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. 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...
  2. H

    What Is the Correct Progressive Wave Equation for a Particle at Origin?

    Homework Statement A wave moving in the positive Ox-direction has displacement of particle of 0 at the origin, O at time = 0.The displacement-distance graph showed a positive sine graph. Write an expression for the variation of the displacement y with time t for the particle at O...
  3. O

    What is the x>0 mean here? standing wave equation find nodes?

    Homework Statement Two waves travel in opposite direction form a standing wave of y= 2sin(31.42x)cos(7854t). Find the positions of first two nodes with x>0. Homework Equations The Attempt at a Solution The question didn't say whether the standing wave starts from node or...
  4. B

    Speed of wave from wave equation.

    Homework Statement A traveling pulse is given by f(x,t)=A{ e }^{ \frac { 2abxt-{ a }^{ 2 }{ x }^{ 2 }-{ b }^{ 2 }{ t }^{ 2 } }{ { c }^{ 2 } } } where A, a, b, c are positive constants of appropriate dimentions. The speed of pulse is: a) \frac { b }{ a } b) \frac { 2b }{ a } c) \frac { cb...
  5. E

    Longitudinal Wave Equation meaning and derivation

    Hi, I have recently been studying waves, and I understand the transversal wave formula y=Asin(w(t-x/v)) which gives the y coordinate of a point at x along the x-axis in the instant t. However, Wikipedia (http://en.wikipedia.org/wiki/Longitudinal_wave) gives this as the equation for...
  6. P

    Need help with argument about wave equation

    in the equation v=f /lambda can you use c instead of v for wavespeed?
  7. P

    Time dependent wave equation trouble

    Time dependent wave equation trouble! Homework Statement I'm having heaps of trouble getting my head around the time dependent wave function and the use of operators to find measurement/probabilities etc... I'm having trouble with something like the following, If have a 1D inf potential...
  8. S

    Method of separation of variables for wave equation

    Homework Statement $$u_{tt} = a^2u_{xx} , 0<x< l , t>0 , $$a is constant $$ u(x,0)=sinx , u_{t} (x,0) = cosx , 0<x< l , t>0 $$ $$ u(0,t)=2t , u(l,t)=t^2 , t>0 $$ Homework Equations The Attempt at a Solution I can solve the eigenvalue problem of X(x), and then solve for T(t), but...
  9. E

    Wave equation for a standing wave - clarification needed

    Both of the highlighted equations deal with a standing wave. However, they are slightly different in the sense that the latter has a phase shift in it. Why ? Also, how does one go from the latter equation for a standing wave to : 2A*sinkx*sinwt. For a string with both ends fixed And...
  10. D

    Solve Polar Wave Equation with Boundary Conditions

    Hi! I've been given the following problem to solve: Consider the azimuthally symmetric wave equation: \frac{∂2u}{∂t2} = \frac{c2}{r}\frac{∂}{∂r}(r\frac{∂u}{∂r}) where u(r,0)=f(r), ut(r,0)=g(r), u(0,t)=1 and u(L,0)=0. Use the separation of variables method to find the solution to this...
  11. E

    Light Interference Wave Equation - Assumption

    Homework Statement Homework Equations In question. The Attempt at a Solution To be clear it's part (vi) that's unclear to me. In order to ignore the cosine term it has to reduce to 1. This can happen, only if k(x1+x2)/2 = ωt Is this a correct assumption ? Also, it is known that k = 2∏/λ...
  12. E

    Wave Equation After Reflection

    What do the components of the following equation represent : http://www.mediafire.com/view/?0we6f9jkw26qi9o To be clear, this represents a wave of the form Acos(kx-wt) after being reflected off a wall. I understand that the ∅ represents the phase change of the wave after hitting the...
  13. E

    Standing waves - Wave Equation

    I don't completely understand how equation 4.4.4 was derived and determined. I understand the derivation behind the basic wave equation 4.3.4 but not what happened in 4.4.4. Why is there a need for all the negative signs ? Would a simple phase change suffice ? Please do be a bit detailed in...
  14. A

    Equation - Wave Equation Derivation Question

    equation -- Wave Equation Derivation Question Hello, my teacher says that if, on a wave equation f(x-ct)=f(e) then \partial_{ee}= \partial_{tt}- c^2 \partial_{xx} but i think that \partial_{t}=\frac{\partial }{\partial e} \frac{\partial e}{\partial t}=-c\frac{\partial }{\partial e} and...
  15. M

