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Finding Standing Waves of a Certain Disperssion

  1. Nov 14, 2009 #1
    1. The problem statement, all variables and given/known data

    I have a system of infinite particles which when stationary are parted with distance a.
    Their movement is described with
    [tex]mu^{..}_{n}=\alpha(u_{n+1}+u_{n-1}-2u_{n}[/tex]

    From which (assuming the solution is an harmonic wave) I got the dispersion:
    [tex]\omega(k)=2\sqrt{\frac{\alpha}{m}}\left|sin(\frac{ka}{2})\right|[/tex]

    where all the constants are given. One of the questions is:
    "Find a frequency w over which the wave is a standing wave"

    2. Relevant equations
    Form of the solution:
    [tex]u_{n}(t)=ue^{i(kna-\omega(k)t)}[/tex]



    3. The attempt at a solution
    I found out that I can bound k to the interval [-2pi/a,+2pi/a]
    (for every other k there's a corresponding k in the interval, for which the solution is the same)
    But I have no idea what to do next
    but maybe if for some k, w(k)=-w(-k) then summing the wave k and wave -k, will
    result in a standing wave. But the only k that satisfies this is k=0, for which there is no wave at all.
     
  2. jcsd
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