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The space of solutions of the classical wave equation

  1. Oct 13, 2013 #1

    ShayanJ

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    Consider the classical wave equation in one dimension:
    [itex]
    \frac{\partial^2 \psi}{\partial x^2}=\frac{1}{v^2} \frac{\partial^2 \psi}{\partial t^2}
    [/itex]
    It is a linear equation and so the set of its solutions forms a vector space and because this space is a function space,its dimensionality is infinite.
    Also,because [itex] \sin{\omega t} [/itex] and [itex] \cos{\omega t} [/itex] are solutions to the aforementioned equation,every other solution can be formed by a fourier series,which means [itex] {\sin{n\omega t}}_1^{\infty}[/itex] and [itex]{ \cos{\omega t}}_1^{\infty} [/itex] form a basis for the vector space of the solutions of the classical wave equation.
    We know that the number of base elements of a vector space shouldn't vary between different bases,but about the classical wave equation,we can tell that every function of the form [itex] f(x+vt)+g(x-vt) [/itex] is a solution.
    If it is also right that every solution of the classical wave equation can be written in the form[itex] f(x+vt)+g(x-vt) [/itex],then it seems that we have a basis with only two elements,in contrast to the sines and cosines which make a infinite set of base elements!and this seems to be a contradiction.
    Can anyone help?
    Thanks
     
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  3. Oct 13, 2013 #2

    UltrafastPED

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    But f,g - you have to use the basis to construct them!
     
  4. Oct 13, 2013 #3

    ShayanJ

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    You're describing a change of basis!
    Every element of a base set can be constructed from a linear combination of the elements of another base set!

    The sentence you quoted means that I can choose e.g. [itex] ln(x-vt) [/itex] and [itex] e^{x+vt} [/itex] as a basis!

    May be there are solutions that are not of the form [itex] f(x+vt)+g(x-vt) [/itex]!
    This solves the problem!
     
  5. Oct 26, 2013 #4
    No, he's right. Every solution is of that functional form. Every solution to the wave equation has a forward traveling wave and a backward traveling wave.

    However, the context of your conclusion solution is what is setting you off. First look at sturm-liouville theory, then learn some real and fourier analysis. The basics are that you construct this f and g from the fourier series, just as you construct any other vector from a basis, which is determined by solving the separable eigenvalue equations to obtain the eigenvectors and applying the boundary conditions to obtain the eigenvalues. The sin(npix/L) and cos(npix/L) sequences form a basis in L^2[[0,1]:1] (if I remember correctly?) which is a Hilbert Space and is infinite dimensional. Pretty much, it is complete in the since that the sum of fourier terms can converge to any periodic function in the interval [0,1] in x. They are a lot like taylor series in that sense.
     
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