered two separate review problems that have to do with finding a value for amplitude and I am really struggling with it. 1. Homework Statement Question 1- A mass of 3kg is free to move on a horizontal frictionless surface and attached to a spring of k=15 N/m. It is displaced...
  47. 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)...
  48. 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...
  49. L

    I Wave equation solution using Fourier Transform

    I'm studying Quantum Field Theory and the first example being given in the textbook is the massless Klein Gordon field whose equation is just the wave equation \Box \ \phi = 0. The only problem is that I'm not being able to get the same solution as the book. In the book the author states that...
  50. Riverbirdy

    Finite difference Method of Wave Equation

    Hi, Physics forum! Just a little push of my doubts I hope somebody could help me with my confusion of one of our home works. I know that all boundary conditions are zero. My doubt is how do I interpret (x,y,0)=0.01 source in the figure? Where is it located in the grid. I am hoping someone...
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