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. J

    What exactly does 'x-vt' mean in the wave equation?

    So the equation is f(x,t) = f(x-vt,0) = A sine k(x-vt). Now Since x is displacement of wave from origin, t is time taken for that,wouldn't 'x-vt' always be equal to 0 ? If x-vt is 0, Y = A sine k(x-vt) will also be zero. So am I writing the equation wrong?
  2. C

    Question regarding the derivation of the wave equation

    Homework Statement There's a derivation here that I'm looking at, and I've hit a snag. At (1) about 15 lines down the page, the author divides by Δx and takes the limit as Δx goes to 0. I understand what he did on the right side of the equation, but on the left side of the equation, by what...
  3. V

    Understanding the Standing Wave Equation for Students

    Homework Statement Homework Equations y (x,y) = 2YmSin(kx)Cos(wt)The Attempt at a Solution I am having trouble at setting up the standing wave equation for this problem. Once I set up the equation, I know that part a b c d is just plugging in the numbers. From what I learned, I know that...
  4. J

    Reflection of wave at open end (boundary condition)

    Hello! I need to understand one seemingly simple thing in wave mechanics, so any help is much appreciated! When a pulse travels to the right toward an open end(like a massless ring that is free to oscillate only in the vertical direction), then when the wave reaches the end it gets reflected and...
  5. thegirl

    Why is the wave equation a second order differential?

    I don't know if this is a silly question? Am I missing simple math? How does a wave depending on amplitude and frequency make it's equation a second order differential equation?
  6. S

    How Does Tsunami Wavelength Affect Particle Displacement and Wave Equation?

    Homework Statement Wavelength of a tsunami: Propagates toward shore as a sinusoidal plane wave. - determine the peak value of the horizontal particle displacement. Is your answer consistent with the assumption that the water momentum is mainly horizontal? Explain by considering the...
  7. Ben.P

    Using The Wave Equation To Solve A Question

    Hi, In class a few days ago, my teacher gave me a challenge question which they wanted me to answer. I have yet to succeed in finding the correct answer so I would be grateful for anybody who might be able to explain where I am going wrong with this problem! Homework Statement Here is the...
  8. L

    What is the condition for a particle to go into a square potential well?

    Hello everyone I am searching for the answer for the condition (related to the total energy of the particle E) for which any particle will go into the square potential well. I have studied Griffiths's quantum mechanics book Introduction to the quantum mechanics Section 2.5 and 2.6) but still...
  9. D

    Fourier series solution of wave equation

    Homework Statement Suppose a horizontally stretched string is heavy enough for the effects of gravity to be significant, so that the wave equation must be replaced by ##u_{tt} = c^2u_{xx} - g## where ##g## is the acceleration due to gravity. The boundary conditions are ##u(0,t) = u(l,t) = 0##...
  10. J

    Dirac Relativistic Wave Equation

    I would like people's opinions on why the negative energy solutions of Dirac's Relativistic Wave equation were simply ignored in 1934 to make things fit. Another related question is with the energy conservation laws as they stand. Why in pair production from a photon at 1.022MeV forming a...
  11. Priyadarshini

    What is the Significance of Schrodinger's Wave Equation in Quantum Mechanics?

    Why would one need to use the Schrodinger's wave equation? What does it give us? I understand that we solve for psi and stuff, but graphically what does it give us, in practical and not theoretical terms? What exactly is a wave function? After solving for psi with the help of the equation, we...
  12. praveena

    Is the One Dimensional Wave Equation Applicable to a String Oscillating in Time?

    Hai PF, I had a doubt in the sector of partial differential equation using one dimensional wave equation. Actually the problems is below mentioned :smile: A string is stretched and fastened at two points x=0 and x=2l apart. motion is strated by displacing the string in the form...
  13. R

    Show that the function is a solution of the wave equation

    Homework Statement Show that the function is a solution of the wave equation utt = a2uxx. u = (t)/(a2t2-x2) Homework Equations Quotient rule (f/g)' = (g*f ' - f*g') / g2 The Attempt at a Solution I began with the first and second partials of u with respect to x...
  14. E

    Hyperbolic partial differential equation

    What is the general solution of the following hyperbolic partial differential equation: The head (h) at a specified distance (x) is a sort of a damping function in the form: Where, a, b, c and d are constants. And the derivatives are with respect to t (time) and x (distance). Thanks in advance.
  15. SuperSpasm

    Why do two textbooks have conflicting equations for the harmonic wave equation?