    Solution of wave equation, 2nd partial derivatives of time/position

    f(z,t)=\frac{A}{b(z-vt)^{2}+1}... \frac{\partial^{2} f(z,t)v^{2} }{\partial z^2}=\frac{-2Abv^{2}}{[b(z-vt)^{2}+1]^{2}}+\frac{8Ab^{2}v^{2}(z-vt)^{2}}{[b(z-vt)+1]^{3}}=\frac{\partial^2 f}{\partial t^2} \frac{-2Abv^{2}}{[b(z-vt)^{2}+1]^{2}}+\frac{8Ab^{2}v^{2}(z-vt)^{2}}{[b(z-vt)+1]^{3}} this...
  16. O

    Understanding the One Dimensional Wave Equation

    Homework Statement Given that the the One Dimensional wave equation is \frac{∂^{2}y(x,t)}{∂x^{2}} = \frac{1}{v^{2}} \frac{∂^{2}y(x,t)}{∂t^{2}} is y(x,t) = ln(b(x-vt)) a solution to the One Dimensional wave equation? Homework Equations Shown above. The Attempt at a Solution So my Professor...
  17. G

    Solving the Wave Equation in semi-infinite domain with easy ICs

    Hi, so the problem is this: I am trying to solve (analytically) the wave equation with c=1: u_{xx}=u_{tt} on x,t>0 given the initial conditions u(x,0)=u_{t}(x,0)=0, u(0,t)=sin(wt) I know how to solve on semi-infinite domains for quite a few cases using Green's Functions, Fourier Transforms...
  18. M

    PDE Wave Equation and Energy Conservation

    Homework Statement Just looking back through my notes and it looks like I'm missing some. Just a few questions. For one example in the notes I have the wave utt-c2uxx + u3 = 0 and that the energy density 1/2u2t + c2/2u2x + 1/4u4 I have that the differential form of energy conservation...
  19. S

    Can a Football Behave Like a Wave at the Same Speed as an Electron?

    i've just learned de broglie wave equation in chemistry which tells that matter can act as wave. if an electron is moving at a certain speed(v) at which its wavelength is comparably in meters. If a football is made to move at the same speed (v),will it behave as a wave? Since Football also has...
  20. J

    What is the Difference Between Sinusoidal and Cosinusoidal Wave Equations?

    Homework Statement This is from the Young and Freedman 13th Ed book, chapter 15 "Mechanical Waves. It's the same question in the book found in this link: https://www.physicsforums.com/showthread.php?t=557243 My problem though is much more fundamental I guess. In the part c, the manual...
  21. G

    Plane wave equation of linear polarization

    Question 1 Basically I have no idea how to calculate the z part of the equation since x and y are assumed to be propagating in the z direction.
  22. S

    Electromagnetic wave equation in Einstein Notation

    Hey! How to transform the equation \bigtriangleup\vec E=\operatorname{div}(\operatorname{grad}(\vec E))=\epsilon_0\cdot\mu_0\cdot\frac{\partial^2\vec E}{\partial t^2} in Einstein Notation? Thank you all for your help!
  23. S

    Best way to solve Schrodinger's wave equation numerically.

    I have been trying to research the best way to solve the Schrodinger wave equation numerically so that I can plot and animate it in Maple. I'd also like to animate as it is affected by a potential. I have been trying for weeks to do this and I don't feel any closer than when I started. I have...
  24. R

    Is f(x,t)=exp[-i(ax+bt)^2] a harmonic wave?

    Homework Statement Dear Guys, Does f(x,t)=exp[-i(ax+bt)^2] qualify as a harmonic waves? Please help! Manish Germany Homework Equations The Attempt at a Solution it is of the form g(ax+bt). which is the general form for harmonic wave. but what bothers me is the...
  25. R

    Solve Harmonic Wave Equation: Manish from Germany

    Dear Guys, Does f(x,t)=exp[-i(ax+bt)^2] qualify as a harmonic wave? Please help! Manish Germany
  26. C

    Transverse Wave equation for a string of changing length?

    I'm trying to learn more about the physics of guitars. I followed through the derivation of the transverse wave equation and that makes sense, but it seems like several of the simplifying assumptions might not apply. There are a lot of approximations with small angles and small slopes. I...
  27. B

    Electromagnetic wave equation not invariant under galilean trans.

