The space of solutions of the classical wave equation

[ShayanJ]

Consider the classical wave equation in one dimension:
$\frac{\partial^2 \psi}{\partial x^2}=\frac{1}{v^2} \frac{\partial^2 \psi}{\partial t^2}$
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 form$f(x+vt)+g(x-vt)$,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 form$f(x+vt)+g(x-vt)$,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.

 

[blue_raver22]

I can provide some clarification on the space of solutions of the classical wave equation. First, it is important to note that the classical wave equation is a partial differential equation that describes the behavior of a wave in one dimension. It is a linear equation, meaning that the superposition principle holds, and the set of its solutions forms a vector space. This means that any linear combination of solutions is also a solution to the equation.

The dimensionality of this vector space is infinite because it is a function space. This means that there are infinitely many possible solutions to the equation. As mentioned in the content, the solutions to the classical wave equation include sine and cosine functions, and these can be used to form a basis for the vector space. However, they are not the only solutions. As you mentioned, any function of the form f(x+vt)+g(x-vt) is also a solution to the equation, and this leads to a different basis with only two elements.

This may seem like a contradiction, but it is actually a result of the different properties of the two bases. The sine and cosine functions are periodic, meaning that they repeat themselves after a certain interval. This leads to an infinite set of solutions. On the other hand, functions of the form f(x+vt)+g(x-vt) are not necessarily periodic, and so they do not form an infinite set of solutions. However, they are still valid solutions to the equation and can be used as a basis for the vector space.

In summary, the space of solutions of the classical wave equation is a function space with infinite dimensionality. While sine and cosine functions can be used as a basis, there are also other valid solutions that can form a different basis with a smaller number of elements. This is not a contradiction, but rather a result of the different properties of the two bases.