The space of solutions of the classical wave equation

[ShayanJ]

Consider the classical wave equation in one dimension:
<br /> \frac{\partial^2 \psi}{\partial x^2}=\frac{1}{v^2} \frac{\partial^2 \psi}{\partial t^2}<br />
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 \sin{\omega t} and \cos{\omega t} are solutions to the aforementioned equation,every other solution can be formed by a Fourier series,which means {\sin{n\omega t}}_1^{\infty} and { \cos{\omega t}}_1^{\infty} 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 f(x+vt)+g(x-vt) is a solution.
If it is also right that every solution of the classical wave equation can be written in the formf(x+vt)+g(x-vt),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

 

[UltrafastPED]

If it is also right that every solution of the classical wave equation can be written in the formf(x+vt)+g(x-vt),then it seems that we have a basis with only two elements

But f,g - you have to use the basis to construct them!

 

[ShayanJ]

But f,g - you have to use the basis to construct them!

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. ln(x-vt) and e^{x+vt} as a basis!

May be there are solutions that are not of the form f(x+vt)+g(x-vt)!
This solves the problem!

 

[epr1990]

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.