Fourier's method, division by zero

[usn7564]

Homework Statement

Solve the BVP for a vibrating string with Separation of Variables/Fourier's method.

$$\frac{\partial ^2}{\partial ^2 t} u(x,t) = c^2 \frac{\partial ^2}{\partial ^2 x} u(x,t)$$

The string is of length L with each end fixed, ie u(0,t) = u(L,t) = 0

The Attempt at a Solution

I know how to do Fourier's method but have a query regarding separation of variables. I make the assumption u(x,t) = X(x)T(t) for some functions X and T and then divide the LHS and RHS by X(x)T(t) to find that both are constant etcetc, but this step has me confused.

Right so X(x)T(t) can't be 0 along the whole string because that would just be a trivial solution so the division is warranted, or that's how I remember being taught it in any case. But won't the string pass u(x,t) = 0? [just thinking physically it kind of has to if the coordinate system is set at u(x,t) = 0 when the string is simply at rest and stretched out, and IIRC the solutions do allow for it]. In that case either X(x) or T(t) will be zero at certain x / certain t so aren't we disregarding any solution that has the string going past u(x,t) = 0 by doing this division?

In short, we divide by X(x)T(t) which takes on the value zero at certain (x, t). Why can we do this?

I'm guessing it's rather simple but can't wrap my head around it, would appreciate some explaining thank you.

 

[Orodruin]

I suggest looking at this post https://www.physicsforums.com/threa...lve-Schrödinger-equation.797405/#post-5007984, which I made in a different thread. It provides a different look on finding eigenfunctions on product form in Sturm-Liouville problems and is also applicable to your case.

 

[ehild]

The Attempt at a Solution

I know how to do Fourier's method but have a query regarding separation of variables. I make the assumption u(x,t) = X(x)T(t) for some functions X and T and then divide the LHS and RHS by X(x)T(t) to find that both are constant etcetc, but this step has me confused.You do not need to divide by zero.
Right so X(x)T(t) can't be 0 along the whole string because that would just be a trivial solution so the division is warranted, or that's how I remember being taught it in any case. But won't the string pass u(x,t) = 0? [just thinking physically it kind of has to if the coordinate system is set at u(x,t) = 0 when the string is simply at rest and stretched out, and IIRC the solutions do allow for it]. In that case either X(x) or T(t) will be zero at certain x / certain t so aren't we disregarding any solution that has the string going past u(x,t) = 0 by doing this division?

In short, we divide by X(x)T(t) which takes on the value zero at certain (x, t). Why can we do this?

I'm guessing it's rather simple but can't wrap my head around it, would appreciate some explaining thank you.

If you search the solution in separable form, u(x,t)=X(x)T(t) and do the differentiations, you get the equation T" X = c² T X". You might want solutions for which T"=AT and X"=BX holds, with A and B constants. So ATX-c²BTX = TX(A-c²B)=0 for all x and t. TX might be zero, but not everywhere and all time. As TX is not zero for all t and x, A-c²B = 0 must hold.

Now you have the equations T"(t)=AT(t) and X"(x)=BX(x) to solve.