    Homework Statement I'm using a couple different textbooks on waves, and it seems they've possibly contradicted one another. I think the problem may be that one deduced the equation for displacement in a transverse harmonic wave y(x,t) through the equation for displacement at t=0 [y(x,0)] and...
  16. V

    Exponential Solution to Wave Equation

    I just learned how to derive the wave equation and now I have some questions about it. In my physics text (first year) it simply says (without reason) that the solution to the wave equation is y(x,t) = Acos(kx-wt), where A is the amplitude of displacement, k is the wave number and w is the...
  17. B

    How can the wave equation be rearranged to include r?

    Hello! The wave equation given: {1\over{c^2}} {\partial^2 \phi\over{\partial t^2}} = \Delta \phi with r = \sqrt{x^2+y^2+z^2} needs to be rearranged, so that {1\over{c^2}} {\partial^2 ( r \phi) \over{\partial t^2}} = {\partial^2 (r \phi) \over{\partial r^2}} . Are there any tricks to obtain...
  18. S

    How to Determine Pulse Time in a Non-Uniform String?

    Homework Statement a non uniform string of length L and mass M, has a variable linear density given by μ=kx where k is the distance measured from one side of the string and k is a costant. a) find that M=(kL2)/2 b) show that the time t required to a pulse generated from one side of the string...
  19. S

    How do we justify the generality of the wave equation?

    We derived the wave equation by applying f=ma to an element of an oscillating string, yielding ∂2y/∂x2 = 1/v2 ⋅ ∂2y/∂t2. In order to get this result it was necessary to assume that the string in question was nearly horizontal, so that the angle formed by the tension vector and the horizontal...
  20. A

    Wave function for transverse waves on a rope

    Homework Statement Serway's Physics for Sciencetists and Engineers with Modern Physics, 9th Edition (current), Chapter 16, problem 19:[/B] (a) Write the expression for y as a function of x and t in SI units for a sinusoidal wave traveling along a rope in the negative x direction with the...
  21. Shahab Mirza

    How to interpret this Wave Equation (Derivation) Help ?

    Hi dear people , Hello I waw studying super position of two Sound Waves , traveling in same medium with same frequency , same wavelength and same amplitude while differing in phase . quick derivation : Wave 1 displacement y1= A sin (kx-vt ) and wave 2 displacement y2= A sin (kx-vt-phase...
  22. Coffee_

    Causality of the wave equation

    Consider the wave equation ##\nabla^{2} f - \frac{1}{c^{2}} \frac{\partial ^{2} f}{\partial t^{2}} = \delta(r) \delta(t) ## where there is no wave before ##t=0## The solution will be something up to a constants like ##f=\frac{\delta(r-ct)}{r}##. So we have a dirac delta function that...
  23. T

    Gas Dynamic to Acoustic wave equation

    Homework Statement Derive from the formulas ##\frac{D^\pm}{Dt}(u \pm F) = 0## where ##\frac{D^\pm}{Dt} = \frac{\partial}{dt} + ( u \pm c) \frac{\partial}{\partial x}## the one-dimensional wave equation in the acoustical limit. \begin{cases} u << c\\ c \approx c0 = const\\ F =...
  24. R

    How to compute phase of the signal?

    Suppose you are given a phase spectrum or (/and) equation of the (main) signal only and you are said that the given (main) signal is formed of 3 other signals. Is it possible to compute phases of these three signals from the equation or (/ and) phase spectrum of the (main) signal? Also,what...
  25. squelch

    Bra-Ket Notation, Wave Equation, Particle States

    Homework Statement A particle is in the state |\psi \rangle = \frac{1}{{\sqrt 3 }}|U\rangle + \frac{{a\sqrt {(2)} }}{{\sqrt {(3)} }}i|D\rangle . The up state |U\rangle = \left( {\begin{array}{*{20}{c}} 1\\ 0 \end{array}} \right) and the down state |D\rangle = \left(...
  26. Ercross

    How the mathematical wave equation was derived

    Please, I've been unable to comprehend how the mathematical wave equation was derived.. Please can anyone explain it to me...
  27. C

    General wave equation conceptual questions

    Homework Statement The general wave equation can be shown as: y(x,t) = ymsin(kx-ωt) Homework Equations See above The Attempt at a Solution My question relates to the variables present in this equation. I understand what the amplitude is, its the magnitude of the maximum displacement of...
  28. bananabandana

    Longitudinal Wave Equation from Transverse One

    Homework Statement Please see attached. Part ii) Homework EquationsThe Attempt at a Solution So I try to conserve volume as it suggests in the hint. I take the initial volume of the region to be given by: $$ h \times \delta x \times l = (\delta x + \eta) (h+\Psi) l $$ Where l is just some...
  29. P

    General solution for a standing wave to the wave equation.