    Homework Statement Prove that the electromagnetic wave equation:  (d^2ψ)/(dx^2) + (d^2ψ)/dy^2) + (d^2ψ)/(dz^2) − (1/c^2) * [(d^2ψ)/(dt^2)]= 0 is NOT invariant under Galilean transformation. (i.e., the equation does NOT have the same form for a moving observer moving at speed of...
  28. V

    How do I solve the Schrodinger wave equation with a differential equation?

    in solving the schrodinger wave equation, there arises this differential equation (d^2/dx^2) ψ + (1/x) (d/dx )ψ + (a/x)ψ + (b/x^2)ψ + cψ = 0 Please any leads on how to solve this equation will be highly appreciated.
  29. A

    Optics: Finding the wave equation given position and amplitude information

    A harmonic wave traveling in +x-direction has, at t = 0, a displacement of 13 units at x = 0 and a displacement of -7.5 units at x = 3λ/4. Write the equation for the wave at t = 0. Homework Equations The equation for a harmonic wave is r = asin(kx-vt+θ) a being the amplitude k...
  30. S

    What is the Wave Equation for a String?

    Homework Statement Hello all, stuck on a question involving a formula for a wave that doesn't make much sense to me. Assuming that a wave on a string is represented by: y(x,t) = y_i*sin((2∏/λ)(vt-x)) Where y is transverse displacement at time t of the piece of string at x. The...
  31. K

    D'alembert's solution to the wave equation, on Chain Rule

    Homework Statement Please have a look at the picture attach, which shows the proof of the D'alembert's solution to the wave equation. If you can't open the open, https://www.physicsforums.com/attachment.php?attachmentid=54937&stc=1&d=1358917223 click onto this...
  32. Y

    Verify about the solution of wave equation of potential.

    I read in the book regarding a point charge at the origin where Q(t)= \rho_{(t)}Δv'\;. The wave eq is. \nabla^2V-\mu\epsilon\frac{\partial^2 V}{\partial t^2}= -\frac {\rho_v}{\epsilon} For point charge at origin, spherical coordinates are used where: \nabla^2V=\frac 1 {R^2}\frac...
  33. S

    Question about the wave equation

    Hello, is it possible for one to assume a straight-line propagation of an e.m. wave and constant velocity c? If so, is it possible to simplify the wave equation utt=c2uxx by expressing the spatial variable x through the time variable t? x must be a function of t, since the motion is...
  34. P

    Solving the PDE Wave Equation - A_n & B_n Terms

    Hello, this is a problem I've been trying to do but I'm not sure it is right. Particularly the A_n and B_n terms. Thanks https://docs.google.com/open?id=0BwZLQ_me50B8M0sxelVrbTBhYVk
  35. Z

    Wave Equation (Fourier Coefficients)

    For the wave equation I managed to get the coefficient of f: a1=2 and the coefficient of g: \frac{12pi}{2pi*2}=B2 Is these answers right, since my B2 does not match the answer I was given. Thank you
  36. D

    Solving Damped Wave Equation with Initial Conditions

    $$ u_{tt} + 3u_t = u_{xx} $$ $$ u(0,t) = u(\pi,t) = 0 $$ $$ u(x,0) = 0\quad\text{and}\quad u_t(x,0) = 10. $$ \begin{alignat*}{3} u(x,t) & = & \exp\left[-\frac{3t}{2}\right]\sin x\left[A_1\cosh\frac{t\sqrt{5}}{2} + B_1\sinh\frac{t\sqrt{5}}{2}\right]\\ & + &...
  37. D

    MHB How to Solve Second Order Damped Wave Equation for PDEs: A Comprehensive Guide

    Does anyone know where I find second order damped wave equation worked where the overdamped, underdamped, and critically damped cases are all taken into account? I found resources where they throughout the overdamped and just focus on the underdamped.
  38. D

    How to Visualize the Wave Equation Solution in Mathematica?

    $$ u(x,t) = \frac{1}{2}\int_{x - t}^{x + t}g(s)ds = \begin{cases} t, & (x,t)\in R_1\\ \frac{1}{2}(1 - x + t), & (x,t)\in R_2\\ \frac{1}{2}(x + t + 1), & (x,t)\in R_3\\ 1, & (x,t)\in R_4\\ 0, & (x,t)\in R_5,R_6 \end{cases} $$ where \begin{alignat*}{3} R_1 & = & \{(x,t):-1 < x - t <...
  39. D

    Mathematica Wave Equation Plot with Mathematica: Solutions for Different Regions and Time

    $$ u(x,t) = \frac{1}{2}\int_{x - t}^{x + t}g(s)ds = \begin{cases} t, & (x,t)\in R_1\\ \frac{1}{2}(1 - x + t), & (x,t)\in R_2\\ \frac{1}{2}(x + t + 1), & (x,t)\in R_3\\ 1, & (x,t)\in R_4\\ 0, & (x,t)\in R_5,R_6 \end{cases} $$ where \begin{alignat*}{3} R_1 & = & \{(x,t):-1 < x - t < 1\text{ and }...
  40. D

    MHB Solution of the Damped Wave Equation under Certain Boundary Conditions

    $$ u_{tt} + 3u_t = u_{xx}\Rightarrow \varphi\psi'' + 3\varphi\psi' = \varphi''\psi. $$ $$ u(0,t) = u(\pi,t) = 0 $$ $$ u(x,0) = 0\quad\text{and}\quad u_t(x,0) = 10 $$ \[\varphi(x) = A\cos kx + B\sin kx\\\] \begin{alignat*}{3} \psi(t) & = & C\exp\left(-\frac{3t}{2}\right)\exp\left[t\frac{\sqrt{9...
  41. E

    How Do You Derive u=p/ρ₀c₀ and ρ=p/c₀² from 1D Wave Equations?

