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. M

    Help me understand wave equation ?

    hi, It is a bit confusing, because when I say "wave equation" the connotation it invokes in my mind is if I apply the equation I will find the function of the wave I'm exploring. Of course I know now this is wrong.. it is the other way around, I have to know my wave-function in advance and...
  2. S

    Wave equation - v speed or velocity?

    Consider the classical wave equation...
  3. A

    Origin of the schroedinger's wave equation

    How did schroedinger arrive at the wave equation? I recently read in a book about the concept of wave packets using Fourier analysis and the wave equation was derived by forming a differential equation of the Fourier integral. But some books say that there is no formal proof of the...
  4. D

    MHB Characteristics and Initial Density Distribution in Traffic Flow Wave Equation

    Suppose that along a stretch of highway the net flow of cars entering (per unit length) can be taken as a constant $\beta_0$. The governing equation of motion is then $$ \frac{\partial\rho}{\partial t} + c(\rho)\frac{\partial\rho}{\partial x} = \beta_0 $$ where $$ c(\rho) = u_{\text{max}}\left(1...
  5. M

    Understanding the Significance of the Linear Wave Equation in Wave Mechanics

    Hello, I am studying wave mechanics and I managed to derive the linear wave equation with a string. Now I don't understand the significance of the equation or why I can use a string oscillating to make it general and apply to all sorts of waves Edit: this one \frac{\partial^2...
  6. D

    MHB Solving the Quasi-Linear 1-D Wave Equation

    Consider the quasi-linear 1-D wave equation $$ \frac{\partial\rho}{\partial t} + 2\rho\frac{\partial\rho}{\partial x} = 0 $$ with the piecewise constant initial conditions $$ \rho(x,0) = \begin{cases} \rho_1, & x < -x_0\\ \rho_2, & -x_0 < x < x_0\\ \rho_3, & x > x_0 \end{cases} $$ where $\rho_1...
  7. D

    What is the Density Wave Equation and How Does it Describe Traffic Flow?

    Traffic is moving with a uniform density of \rho_0. $$ \frac{\partial\rho}{\partial t} + c(\rho)\frac{\partial\rho}{\partial x} = \beta_0 $$ where $$ c(\rho) = u_{\text{max}}\left(1 - \frac{2\rho}{\rho_{\text{max}}}\right). $$ Show that the variation of the initial density distribution is given...
  8. R

    Determine the direction and speed of the wave from a given wave equation

    Homework Statement Given an equation for a wave \psi(x,t) = A e^{-a(bx+ct)^{2}} determine the direction of its propagation if you know \psi(x,t) = f(x \pm vt) and use this to find its speed. Homework Equations The Attempt at a Solution I figured I would just rearrange the expression in...
  9. Duderonimous

    Show that y(x,t)=y1(x,t)+y2(x,t) is a solution to the wave equation

    Homework Statement Let y_{1}(x,t)=Acos(k_{1}x-ω_{1}t) and y_{2}(x,t)=Acos(k_{2}-ω_{2}t) be two solutions to the wave equation \frac{∂^{2}y}{∂x^{2}}=\frac{1}{v^{2}}\frac{∂^{2}y}{∂t^{2}} for the same v. Show that y(x,t)=y_{1}(x,t)+y_{2}(x,t) is also a solution to the wave equation. Homework...
  10. M

    Diagonalization of 2D wave equation

    Homework Statement I've just derived the 1D wave equation for a continuous 1D medium from a classical Hamiltonian. I simply wrote Hamilton's equations, where the derivatives here must be functional derivatives (e.g. δ/δu(x)) since p and u are functions of x, and I got the wave equation (see...
  11. sunrah

    Fundamental solution of wave equation

    Homework Statement show that E(x,t):= \frac{1}{2} \left\{ \begin{array}{ll} 1 & \mbox{if $|x|<t $};\\ 0 & \mbox{else}.\end{array} \right. is a fundamental solution of the wave equation. Homework Equations LE = E_{tt} - \Delta E = \delta The Attempt at a Solution firstly...
  12. U

    Wave equation: intial conditions

    Homework Statement Solve the initial boundary value problem u_{tt}=c^2u_{xx} u(-a,t)=0,\quad u(a,t)=0,\quad u(x,0)=\sin(\omega_1 x)-b\sin(\omega_2x) where a, b, \omega_1, \omega_2 are positive constants. Homework Equations d'Alembert's solution The Attempt at a Solution...
  13. M

