Your text should have a section on this. I'll walk you through the general solution for both initial displacement AND initial velocity.
Do your seperation of variables and equate it to -lamda. We will get
If then let the equation for F then becomes a zero order Bessel equation, with general solution
Select
From regular methods, let T we find that
Of course d,k will be taken care in our later summation so what we have is
Note that Bw is not B. Can you solve from here?
I can't believe that your prof would assign this question without first going threw the basic heat/wave equations. That just doesn't make sense.
Also, if you can pick up a book on advanced engineering mathematics, they pretty much have all the information you're looking for (in terms of DEs). I'm working out of "Advanced Engineering Mathematics" by O'Niel. If you want, I can scan some pages on this topic and send them your way. They will describe all you need to know. But for now,
I'll walk you through the seperation of variables in a basic wave equation, and then you must apply it to this situation.
Let us suppose that,
for
and for
For our purposes, we will let
In words this means we are letting our function equal some function only dependent on X multiplied by some function only dependent on T.
Let us get this into the form of the equation above.
Subbing back into
We get
Which we can manipulate to
Of course what this means is, we have one side only dependent on T and one side only dependent on X. This will only happen if these are both equal to a constant, so we will let that constant equal -lambda
We then get 2 equations from this
So let us solve our equation involving X. We will let
Subbing this into our equation we get
Of course this means,
Lets simplify this by
So then we get
But we have initial conditions that state
So,
Again from initial conditions,
So,
This is only true when,
This means,
And finally,
Do the same for T.
yes, if you are able to scan some pages by tonight that would be useful. None of this work that I am trying to understand was taught in the module. Our lecturer was away for about a month, so we missed about 10hrs of lectures, and he recently got replaced so we are moving on to a different topic. This all seems mind boggling. The new lecturer said that this Vibrating Membrane problem I am trying to solve is pretty straight forward, but the way you are explaining it is very confusing.
Actually what I just explained is not a membrane problem. What I explained is how to solve the wave equation in 2D. Much easiar then the membrane question, but the membrane one follows a similar pattern.
What is it that you don't understand about the change of variables?
I'll have some stuff for you tonight.