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Classical mechanics

  1. May 7, 2006 #1
    in wave equations sq rt of -1 appears. could you kindly explain why.

    cecilgamini
     
  2. jcsd
  3. May 7, 2006 #2

    Fermat

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    I helps to simplify the manipulation and solution of equations.

    d'Alembert's solution to the wave equation would be,

    u(x,t) = F(x-ct) + G(x+ct)

    which can be re-written as

    u(x,t) = f(x,t) + g(x,t)

    and we can write

    f(x,t) = A.exp[i(kx - wt)]
    g(x,t) = B.exp[i(kx + wt)]

    where

    w = kc.

    Although the deformation u(x,t) will clearly have to be a real number for all values of x and t, it turns out to be useful to consider complex soutions to the wave equation as well. The real and imaginary parts of a complex solution will individually satisfy the wave equation, so a complex solution encodes two real solutions.

    Where a solution involves trigonometric functions, e.g.

    u(x,t) = A.cos(x - ct) + B.cos(x + ct)

    rather than a complex exponential function, then the solution and algebraic manipulation of the latter is often much easier than the former.
     
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