    Wave equation with nonhomogenous neumann BC

    I've been searching online for the past week but can't seem to find what I am looking for. I need the analytic solution to the wave equation: utt - c^2*uxx = 0 with neumann boundary conditions that are not homogeneous, i.e. ux(0,t) = A, for nonzero A. also, the domain i require the...
  14. V

    Expressions of travelling harmonic wave equation

    Hi all, apologies if this has been answered elsewhere - I was unable to find an answer using the search function. Homework Statement "Expressed in terms of wavenumber and angular frequency, the equation for a traveling harmonic wave is: y = Asin(kx-ωt). Express this function in terms of (a)...
  15. R

    The Wave Equation (vibrating string)

    Homework Statement The differential equation describing the motion of a stretched string can be written \frac{\partial ^2 y}{\partial x^2} = \frac{\mu}{T} \frac{\partial^2 y}{\partial t^2} μ is the the mass per unit length, and T is the tension. (i) Write down the most general solution you...
  16. V

    Can the Schrondinger Wave Equation Be Used to Solve Normalized Cases?

    Homework Statement Solving Normalized case of schrondinger wave equation Homework Equations The Attempt at a Solution This type of question is not normalized case of solving using schrondiger equation. Any example of solving normalized case using schrondinger equation ? How...
  17. H

    MHB Wave Equation PDE: Help Solve Test Problem

    Hi, I have a test tomorrow and I'd like you to guys help me please. Solve the following: $\begin{align*} & {{u}_{tt}}={{u}_{xx}}+1+x,\text{ }0<x<1,\text{ }t>0. \\ & u(x,0)=\frac{1}{6}{{x}^{3}}-\frac{1}{2}{{x}^{2}}+\frac{1}{3},\text{ }{{u}_{t}}(x,0)=0,\text{ }0<x<1. \\ &...
  18. T

    Induction on the n-dimensional, radially symmetric wave equation

    Homework Statement Consider the radially symmetric wave equation in n dimensions u_{tt} = u_{rr} + \frac{n-1}{r}u_r Use induction to show that the solution is u = \left(\frac{1}{r}\frac{\partial}{\partial r}\right)^{(n-3)/2} \frac{f(t-r)}{r} for n odd and u =...
  19. L

    Conformal inv of scalar wave equation

    I'm trying to prove the conformal invariance (under g_{\mu\nu}\to\omega^2 g_{\mu\nu}) of \bar{\Box}{\bar{\phi}}+\frac{1}{4}\frac{n-2}{n-1}\bar{R}\bar{\phi} I've found that this equation is invariant upto a quantity proportional to...
  20. K

    Reducing the Wave Equation: Change of Variables

    Homework Statement Show that the wave equation u_{tt}-\alpha^{2}u_{xx}=0 can be reduced to the form \phi_{\xi \eta}=0 by the change of variables \xi=x-\alpha t \eta=x+\alpha tThe Attempt at a Solution \frac{\partial u}{\partial t}=\frac{\partial \xi}{\partial t}\frac{\partial...
  21. K

    Relfected wave equation for free and fixed end

    Homework Statement Hello, I have problems with expressing a reflected wave mathematically. In my printed notes I found the following formulas for reflected waves: a) For a fixed end: incoming wave: y_1(x,t)=e^{-i(kx+ωt)} reflected wave: y_2(x,t)=re^{i(kx-ωt)} where r is the reflection...
  22. K

    Intro Electromag Question - Wave Equation

    Homework Statement Homework Equations The Attempt at a Solution b) I could figure it out if kz was changed to kx... Double Derivative of E(r, t) with respect to x is = 0 Double Derivative of E(r, t) with respect to t is = -ω2*E0*cos(kz - wt + ∅0) Multiply the second term by k2/ω2...
  23. L

    Compression/longitudinal wave equation

    This is not really a homework problem, I do not understand the concepts. The attachment below is as informative as the slides provided are. The course seems to be based on H.J. Pain's book "waves and vibrations". I know that the information provided is limited, but there is little I can...
  24. O

    Simpliest possible linear wave equation?