    Given The 1D wave equations p_{x}'' - (1/c_{0}^2)p_{t}'' = 0 u_{x}'' - (1/c_{0}^2)u_{t}'' = 0 ρ_{x}'' - (1/c_{0}^2)ρ_{t}'' = 0 and linearised continuity and momentum equations ρ_{t}' = -ρ_{0}u_{x}', ρ_{0}u_{t}'=-p_{x} how may one derive the following two equations? u=p/ρ_{0}c_{0}...
  42. D

    MHB Wave equation soln check and plot question

    \begin{alignat*}{3} u_{tt} & = & c^2u_{xx}\\ u(0,t) & = & 0\\ u_x(L,t) & = & 0\\ u(x,0) & = & \frac{x}{L}\\ u_t(x,0) & = & 0 \end{alignat*} Let's start with $u_t(x,0) = 0$. Then $$ u_t(x,0) = \sum_{n = 1}^{\infty}B_n\frac{\pi c}{L}\left(n + \frac{1}{2}\right)\sin\left[\frac{\pi x}{L}\left(n +...
  43. A

    Does the wave equation have unique solutions?

    Well title says it all pretty much. My question is if one set of boundary conditions uniquely specifies the solutions to the wave equation. My speculation comes from the fact that my book introduces electromagnetic in a bit weird way I think. It shows how Maxwells equations lead to the wave...
  44. D

    Heat equation and Wave equation problems

    1. Solve the Heat equation u_t = ku_xx for 0 < x < ∏, t > 0 with the initial condition u(x, 0) = 1 + 2sinx and the boundary conditions u(0, t) = u(∏, t) = 1 (Notice that the boundary condition is not homogeneous) 3. Find the solution of the Wave equation u_tt = 4 u_xx with u(0...
  45. M

    Solving boundary value problem (Wave Equation)

    Homework Statement Show that the boundary-value problem $$u_{tt}=u_{xx}\qquad u(x,0)=2f(x)\qquad u_t(x,0)=2g(x)$$ has the solution $$u(x,t)=f(x+t)+f(x-t)+G(x+t)-G(x-t)$$ where ##G## is an antiderivative/indefinite integral of ##g##. Here, we assume that ##-\infty<x<\infty## and ##t\geq 0##...
  46. O

    D'Alembert solution of wave equation with initial velocity given

    Hi there, This is a problem concerning hyperbolic type partial differential equations. Currently I am studying the book of S. J. Farlow "Partial differential equations for scientists and engineers". The attached pages show my problems. Fig. 18.4 from case two (which starts in the lower part...
  47. G

    Solutions to 2D wave equation using 1D equation solution.

    http://imageshack.us/a/img824/1121/asdasdaw.png I am having trouble completely understanding what the question wants. I know it is quite clear but the part I am having trouble is the following. It says 'pretend' w(x,t) is a solution to the 2D equation, just independent of y, then to...
  48. S

    Understanding the 1D Wave Equation for Free Particles in Quantum Mechanics

    I was studying a book on QM and found out that the wave function for a free particle of completely undetermined position traveling in positive x direction is given by e^(2(pi)i(kx - nt)) where n is the frequency i have been trying a lot to derive it but till now i can't . Can anyone help me
  49. C

    Simple 2-D Wave Equation Problem

    Homework Statement Solve the boundary value problem \frac{\partial ^2 u}{\partial t^2} = c^2 (\frac{\partial ^2 u}{\partial x^2}+\frac{\partial ^2 u}{\partial y^2}), 0<x<a, 0<y<b, and t>0 for the boundary conditions u(0,y,t) = 0 and u(a,y,t) = 0 for 0 \leq y \leq b and t\geq0...
  50. N

    What is the purpose of the Schrodinger wave equation in quantum mechanics?

    I am new to quantum mechanics and trying to combine the pieces. If I am looking into the quartum world, first I prepare a mechanism with which i can bring the properties and behavior of the particles i.e. an experiment to study them, but the information i emphasize to look on in the experiment...
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