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. mr_sparxx

    Very simple: second order derivative in wave equation

    In the equation regarding an array of masses connected by springs in wikipedia the step from $$\frac {u(x+2h,t)-2u(x+h,t)+u(x,t)} { h^2}$$ To $$\frac {\partial ^2 u(x,t)}{\partial x^2}$$ By making ##h \to 0## is making me wonder how is it rigorously demonstrated. I mean: $$\frac {\partial ^2...
  2. N

    What is the Fault in My Derivation of the Wave Equation in a Conductor?

    Homework Statement http://web.phys.ntnu.no/~ingves/Teaching/TFY4240/Exam/Exam_Dec_2008_tfy4240.pdf problem 2a) Homework EquationsThe Attempt at a Solution Hi. In problem 2a I was supposed to find a wave equation, however while digging around in maxwell's equations, I found this result...
  3. S

    Numerical method for wave equation

    Hi, I am trying to plot a function subjected to a nonlinear wave equation. One of the method I found for solving the nonlinear schrodinger equation is the split step Fourier method. However I noticed that this method only works for a specific form of PDE where the equation has an analytic...
  4. M

    What Can Be Done With the Expression for Total Energy in a Vibrating String?

    Hi PF! SO we have defined energy per unit mass as $$E(t) = \int_0^L \frac{1}{2} u_t^2 + \frac{c^2}{2} u_x^2 dx$$. We are given a vibrating string that exhibits ##u_x(0,t) = 0## and ##u(L,t)=0##. I am trying to figure out what is happening with total energy, ##E(t)##. My work is $$\int_0^L...
  5. V

    General solution to the wave equation of electromagnetic field

    Suppose that we have the four-vector potential of the electromagnetic field, [texA^i[/tex] The wave equation is given by $$(\frac {1}{c^2} \frac {\partial^2}{\partial t^2}-\nabla^2) A^i=0$$ Now the solution, for a purely spatial potential vector, is given by $$\mathbf{A}(t...
  6. V

    Fourier Transform Real Function Wave Equation

    Hello, I hope somebody can help me with this. 1. Homework Statement I am supposed to show that if there is a function \phi(x,t) which is real, satisfies a linear wave equation and which satisfies \phi(x,0)=0 for x<0 then the Fourier Transform \tilde{\phi}(k) of \phi(x,0) is in the lower...
  7. M

    One dimensional wave equation

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  8. baby_1

    Simple question in Del operator on plane wave equation

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  9. PeSoberbo

    What is the Second Step in Feynman's Deduction of the Sound Wave Equation?

    I am studying the sound wave equation deducted by Feynman in his lectures. In section 47-3: P0 + Pe = f(d0 + de) = f(d0) + de f'(d0) Where f'(d0) stands for the derivative of f(d) evaluated at d=d0. Also, de is very small. I do not understand the second step of the equality. Can anyone help...
  10. R

    Electromagnetic Wave Equation

    Homework Statement Show that the solution \textbf{E}=E(y,z)\textbf{n}\cos(\omega t-k_xx) substituted into the wave equation yields \frac{\partial^2 E(y,z)}{\partial y^2}+\frac{\partial^2 E(y,z)}{\partial z^2}=-k^2E(y,z) where k^2=\frac{\omega^2}{c^2}-k_x^2 Homework Equations See above. The...
  11. D

    MHB Wave equation of string under tension with a mass at x(a)

    We have a string of length \(\ell\) with fixed end points. At \(x(a)\), we have a mass. We can break the string up into two sections \(a + b = \ell\); that is, a is the distance up to the mass and b afterwards. The string is under tension \(T\). My question is why is the DE then \[...
  12. D

    Problem interpreting course notes - 3D wave equation

    I've been stuck trying to figure out what's going on in a particular section of my notes for the last couple days. The biggest issue is the lecturer has just not explained where the example has come from and what it represents. I thought I would post the relevant section here and see if anyone...
  13. R

    Uncovering the Mysteries of Electromagnetic Wave Equations in Antenna Theory

    Is it possible to solve these partial differential equations directly, relating to Antenna Theory; ∇^2 E - μ_0 ε_0 \frac{∂^2E}{∂t^2} = -μ_0 \frac{∂J}{∂t}. AND ∇^2 B - μ_0 ε_0 \frac{∂^2B}{∂t^2} = -μ_0 ∇ x J. I don't like the idea of having to make up fields that don't exist in order to make...
  14. S

    Are normalization constants of wave equation time dependent?

