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. W

    Inhomogeneous Electromagnetic Wave Equation

    Homework Statement Consider a medium where \vec{J_f} = 0 and {\rho_f}=0, but there is a polarization \vec{P}(\vec{r},t). This polarization is a given function, and not simply proportional to the electric field. Starting from Maxwell's macroscopic equations, show that the electric field in...
  2. P

    Solving Wave Equation / Imaginary Numbers

    Homework Statement Consider the simplified wave function: \psi (x,t) = Ae^{i(\omega t - kx)} Assume that \omega and \nu are complex quantities and that k is real: \omega = \alpha + i\beta \nu = u + i\omega Use the fact that k^2 = \frac{\omega^2}{\nu^2} to obtain expressions for \alpha and...
  3. P

    Wave Equation / Damping / Phase Velocity

    Homework Statement Consider the simplified wave function: \psi (x,t) = Ae^{i(\omega t - kx)} Assume that \omega and \nu are complex quantities and that k is real: \omega = \alpha + i\beta \nu = u + i\omega Show that the wave is damped in time. Use the fact that k^2 =...
  4. tandoorichicken

    Is This General Solution Valid for the Wave Equation?

    Hello everyone. I'm asked in a problem to prove that a given general solution is valid for the wave equation \nabla^2 p - \frac{1}{c_0^2} \frac{\partial^2 p}{\partial t^2} = 0 . The given solution was p(x, t) = A_1 f_1 (x - c_0 t) + A_2 f_2 (x + c_0 t). I just need a check of work here. I...
  5. S

    Wave equation and graphing wave instance

    If the end of a string is given a single shake, a wave pulse propagates down the string. A particular wave pulse is described by the function y(x,t) = (A^3/(A^2 + (x - vt)^2)) where A = 1.00 cm, and v = 20.0 m/s. a) Sketch the pulse as a function of x at t = 0. How far along the string...
  6. S

    How is Wave Propagation Time Calculated for a Rope in a Mine Shaft?

    Hello, I have a couple of questions about my assignment. Here are the assigned questions plus my attempts. 1) One end of a rope is tied to a stationary support at the top of a vertical mine shaft 80.0 m deep. The rope is stretched taut by a box of mineral samples with a mass of 20.0 kg...
  7. S

    How Do You Solve the Wave Equation on a Half-Line?

    lol my head is about to explode! :P i think this is similar to a previous question i asked but i can't quite get it none the less... http://img137.imageshack.us/img137/7796/picture11gf9.png now what i did was to following this ...
  8. S

    Solving the Wave Equation PDE: A General Solution Approach

    Hi everyone, I'm having a bit of trouble with this pde problem: http://img243.imageshack.us/img243/9313/picture3ui3.png i get the answer to be u(x,t)=0 but i am guessing that's not right. is the general solution to this problem: u(x,t) = f(x+ct) + g(x-ct) ?? thanks sarah :)
  9. B

    Number of Positive Solutions for Omega?

    Could someone help me out on the following questions? Q. Consider the free vibrations of a string of length L clamped at x = 0 and constrained at x = L such that u_x \left( {L,t} \right) = - ku\left( {L,t} \right),k > 0. (a) Show that the eigenvalues are given by the positive roots of...
  10. I

    How to Solve PDE Problems Involving Wave Equation

    hello, can you guys give me a good resource(websites, etc) on how to solve this type of problem? The thing is, I'm not sure what methods are appropriate for solving this problem. I believe this is a PDE problem involving the Wave equation, but I don't know how to start. I would like to say...
  11. C

    Series of Wave Equation problem

    A series of waves traveling at 200m/sec are being generated by a 50hz source. A point at the very top of the crest of a certain wave is ? meters away from a corresponding point 4 crests away. The only equation I have is V=lambda*F. Is there a way to get the distance w/ this or am I missing...
  12. T

    Wave equation, D'Alembert's Solution

    I am having trouble understanding the solution to the wave equation: this is thought of as the final solution to the PDE: but I see that: is a solution to the function. But what I don't get is why D'Alembert's Solution isn't in terms of sines and cosines like that solution right above...
  13. T

    Deriving the 3D Wave Equation: A Comprehensive Explanation

    I would like to know how to derive the wave equation for a 3 dimensional case, I was looking it up on wikipedia, and their explination wasn't very comprehensive, I was wondering if anyone knew of any other website that would be able to let me fullly understand it. Edit: not the shrodinger wave...
  14. Amith2006

    Is y = a[log(x - vt)] a Traveling Wave? Understanding the General Wave Equation

    Sir, Does the equation, y = a[log(x – vt)] Represent a traveling wave? I think the answer is No, because it is not a harmonic function. Is it right?
  15. Amith2006

    What Is the Equation of a Wave Reflected from a Rigid Boundary?