    Homework Statement Hello, as in topic, i am looking for simpliest wave equation i can get, i don't really need to know what it is etc., i only need it to be as simple as possible. Homework Equations The Attempt at a Solution I think that simpliest wave equation will be just y =...
  25. H

    Deriving Wave Equation - Electric Field Inside Metal

    Homework Statement Consider an electromagnetic wave hitting a metallic surface with conductivity σ at normal incidence. a) Derive the wave equation describing this situation. Hint: Use Ohm’s law, J = σE to eliminate the current. b) Solve the wave equation for the electric field to...
  26. L

    What Is the Collapse of the Wave Function in Quantum Mechanics?

    i am an A-level physics student (high school if you're american) and for a research topic i have chosen wave particle duality. i have been able to explain the ideas of diffraction, double slit experiments, photo electric effects and electron diffraction easily enough, but we are expected to take...
  27. S

    Wave equation derivation for ocean waves

    Hi there, I'm a mechanical engineering student who's extremely interested in going into physical oceanography after finishing undergrad. I'm trying to find a good source for the wave equation as it relates to physical oceanography, as well as orbital paths of particles, and have yet to find...
  28. D

    Sum of two waves & wave equation

    This thing is driving me mad, I thought I figured it out already, but it seems I was wrong. Any help would be appreciated. Homework Statement "Under what conditions does the sum of two sinusoidal waves also satisfy the wave equation?" The sum wave is D(x,t) = A_{1}sin(k_{1}...
  29. M

    MHB Fourier transform to solve the wave equation

    I need to use the Fourier transform to solve the wave equation: $\begin{aligned} & {{u}_{tt}}={{c}^{2}}{{u}_{xx}},\text{ }x\in \mathbb{R},\text{ }t>0, \\ & u(x,0)=f(x), \\ & {{u}_{t}}(x,0)=g(x). \end{aligned} $ So I have $\dfrac{{{\partial }^{2}}F(u)}{\partial...
  30. E

    What Is the New Equation of a Wave After Reflection?

    Hello all, Homework Statement We can represent a mechanical transverse wave by Y=Asin(kx-wt+∅). Now imagine this wave traveling (towards right as velocity is positive) and meeting up with two cases Case 1) Rigid wall. Case 2) Free end. The way gets reflected completely( ignoring transmission...
  31. M

    MHB Making homogenous a wave equation

    Solve $\begin{aligned} & {{u}_{tt}}={{u}_{xx}}+t,\text{ }t>0,\text{ }x\in \mathbb R, \\ & u(x,0)=x \\ & {{u}_{t}}(x,0)=1. \end{aligned} $ Okay first I should set $v(x,t)=u(x,t)-\dfrac16 t^3,$ then $u(x,t)=v(x,t)+\dfrac16 t^3$ so $u_{tt}=v_{tt}+t$ and $u_{xx}=v_{xx}$ so...
  32. M

    MHB Wave equation and weird notation

    I have $u_{tt}=u_{xx},$ $x\in\mathbb R,$ $t>0,$ $u(x,0)=0$ and $u_t(x,0)=\chi_{[-1,1]}(x).$ What does mean the last condition? In such case, how to solve the equation then? Thanks!
  33. M

    MHB Wave equation and multiple boundary conditions

    I need to apply D'Lembert's method but in this case I don't know how. How to proceed? Determine the solution of the wave equation on a semi-infinite interval $u_{tt}=c^2u_{xx},$ $0<x<\infty,$ $t>0,$ where $u(0,t)=0$ and the initial conditions: $\begin{aligned} & u(x,0)=\left\{ \begin{align}...
  34. L

    Sketching wave equation solutions.

    Homework Statement An infinite string obeys the wave equation (d2z/dx2)=(ρ/T)(d2z/dt2) where z is the transverse displacement, and T and ρ are the tension and the linear density of the string. What is the velocity of transverse traveling waves on the string? The string has an initial...
  35. H

    Linear wave equation vs. linear system

    Hi, I have a general question about linear wave equation: Is solving the linear wave equation is equivalent to compute the output of a linear system where source is the input? Thanks in advanced! Chao
  36. D

    Boundary conditions of solution to the wave equation

    Homework Statement Because q(x,t) = A*exp[-(x-ct)2/σ2] is a function of x-ct, it is a solution to the wave equation (on an infinite domain). (a) What are the initial conditions [a(x) and b(x)] that give rise to this form of q(x,t)? (b) if f(x) is constant, then Eq. (2) shows that solution is...
  37. S

    Eigen Value, Eigen function -Schrodinger wave equation

    Hi, I have a understanding of what is eigen value and eigen function. But I am unable to correlate the same with Schrodinger wave equation. Can you please help me to clarify the concept? Thanks!
  38. fluidistic