    The wave function solution psi is a function of time and position. Hence the integral of its square over all x will, in general, give a function of time. To normalize this, we must multiply with the inverse of the function. Therefore it seems that the normalization constant does not remain...
  15. I

    Using Wave Equation to Prove that EM Waves are Light

    Homework Statement I'm working on using the wave equation to prove that EM waves are light. Homework Equations Here's what I'm working with: E = Em sin(kx-wt) B = Bm sin(kx-wt) ∂E/∂x = -∂B/∂t -∂B/∂x = μ0ε0 ∂E/∂t and the wave equation: ∂2y/∂x2 = 1/v^2(∂2y/∂t2) The Attempt...
  16. J

    Graphing the Wave Equation with Quadratic Functions

    Homework Statement set \phi = f(x-t)+g(x+t) a) prove that \phisatisfies the wave equation : \frac{\partial^2 \phi}{\partial t^2} = \frac{\partial^2 \phi}{\partial x^2} b) sketch the graph of \phi against t and x if f(x)=x^2 and g(x)=0The Attempt at a Solution part a, I have already gotten the...
  17. L

    PDE (wave equation) used to find acoustic pressure in a a pipe

    Homework Statement Assume that the wavelength of acoustic waves in an organ pipe is long relative to the width of the pipe so that the acoustic waves are one-dimensional (they travel only lengthwise in the pipe). Therefore, the equation governing the pressure in the wave is: ∂2p/∂t2-c2*∂2p/∂x2...
  18. F

    Mobile Phone Tower Question - Wave Equation and No. of Photons Per Sec

    Homework Statement A mobile phone signal with a frequency of 1945Mhz is being broadcast from a transmitter with a peak output of 3kW. A: What part of the EM spectrum is the signal. Classify it in terms of its orientation of oscillation and propagation. B: Write a general equation for the...
  19. P

    Wave equation and fourier transformation

    Homework Statement utt=a2uxx Initial conditions: 1)When t=0,u=H,1<x<2 and u=0,x\notin(1<x<2) 2)When t=0,ut=H,3<x<3 and u=0,x\notin(3<x<4) The Attempt at a Solution So I transformed the first initial condition \hat{u}=1/\sqrt{2*\pi} \int Exp[-i*\lambda*x)*H dx=...
  20. S

    Deriving wave equation from Lagrangian density

    Hi, This is a worked example in the text I'm independently studying. I hope this isn't too much to ask, but I am stupidly having trouble understanding how one step leads to the other, so was hoping someone could give me a little more of an in-depth idea of the derivation. Thanks. Homework...
  21. xortdsc

    How to compute the energy of scalar wave equation

    Hi, considering the scalar wave equation $$ { \partial^2 u \over \partial t^2 } = c^2 \nabla^2 u $$ (where ∇^2 is the (spatial) Laplacian and where c is a fixed constant) how can I derive the potential and kinetic energy for a given state u and u' ? Thanks and cheers
  22. C

    Relationship between Simple Harmonic Motion Equation and Wave Equation

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  23. E

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  24. B

    Deriving wave equation with single string and small amplitude wave.

    1.This equation in the link below refers to the small angle approximation regarding deriving the wave equation from Newtons laws from small amplitude waves in a single string with fixed tension. 2.http://imgur.com/NGSwzcl 3. I'm a bit rusty on the maths and have no idea how these...
  25. X

    Wave equation and odd extension with constant IC

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  26. W

    Solving 2nd ODE and Multivariable Calculus for Wave Equation

    Hello guys, I would like to ask some questions regarding my coursework, which is about 2nd ODE and multivariable calculus. Since we have the one-dimensional wave equation and values for the string stretched between x=0 and L=2: 0≤x≤L, t≥0 The string is fixed at both ends so we have ...
  27. K