    Sir, Consider a progressive wave represented by the equation, y = A[sin(wt – kx)] If it is reflected from a wall, what will probably be the equation of the reflected wave? I think it is y = A1[sin(wt + kx)] Is it...
  16. L

    Using a Fourier transform on the wave equation

    Hi, I want to know how to get rid of the time part of the homogeneous wave equation: \newcommand{\pd}[3]{ \frac{ \partial^{#3}{#1} }{ \partial {#2}^{#3} } } \nabla^2\psi-c^{-2}\pd{\psi}{t}{2} = 0 I've read that this can be done using a Fourier transform, with the following given as the...
  17. F

    How to put magnetic field in the wave equation

    Could someone please help, i think this is gauss's law but I am not sure how to answer it as they give me a wave equation and i don't know how to put magnetic field in as well. Help would be much appreciated Question suppose that an appropriate device is used to genereate an...
  18. Y

    Exploring the Role of the Wave Equation in Describing Wind and Particle Movement

    I believe the PDE Utt-v^2*(Uxx+Uyy+Uzz) can be used to desribe air pressure, and how this pressure changes during time (with a given pressure in t=0) My question is, does this equation also takes into account effects of movement of particles? Such as wind? Can it describe wind? Or do we need...
  19. M

    Exploring the Effects of Changing the Power in the Wave Equation

    What happens when you make the x varible in the wave egn to some power m(dx/dt) + k x^n =0 What happens when n increases/decreases?
  20. M

    Calculating Frequency and Wavelength of a Sinusoidal Wave on a String

    Hello everyone! I'm having troubles understanding why I'm not getting this right...the problem is: A sinusoidal wave is traveling on a string with speed 10. cm/s. The displacement of the particles of the string at x = 25 cm is found to vary with time according to the equation y = (5.0 cm)...
  21. V

    Mechanical Waves & Wave Equation

    Hi, I'm having a little bit of difficulty understanding exactly what to do to get to an answer in section a of this problem. It asks to show that the given function satisfies the wave equation... I have the wave equation. How do I go about 'showing' that it satisfies the wave equation? Do I...
  22. I

    How Do You Calculate Wave Speed Using the Wave Equation?

    Wave Equation problem..! Help...! Hi can anyone solve this?? I couldn't figure out how to use it Question Use the wave equation to find the speed of wave given by y(x,t)=(3.00 mm) sin [(4.00/m)x-(7.0/s)t) ] I guess the wave equation is , (d^2y) 1 (d^2y)...
  23. S

    Solving 1D Wave Equation w/ Initial Values

    given the initial boundary value problem u_{tt} - u_{xx} = 0 , 0<x<1, t>0, u(x,0) =1 , 0\leq x \leq 1 , u_{t}(x,0) = \sin(\pi x) , 0\leq x \leq 1 , u_{x} (0,t) = 0 , t \geq 0 , u_{x} (1,t) = 0 . find u(0.5,1). Where u is d'Alembert's solution for the 1D wave equation well f(x)...
  24. P

    Wave Equation: U is Amplitude Vector?

    In the wave equation, what is u, is it the amplitude? If it is, how can it be a vector?
  25. E

    Solving 1D Wave Equation PDE with f(x), g(x) and nL<=x<(n+1)L

    I have formula for 1D wave equation: (*) u(x, t) = 1/2 [ f(x + ct) + f(x - ct) ] + 1 / (2c) Integral( g(s), wrt s, from x-ct to x+ct ) I am trying to find u(1/2, 3/2) when L = 1, c = 1, f(x) = 0, g(x) = x(1 - x). However, for (*) to work, the initial position f(x) and initial...
  26. T

    Vibrations and the wave equation

    An infinite string vibrates according to the homogenenous wave equation u_{tt}-u_{xx} = 0 with initial data given by u(x, 0) =f(x) and u_{t}(x, 0) = g(x) for -infinity<x<infinity where both f and g are smooth functions that are positive on the intervals -4<x<-3 and 2<x<3 and both zero...
  27. S

    Solving Wave Equation with Elastic Rods: u(x,t)

    A rod of uniform elastic material of length 1/2 lies along the X axis with its left end fixed at x=0. At time t=0, an identical rod hits the riht end of the first rod with a speed of v. The second rod is thereafter kept alongside the first rod, and neither end is fixed. If the Young's Modulus...
  28. S

    Can the Free Particle Wave Equation Be Solved for Periodic Boundary Conditions?

    for a free particle, the wave equation is a superposition of plane waves, \psi(x,0)= \int_{-\infty}^{\infty}g(k)\exp(ikx)dk and g(k)= \int_{-\infty}^{\infty}\psi(0,0)\exp(-ikx)dx one is the Fourier transform of the other. some cases to solve this is when we assume a small delta k, so...
  29. D

    Question about Wave equation for light in nonflat spaces

    Please correct me if I am wrong. Solutions to the linear wave equation: \large\frac{\partial \Psi}{\partial t} = \frac{1}{c^{2}}\frac{\partial \Psi}{\partial x^{i}} are sinusoidal waves of constant wavelength, i.e. they describe light traveling in a flat space. But when light...
  30. C

