piano.lisa said:
Before I can discuss the driven part... how do I do the first part with the normal modes?
At t=0, the mode is n=3... how about some other time? How can I find it?
I thought you had gotten past the first part. Let's go back to your series
iii. q(x,t) = Σ[μsin(nπx/L)cos(ωt)]
iv. ω = (nπ/L)[(T/μ)^(1/2)]
<the sum is from n=1 to infinity, and the ω is different for each n>
<μ is a function of the amplitude>
Each sin(nπx/L)cos(ωt) term in the series is one possible mode of vibration of the string. The use of μ in your series is an unusual symbol for represnting the coefficients because it was also used for the mass density. Whatever letter you use, it also needs a subscript. This just represents the amplitude of the contribution from one mode of oscillation. I am going to use a more conventional representation of b_n for the coefficient. What this equation represents is a superposition of all possible modes of oscillation of the string, and should be written as
q(x,t) = \sum\limits_{n = 1}^\infty {b_n \sin (n\pi x/L)\cos (n\pi vt/L)} = \sum\limits_{n = 1}^\infty {b_n \sin (n\pi x/L)\cos (\omega _n t)}
where v = [(T/μ)^(1/2)] is the velocity. The factors {b_n \sin (n\pi x/L} ) are position dependent amplitudes of the harmonic motion of each little bit of mass dm at position x of the string. At positions where the sine is zero, there is no motion (node) associated with mode n, and where the sine is 1, there is maximum amplitude (antinode) associated with that mode.
The transverse velocity at each position x is given by
q'(x,t) = - \sum\limits_{n = 1}^\infty {b_n \omega _n \sin (n\pi x/L)\sin (\omega _n t)}
Your conditions
i. q(x,0) = Asin(3πx/L)
ii. q'(x,0) = 0
are saying that at time zero every point of the string is at rest (zero velocity) and that the momentary shape of the string is described by Asin(3πx/L). What you need to do and perhaps already have done is set the initial position function equal to the time dependent series at the particular time (zero) the shape of the string is known
q(x,0) = A\sin (3\pi x/L) = \sum\limits_{n = 1}^\infty {b_n \sin (n\pi x/L)}
and use this equality to determine the values of the coefficients in the series. This is particularly easy in this case, because the function on the left that describes the initial shape is identical to
one of the functions in the series. It can be shown that in such cases, only one of the terms of the series has a non-zero coefficient, the one in the term that matches the function on the left.
In general you would have to also consider the initial velocities. If q'(x,0) had been other than zero, then you would not have been able to express the time dependence as a pure cosine function. You would have to add a phase factor to the cosine or use a sum of sine and cosine functions for the time dependence.
The first part of your problem is essentially done. b_3 = A and all other b_n are zero. The series bcomes a single term.
This determines the motion of the string for all time. The string vibrates in the n = 3 mode, and that is all there is to it. You have one product of the position dependent sine and the time dependent cosine with n = 3 multiplied by A.
When you move on to the driven part, you are going to have to learn how to find all the coefficients in the series. This is what Fourier analysis is all about. You will need to impose some condition on the shape of the string and its velocity at some point x. There are many possible ways to drive a string, so the problem statement is pretty open ended. I suggest that if this is your first look at such a problem you extend the simple first part of this problem to a different shape at time zero, but with every part of the string still at rest. One such shape is a triangle of height A with a vertex at some x value and the other two vertices at x = 0 and x = L where the string is held fixed. This corresponds to plucking a string.
Take a look at what you have seen to do the Fourier analysis of a periodic function, and see if you can come up wih the equations for determining the coefficients in this case. The problem is trying to get you to prove than no mode that has a node at the driving position can have a non-zero coefficient.