    Solution to homogeneous wave equation

    Homework Statement Prove by direct substitution that any twice differentiable function of (t-R\sqrt{με}) or of (t+R\sqrt{με}) is a solution of the homogeneous wave equation. Homework Equations Homogeneous wave equation = ∂2U/ ∂R2 - με ∂2U/∂t2 = 0 The Attempt at a Solution Could you...
  28. T

    Java Sine wave equation into Java Code

    I have been trying to implement this Wave equation into java: A = amplitude of wave L = wave length w = spatial angular frequency s = speed wt = temporal angular frequency d = direction FI = initiatory phase Y（x，y，t）=A*cos（w *（x，y）+ wt*t + FI; I...
  29. A

    Useful use of the wave equation

    Homework Statement Our teacher asked us to find a useful use of the wave equation, in which we must explain how the wave equation is used in the application we chose, in which will be a 2 minutes at max speech. Homework Equations Wave equation v=f*λ The Attempt at a Solution I...
  30. L

    Proving an exponential function obeys the wave equation

    Homework Statement Prove that y(x,t)=De^{-(Bx-Ct)^{2}} obeys the wave equation Homework Equations The wave equation: \frac{d^{2}y(x,t)}{dx^{2}}=\frac{1}{v^{2}}\frac{d^{2}y(x,t)}{dt^{2}} The Attempt at a Solution 1: y(x,t)=De^{-u^{2}}; \frac{du}{dx}=B; \frac{du}{dt}=-C 2...
  31. M

    Help with a wave equation derivation

    Hi, Apologises if I have submitted this issue into the wrong Math forum. However, I was wondering if anybody could help me with 2 steps in a derivation of an equation. Simply by way of background, the derivation is linked to formation of a superposition wave subject to a Doppler effect [1]...
  32. P

    D'Alembert's Solution to wave equation

    Hello, How does the change of variables ## \alpha = x + at , \quad \beta = x - at ## change the differential equation $$ a^2 \frac{ \partial ^2 y}{ \partial x^2 } = \frac{ \partial ^2 y} {\partial t ^2} $$ to $$ \frac{ \partial ^2 y}{\partial \alpha \partial \beta } = 0$$ ? I'm having a...
  33. F

    Wave Equation for a Vibrating String

    Homework Statement A string of length l has a zero initial velocity and a displacement y_{0}(x) as shown. (This initial displacement might be caused by stopping the string at the center and plucking half of it). Find the displacement as a function of x and t. See the following link for...
  34. A

    Multivariable Calculus Chain Rule Problem: Wave equation

    Homework Statement Show that any function of the form ##z = f(x + at) + g(x - at)## is a solution to the wave equation ##\frac {\partial^2 z} {\partial t^2} = a^2 \frac {\partial^2 z} {\partial x^2}## [Hint: Let u = x + at, v = x - at] 2. The attempt at a solution My problem with this is...
  35. F

    Solving 1D wave equation for a flag (not attatched at both ends)

    Hi all, I have the question: Consider a flag blowing in the wind. Assume the transverse wave propagating along the flag is one dimensional. Solve the wave equation for the wave on the flag, assuming the displacement of the flag is zero at the flag pole and the other end of the flag is...
  36. T

    How to Determine Parameters for a Gaussian Wave Pulse on a String?

    Homework Statement One end of a long horizontal string is attached to a wall, and the other end is passed over a pulley and attached to a mass M. The total mass of the string is M/100. A Gaussian wave pulse takes 0.12 s to travel from one end of the string to the other. Write down the...
  37. A

    How Does v^2 in the Wave Equation Represent Wave Speed?

    The wave on the string could be described with wave equation. Wave equation has a factor v^2 = Tension/linear density. It has dimensions of speed, but from where exactly does it follow that this is actually speed of propagation of the wave?
  38. P

    Solving a Wave Equation

    Homework Statement The attempt at a solution I'm using the method of separation of variables by first defining the solution as u(x,t) =X(x)T(t) Putting this back into the PDE I get: T''X = x^{2}X''T + xX'T which is simplified to \frac{T''}{T} = \frac{x^{2}X'' + xX'}{X} = -\lambda^{2} The...
  39. T

    Solving the Wave Equation (PDEs)