    Solving wave equation with Fourier transform

    Homework Statement Use Fourier transforms to calculate the motion of an infinitly large stretched string with initial conditions u(x,0)=f(x) and null initial velocity. The displacements satisfy the homogeneous wave equation. Homework Equations \frac{\partial ^2 u }{\partial t^2...
  39. H

    Wave Equation in 1-d Proof/Verify

    Homework Statement Verify that Acos(kx-ωt) and Bsin(kx-ωt) are solutions of the one dimensional wave eqn. if v=ω/k. Does f(x,t)=(ax+bt+c)^2 represent a propagating wave? If yes what is its velocity? Homework Equations I know the partial differ. eqns. for the wave equation are d^2...
  40. N

    Causality and the inhomogeneous wave equation with a moving source

    Hi, I am confused about my solutions to the following governing equation: u_{tt}-c^2u_{xx}=F_{xx} For F=A(x)sech^2\left(\frac{x-c_gt}{B}\right) Where c,c_g,B \in \mathbb{R} and A(x) is a linear function. Also, we have c_g<c. Substituting physical values for the parameters, I...
  41. D

    Solve PDE Wave Equation for Stretched String of Length 2

    Homework Statement Having exams on monday but still having problems with PDE. I thought i got it until i saw the teacher's solution different from mine and i have no idea wtf she doing also.. Contacting her is not an option. Please give me a hand with this.. Much appreciated. Solve the...
  42. A

    Harmonic Wave Equation: What Are x & t (Length & Time Units)?

    the harmonic wave equation is given by y(x,t)=Rsin{2\pi/\lambda(vx-t)+\phi} where R is amplitude \lambdais wavelength v is velo of wave \phi is initial phase. Could you please tell as well as explain me what are the parameters x and t where there in units of length and time respectively.
  43. W

    Understanding Wave Function and Schrodinger Wave Equation

    hi , can anyone please explain to me what is wavefunction - and can we apply schordinger wsve equation to simple pendulum. thanks
  44. D

    Problem applying divergence theorem to wave equation

    I'm an undergrad doing research in PDE and my adviser gave me some material to read over the holiday. But I'm getting stuck at the beginning where the divergence theorem is applied to a calculation. Maybe somebody can help me? Without getting too detailed about the context of the problem...
  45. N

    Energy considerations and the inhomogeneous wave equation

    Hi, If I have a forced wave equation u_{tt}-c^2u_{xx}= f(x,t) what is my associated energy law?For instance, in the homogeneous case \Box u=0 I know that E(t)=\frac{1}{2}\int u_t^2 +c^2|u_x|^2 \ dx which implies that \frac{d E(t)}{d t} is equal to zero. (just use integration by...
  46. C

    Transverse harmonic wave Equation

    Homework Statement A transverse harmonic wave travels on a rope according to the following expression: y(x,t) = 0.18sin(2.2x + 17.1t) The mass density of the rope is μ = 0.146 kg/m. x and y are measured in meters and t in seconds. Homework Equations I do not know what to use...
  47. M

    Show ψ(x,t) = f(x-at) + g(x+at) satisfies the wave equation

    Homework Statement Show ψ(x,t) = f(x-at) + g(x+at) satisfies the wave equation (a^2)*(∂^2ψ/∂x^2)-(∂^2ψ/∂x^2)=0 Homework Equations The Attempt at a Solution I think i just take the derivative twice and end up with something like the second derivative = a^2*second derivative...
  48. King Tony

    PDE Separating Variables for 3d spherical wave equation

    Homework Statement Just need someone else to double check more work. I just want to know if I'm separating these variables correctly. Homework Equations \frac{\partial^2u}{\partial t^2} = c^2\nabla^2u The Attempt at a Solution Allow u(\rho, \theta, \phi, t) = T(t)\omega(\rho...
  49. C

    Helmholtz wave equation

    If I was only considering the "elliptical arc" as 1/6 of a circle and all I was concerned with was the radial and angular dependencies (w.r.t.) and two sinusoidal sources acting in unison as the forcing term which is actually 2 sin(60t), how would this wave equation be set up and what would be...
  50. G

    D'Alembert's solution of the wave equation

    For a wave equation η(u,v) = f1(u) + f2(v) where u = x - ct and v = x + ct, consider an initial displacement η = η0(x) and an initial velocity ∂tη = \dot{η_{0}}(x). I'm a little confused with the velocity initial condition; shouldn't the time derivative of η0(x) be 0?
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