    Homework Statement If ##u(x,t)\in \mathbb R, x\in[-\pi,\pi] ## represents a standing wave with ##u(\pm\pi,t)=0## Then what is the most general solution u(x,t)? Homework EquationsThe Attempt at a Solution [/B] Using the separation of variable technique: u(x,t)=P(x)Q(t) I get ##P(x)...
  30. M

    How Was the Equation v=fλ Derived in Wave Physics?

    Hi, We were recently testing out waves on a string and completing a lab based on it, and I wanted to provide background information on how the equation v=fλ was formulated. I would greatly appreciate it if someone could direct me to a source on it. Thanks.
  31. R

    Wave dispersion and the bandwidth theorem

    Homework Statement Consider a propagating wave packet with initial length L0. Use the bandwidth theorem to show that the minimum range of angular frequencies present in the wave packet is approximately: \begin{equation} \Delta \omega = \frac{v_{g}}{L_{0}} \end{equation} where vg is the group...
  32. R

    Quantum Mechanics: Wave Equation Probability

    Homework Statement Normalize the wave function $$ \langle x|\psi\rangle = \left\{ \begin{array}{l l} Ne^{-kx} & \quad x>0\\ Ne^{kx} & \quad x<0 \end{array} \right..$$ Determine the probability that a measurement of the momentum p finds the momentum between ##p## and ##p + dp## for this wave...
  33. K

    Dependence of Phase Velocity on Wavelength

    Homework Statement This is Problem 7.6 from Electronic Properties of Engineering Materials by Livingston. "Over a wide range of frequencies, the dielectric constant of a polymer is found to be proportional to the inverse square root of frequency. (a) How does the phase velocity of EM-waves...
  34. rakeru

    Finding Constants to Solution Given 2nd Order, Nonlinear DE

    Homework Statement Okay, here's the deal:I have been given a second order nonlinear differential equation, and I have also been given the general solution with constants A and B. I am supposed to find the constants A and B. The solution represents a fermion at rest, since the solution does not...
  35. M

    Why is the general form of the wave equation a second order partial derivative?

    When I deduct the the general form of wave equation, I noticed it has a second order partial derivative form. I am wondering why wave equation has a second order partial derivative form nor a first order partial derivative form?
  36. N

    Wave equation given a cosmological inflationary metric

    Hi everybody! Can you explain me how I can obtain wave equation given a metric? For example, if I have this metric $$g_{μν}=diag(−e^{2a(t)},e^{2b(t)},e^{2b(t)},e^{2b(t)})$$, how can derive the relation $$\frac{1}{\sqrt{g}}\partial _t(g^{00}\sqrt{g}\partial _t...
  37. V

    Basic question on wave equation - need a reminder

    Greetings all and new to the forum here. It's been many years and I've forgotten how to do it, and it should be a basic question, but assuming we have an equation Ex=E_0*cos(wt-kz), how do we translate to sine? I've seen it written sin(kz-wt) or sin(wt-kz), but I've just plainly forgotten how...
  38. E

    Help on the proof related to the vectorial wave equation

    Dear forum users, I need some help on the following proof that appears in a book (pp. 84 in Bohren' Absorption and Scattering of light by Small Particles). This is no a home work problem. The problem statement: \vec{M} = \nabla\times\vec{c}\psi, where \vec{c} is some constant vector and \psi...
  39. A

    Derivation of wave equation and wave speed

    Hi, I'm trying to wrap my head around the derivation of the wave equation and wave speed. For starters I'm looking at the derviation done on this site: http://www.animations.physics.unsw.edu.au/jw/wave_equation_speed.htm I could maybe explain what I understand at this point Given a string with...
  40. U

