1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Wave equation and solution

  1. Dec 29, 2015 #1
    $$\frac{\partial^2}{\partial t^2}u(x,t)=c^2\Delta u(\vec{x},t)\qquad \vec{x}\in \mathbb{R}^n$$
    is known as the wave equation. It seems not very trivial, so is there any derivations or inspirations of it?

    To solve this equation, we have to know the initial value and boundary conditions:
    \begin{equation*}
    \begin{cases}
    u(0,t)=u(\vec{l},t)=0\\
    u(\vec{x},0)=f(\vec{x})\\
    u_t(\vec{x},0)=g(\vec{x})\\
    \end{cases}
    \end{equation*}
    This above can be solved uniquely, with separation of variables.
    And also see these conditions:
    \begin{equation*}
    \begin{cases}
    u(\vec{x},0)=f(\vec{x})\\
    u_t(\vec{x},0)=g(\vec{x})\\
    \end{cases}
    \end{equation*}
    Why this above can be also solved uniquely with d'Alembert or Kirchhoff's method? Why the boundary conditions can be removed easily? So it seems that it has no influence???
     
  2. jcsd
  3. Dec 29, 2015 #2

    SteamKing

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper

    Obviously, there are. Have you tried looking for them?

    Google "wave equation"
     
  4. Dec 29, 2015 #3

    mathman

    User Avatar
    Science Advisor
    Gold Member

    If you look at the one dimensional version it become more transparent that sine or cosine of the appropriate arguments satisfy the equation.
     
  5. Dec 30, 2015 #4
    Im just wondering why the different initial conditions as i posted give both an unique solution?
     
  6. Dec 30, 2015 #5
    They first of the three is telling you that the ends are fixed, so it is relevant.
     
  7. Dec 30, 2015 #6

    vanhees71

    User Avatar
    Science Advisor
    2016 Award

    To make the solution unique you need all three conditions (the first is called a boundary and the 2nd and 3rd initial contitions). The (1+1)D case is indeed most simple. First you should show that the general solution of the wave equation in this case reads
    $$u(t,x)=u_1(x-c t)+u_2(x+c t),$$
    where ##u_1## and ##u_2## are arbitrary functions that are at least twice differentiable.

    Hint: Introduce the new independent variables ##\xi=x-c t## and ##\eta=x+c t## and show that the wave equation is equivalent to
    $$\frac{\partial^2}{\partial \xi \partial \eta} u=0.$$

    Now think about the boundary conditions and how to work in the initial conditions.

    Hint: You should start with the initial conditions, plugging in the above given general solution. What conclusions can you draw from them on the definition of the functions on the interval ##[0,L]##? Then you should think about how to periodically continue the function to the entire real axis to fulfill also the boundary conditions.

    That's called the d'Alembertian approach. Another very illuminating way is to use Fourier series, starting with the boundary condition, then using the wave equation to constrain the coefficients and finally use the initial conditions to fully determine them. Of course, both ways lead to the same result!
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Wave equation and solution
  1. Wave equation solution (Replies: 5)

Loading...