- #1

ShayanJ

Gold Member

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## Main Question or Discussion Point

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

[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