    Energy on the RHS of Schrodinger wave equation

    Hi, So in the steady state Schrodinger equation (SE), the E on the RHS (see http://scienceworld.wolfram.com/physics/SchroedingerEquation.html) is the sum of the kinetic and potential energy of the electron. However, is this the same as the 'internal energy'? The statistical distribution used...
  41. P

    Derivation of the wave equation satisfied by E and B

    Homework Statement given a medium in which p=0, j=0 but where the polarization vector P=P(r,t). Derive the wave equation satisfied by E and B.Homework Equations i started with the 4 basic Maxwells equations ∇ · D = ρ (1) ∇ · B = 0 (2) ∇ × E = −∂B/∂t (3) ∇ × H = J + ∂D/∂t (4) and with the...
  42. rumborak

    Boundary absorption when simulating wave equation

    I wrote a wave equation simulation in C# a while ago, and while everything works fine, I am running into the expected problem that my simulation boundaries (ie the edges of the grid) reflect the waves coming to them. Obviously I want to keep the grid of reasonable size, so I looked into what...
  43. J

    Sketches for t=0.5, t=1, and t=1.25:for t=0.5for t=1for t=1.25

    Homework Statement The question is to sketch for values of t = 0, t =0.25, t = 0.5, t=1 and t = 1.25 for the wave equation Ytt = Yxx in the infinite domain with initial conditions y(x,0) = f(x) = \begin{cases} 1 & \text{if } 0 ≤ x ≤ 1 \\ & 0 (otherwise) \end{cases} Yt(x,0) = 0 Homework...
  44. J

    MATLAB Help please in matlab -- plotting a D'Alembert wave equation

    Can someone please help me in plotting a D'Alembert wave equation solution in MATLAB? I am so confused as how to plot it in MATLAB I need to plot a graph like the one below
  45. J

    How Do You Sketch the Solution of a Wave Equation with Given Initial Conditions?

    Homework Statement Ytt = 1 Yxx with initial conditions of yT(x,0) = 0 y(x,0) = \begin{cases} 1 & \text{if } x \geq 0 \ & \text{if } x \leq 1 \\ 0 & \text{if } otherwise \end{cases} Sketch the solution of this wave equation for 5 representative values of t, when the solution of the wave is...
  46. Ahmad Kishki

    Thinking about the Schrodinger wave equation

    so H(psi) = E(psi) is the schrodinger equation such that psi is the eigen function of the hamiltonian operator, and since E is the eigen value of the hamiltonian, then this E is the measured E. This is what i understand so far, and i am building on this here. All psi must be eigen function of...
  47. J

    Wave equation boundary problem

    Homework Statement The question is Ytt- c^2Yxx =0 on the doman 0<x< +infinity where initia conditions are y(x,0) = e^-x^2 = f(x) , Yt(x,0) =x*e^-x^2 = g(n) and boundary condition is y(0,t) = 0 and c = 2 Homework Equations D'Almbert solution 1/2(f(x+ct)+f(x-ct))+1/2c∫ g(n) dn over the...
  48. Chemer

    Solving of schrodinger's wave equation

    It's the application of Shrodinger wave equation to H-atom and I can't solve the first step. Please help me solve this. I'm not maths student so it's really hard to solve it:( d^2 psi/dx^2+d^2psi/dy^2+d^2psi/dz^2+8π^2m/h^2(E-V)=0 Where x= rsin(theta)cos(phi) y=rsin(theta)sin(phi) z=rcos(theta)...
  49. J

    Solve Wave Equation: e^(-x^2), x*e^(-x^2), -infinity<x<infinity

    Homework Statement So it says solve this wave equation : [y][/tt] - 4 [y][/xx] = 0 on the domain -infinity<x<infinity with initial conditions y(x,0) = e^(-x^2), yt(x,0) = x*(e^(-x^2)) Homework Equations I used the D Alembert's solution which is 1/2(f(x+ct)+f(x-ct)) + 1/2c ∫ g(z) dz The...
  50. B

    Inhomogeneous damped wave equation

    Homework Statement Solve u_{tt}(x,t)+2u_{t}(x,t)-u_{xx}(x,t)=18\sin\left(\dfrac{3\pi x}{l}\right) \omega=\frac{\pi n c}{l} Boundary conditions u(0,t)=u(l,t)=0 l<\pi Initial conditions u(x,0)=u_t(x,0)=0 Homework Equations [/B] The general inhomogeneous damped wave equation is u_{tt}(x,t)+2\mu...
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