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One-dimensional wave equation with non-constant speed

  1. Aug 19, 2013 #1
    1. The problem statement, all variables and given/known data

    The cross-section of a long string (string along the x axis) is not constant, but it changes wit the coordinate x sinusoidally. Explore how a wave, caused with a short stroke, spreads through the string.


    2. Relevant equations

    Relevant is the one-dimensional wave equation, where the wave speed c is not a constant (i.e. c=√T/ρS, where T is the string tension, ρ is the density of the string, and S is the cross-section).

    The cross-section:
    S=S1+S2*Sin[x]


    3. The attempt at a solution

    I thought about using Laplace transformation so that I get an ordinary differential equation. I also have trouble with the initial conditions, I don't know what is meant by short stroke, or if it really matters.
     
  2. jcsd
  3. Aug 19, 2013 #2

    haruspex

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    I interpret 'short stroke' as simply meaning it is a small perturbation.
    Can you write down the wave equation?
     
  4. Aug 19, 2013 #3
    The wave equation:

    (∂^2 u)/(∂t)^2 = c^2 (∂^2 u)/(∂x)^2,

    where u is displacement of the string and c is the wave speed. c is not a constant, because c^2 = T/S, where T is the string tension and S is the cross-sectional area and is dependant on x. S[x]=S1+S2*Sin[x].

    (Sorry about the formatting. The ∂ stands for derivative.)
     
  5. Aug 19, 2013 #4

    haruspex

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    In LaTeX: ##\frac{\partial^2 u}{\partial t^2} = c^2\frac{\partial^2 u}{\partial x^2}##.
    So, plugging in the expression for c(x), can you apply e.g. separation of variables?
     
  6. Aug 20, 2013 #5
    I think not, because the initial conditions are probably going to be in the form u(t=0,x)=f(x) and [itex]\frac{∂u}{∂t}[/itex](t=0,x)=g(x), because it is an infinite string (no boundary conditions). But if I we separate variables (i.e. u(x,t)=X(x)*T(t)), we have to put a single value and not a function for initial conditions (example: X(0)=value, instead of u(t=0,x)= function).

    I was wandering if d'Alembert's formula applies if speed of propagation (also c in the link) is not constant?
     
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