    Resonance pde wave equation u(\phi,t) involving lagrange polynomials

    1/sin(phi) * d/d\phi(sin(phi) * du/d\phi) - d^2u/dt^2 = -sin 2t for 0<\phi < pi, 0<t<\inf Init. conditions: u(\phi,0) = 0 du(\phi,0)/dt = 0 for 0<\phi<pi How do I solve this problem and show if it exhibits resonance? the natural frequencies are w = w_n = sqrt(/\_n) =2...
  31. quasar987

    Wave Equation: Exploring Motion w/ Angles Near 90°

    If you remember, when in textbooks* they derive the wave equation by considering a small element of string and applying Newton's 2nd law on it, they make the assumption that the angles the tension makes at the two ends of the element with the horizontal is smallish, such that sin ~ tan. Without...
  32. quasar987

    Periodic Wave Solutions of the Wave Equation

    Every "spacially periodic" function [i.e. s.t there exist P s.t. f(x+P,t) = f(x,t)] of the form f(x,t) = X(x)cos(wt) is a solution of the wave equation.
  33. C

    Adding a constant V to the wave equation

    The question says to show that the wave function picks up a time-dependent phase factor, e^\left(-i V_0 t / \hbar \right) , when you add a constant V_0 to the potential energy. And then it asks what effect does this have on the expecation value of a dynamical variable? Assuming I only...
  34. H

    Verifying Answer to Wave Equation and Associated B Field

    Hi, I have the following question on my problem sheet, and I just want to check that my answer to it is correct as I need to use the result in a later problem. If someone could confirm it is correct, or point out mistakes/erros that would be great. ======= Q. Derive the wave equation for E...
  35. S

    The Wave Equation IS the electron

    "The Wave Equation IS the electron" Hello everyone, I've heard it said that well, "the Wave Equation IS the electron". Can anyone explain this to me? I know what the Wave Equation is (even solved a few of them) and have a degree in Chemistry but I probably would not be able to follow...
  36. M

    Why does F'(x-ct) equal Dy/Dx in the wave equation for a string?

    Assume the well known PDE of an infinite length string D^2(y)/Dt^2 = c^2* ( D^2(y)/Dx^2) where y=y(x,t) is the transverse displacement of the string. D/Dx= partial derivative with respect to x D/Dt= partial derivative with respect to t c= velocity of the wave According to Morse and...
  37. F

    How Does Magnetic Flux Density B Relate to E and H in Electromagnetic Waves?

    Hi, We were told to show that the magnetic flux density B obeys a homogenous wave equation. This case applies to electromagnetic waves in a homogenous, linear, uncharged conductor. Now I know that the wave equation for magnetic flux density is as follows. [ tex ] \nabla^2-\epsilon\mju \frac...
  38. L

    Is f(x,t)=Acos(K(x-vt)+phi) a wave equation solution?

    Can someone show me that f(x, t) = A\cos(K(x-vt) + \phi) is in fact a solution of the wave equation? I kind of know how to show it by using calculus, but is there other way to show it? Thank you very much!
  39. P

    Proof of 1D Wave Equation: Simplified Explanation

    can anyone give me the most simple for someone with little diffrential equations back ground, on the proof for where the 1D wave equation was developed, i mean what's the proof for it. The 1D wave equation is a partial diffrential equation \frac{\partial^2\psi}{\partial x^2} =...
  40. C

    What are the questions surrounding the De Broglie wave equation and its proof?

    Hi, I recently came across a proof for the De Broglie wave equation in a book, which went as follows: E of photon = mc2 = m*c*c = (m*c)c = (p)c ( ie - momentum*speed of light) = (p)(f*lamda) Therefore, hf =...
  41. maverick280857

    Wave Equation for the Helium Atom/No of terms = 2 or 3?

    Hi friends... Sometime back, I encountered the Self Consistent Field Method in Quantum Mechanics, which is used to compute wave functions in complex atoms. The book I read this from is "Practical Inorganic Chemistry" by Clyde and Day. The method is explained through an argument about the...
  42. S

    Does the Schrodinger equation account for the characteristic smell of a soap?

    according to quantum mechanics there are many possiblities for a anything to happen...for example if there is a soap on the table..it exists only when we see it...only when we 'actualize' the wave-function...but what about the characteristic smell of the soap doesn't that make it exist? Does...
  43. U

    How Do You Solve the 1D Wave Equation with Gravity and Nonhomogeneous Terms?

    I have a wave equation Ytt=c^2 Yxx - g where g is a constant. The boundary conditions are Y(0,t)=Y(L,t)=0 with initial conditions Y(x,0)=0 and Yt(x,0)=0 I tried to solve it by Laplace transfoming the PDE in time and everything worked fine until I got to the point where I had to inverse the...
  44. W

    How Was the Schrödinger Wave Equation Derived?

    Hello, everyone I am new to this forum, I hope I am posting this at the right place. I am in my first year of college at Concordia University, and taking chemistry right now. But my main interest is physics. So when we were learning about the equation, I wanted to know more detail information on...
  45. E

    The History and Development of Schrodinger's Wave Equation

    "The History and Development of Schrodinger's Wave Equation" i am taking a class in "the history of chemistry" and i chose to write a paper on "the history and development of schrodinger's wave equation"... does anybody have any suggestions on what kind of topics i should include?
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