    Solve ##u_{xx} - 3u_{xt} - 4u_{tt} = 0##, ##u(x,0) = x^{2}##, ##u_{t}(x,0) = e^{x}##. (Hint: Factor the operator as we did for the wave equation.) (From Partial Differential Equations An Introduction, 2nd edition by Walter A. Strauss; pg. 38) This is the first of a set of three exercises on...
  40. ShayanJ

    The space of solutions of the classical wave equation

    Consider the classical wave equation in one dimension: \frac{\partial^2 \psi}{\partial x^2}=\frac{1}{v^2} \frac{\partial^2 \psi}{\partial t^2} It is a linear equation and so the set of its solutions forms a vector space and because this space is a function space,its dimensionality is...
  41. I

    Wave Equation applied to a spring with longitudinal waves?

    Homework Statement A spring of mass m, stiffness s and length L is stretched to a length L + l. When longitudinal waves propagate along the spring the equation of motion of a length dx may be written pdx second partial derivative of n with respect to t = partial derivative of F with respect to...
  42. R

    Solutions To Spherical Wave Equation

    If the solution to the electric part of the spherical wave equations is: E(r, t) = ( A/r)exp{i(k.r-ωt) What happens when t=0 and the waves originates at the origin, i.e. r=0 ... which I assume can't be right as you of course cannot divide by zero. Thanks!
  43. R

    Solutions To The Spherical Wave Equation

    If the solution to the electric part of the spherical wave equations is: E(r, t) = ( A/r)exp{i(k.r-ωt) What happens when t=0 and the waves originates at the origin, i.e. r=0 ... which I assume can't be right as you of course cannot divide by zero. Thanks!
  44. D

    Why does the wave equation support wave motion?

    If motion of an object obeys the wave equation, then it will display wave like behaviour. If you solve the wave equation, you get things like y = Asin \frac{2∏}{\lambda}(x - vt) which is a sinosodial wave. But from the second order differential equation v^{2}\frac{d^{2}y}{dx^{2}} =...
  45. fluidistic

    Wave equation invariance under Lorentz transform

    Homework Statement I must show that the one dimensional wave equation ##\frac{1}{c^2} \frac{\partial u}{\partial t^2}-\frac{\partial ^2 u}{\partial x^2}=0## is invariant under the Lorentz transformation ##t'=\gamma \left ( t-\frac{xv}{c^2} \right )## , ##x'=\gamma (x-vt)##Homework Equations...
  46. J

    One-dimensional wave equation with non-constant speed

    Homework Statement The cross-section of a long string (string along the x axis) is not constant, but it changes wit the coordinate x sinusoidally. Explore how a wave, caused with a short stroke, spreads through the string. Homework Equations Relevant is the one-dimensional wave...
  47. T

    PDE: Wave equation with first order derivative

    Homework Statement Solve using separation of variables utt = uxx+aux u(0,t)=u(1,t)=0 u(x,0)=f(x) ut=g(x) The Attempt at a Solution if not for the ux I'd set U=XT such that X''T=TX'' and using initial conditions get a solution. In my case I get T''X=T(aX'+X'') which is...
  48. Y

    D'Alembertian and wave equation.

    I am studying Coulomb and Lorentz gauge. Lorentz gauge help produce wave equation: \nabla^2 V-\mu_0\epsilon_0\frac{\partial^2V}{\partial t^2}=-\frac{\rho}{\epsilon_0},\;and\;\nabla^2 \vec A-\mu_0\epsilon_0\frac{\partial^2\vec A}{\partial t^2}=-\mu_0\vec J Where the 4 dimensional d'Alembertian...
  49. R

    PDE Wave equation with phi(x) as initial boundaries

    Homework problem: For the wave equation: Utt-Uxx=0, t>0, xER u(x,0)= 1, |x|<1 0, |x|>1 sketch the solution u as a function of x at t= 1/2, 1, 2, and 3 I am able to use d'Alemberts and solve for u however the boundaries and the odd/even reflections are throwing me off and...
  50. IridescentRain

    Solution to the scalar wave equation in cylindrical coordinates

    Hello. I don't know how to prove that a certain function is a solution to the scalar wave equation in cylindrical coordinates. The scalar wave equation is \left(\nabla^2+k^2\right)\,\phi(\vec{r})=0,which in cylindrical coordinates is...
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