Question 7.3: When solving the determinant equation (7.19), we have made a...
When solving the determinant equation (7.19), we have made a mathematical restriction for c by setting c = 2 cos Θ.
Show that for the cases
(a) |c| = 2,
(b) c <−2
he eigenvalue equation D_{N} = 0 cannot be satisfied. Clarify that thereby the special choice of the constant c is justified.
D_{N} = cD_{N−1} −D_{N−2}, if N ≥ 2. (7.19)
(a)
D_{n} = cD_{n−1} −D_{n−2}, D_{1} = c=±2, D_{0} = 1. (7.31)
We assert and prove by induction
|D_{n}| ≥ |D_{n−1}|. (7.32)
Induction start: n = 2, |D_{0}| = 1, |D_{1}| = 2, |D_{2}| = 3.
Induction conclusion from n −1, n−2 to n:
|D_{n}|^{2} = 4|D_{n−1}|^{2} ±4|D_{n−1}||D_{n−2}| + |D_{n−2}|^{2}
≥ 4|D_{n−1}|^{2} + |D_{n−2}|^{2} − 4|D_{n−1}||D_{n−2}|
⇒ |D_{n}|^{2} − |D_{n−1}|^{2} ≥ 3|D_{n−1}|^{2} + |D_{n−2}|^{2} − 4|D_{n−1}||D_{n−2}|.
According to the induction condition,
|D_{n−1}| = |D_{n−2}| + \epsilon with \epsilon ≥ 0.
From this, it follows that
|D_{n}|^{2} − |D_{n−1}|^{2} ≥ 4|D_{n−2}|^{2} +6\epsilon|D_{n−2}| + 3\epsilon^{2} −4\epsilon|D_{n−2}| − 4|D_{n−2}|^{2}
≥ 2\epsilon|D_{n−2}|
≥ 0
⇒ |D_{n}| ≥ |D_{n−1}|. (7.33)
Since |D_{n}| monotonically increases in n, and |D_{1}| = 2 > 0, we have |D_{N}| > 0. Therefore D_{N} = 0 cannot be satisfied. ω = 0 and ω =\sqrt{2T/ma} are not eigenfrequencies of the vibrating chain.
(b) By inserting the ansatz D_{n} = Ap^{n}, p ≠ 0, we also find the solution of the recursion formula D_{n} = cD_{n−1} −D_{n−2}, D_{1} = c, D_{0} = 1:
\left. \begin{matrix} p_{1} = \frac{1}{2} \left(c +(c^{2} −4)^{1/2}\right)< 0 \\ p_{2} = \frac{1}{2} \left(c -(c^{2} −4)^{1/2}\right)< 0 \end{matrix} \right\} \quad \quad 0>p_{1}>p_{2} (7.34)
The general solution for incorporating the boundary conditions D_{0} = 1, D_{1} = c reads
D_{n} = A_{1}p^{n}_{1} + A_{2}p^{n}_{2} . (7.35)
With D_{0} = 1, D_{1} = c, it follows that
A_{1} +A_{2} = 1,
\frac{A_{1}}{2} \left(c + (c^{2} −4)^{1/2} \right)+ \frac{A_{2}}{2} \left(c − (c^{2} −4)^{1/2}\right)= c,
A_{1} =\frac{ c +(c^{2} −4)^{1/2}}{2(c^{2} − 4)^{1/2}}⇔ A_{2} =\frac{−c +(c^{2} −4)^{1/2}}{2(c^{2} − 4)^{1/2}} . (7.36)
One then has
D_{n} = \frac{1}{2} \frac{c +(c^{2} −4)^{1/2}}{(c^{2} −4)^{1/2}} p^{n}_{1}+ \frac{1}{2} \frac{(c^{2} −4)^{1/2} −c}{(c^{2} −4)^{1/2}} p^{n}_{2}
= \frac{1}{(c^{2} − 4)^{1/2}}\left(p^{n+1}_{1}−p^{n+1}_{2}\right) . (7.37)
To determine the physically possible vibration modes, we had required that D_{N} = 0:
D_{N}= 0 ⇒ \left(\frac{p_{2}}{p_{1}}\right)^{N+1}= 1. (7.38)
But now 0 > p_{1} > p_{2}, hence (p_{2}/p_{1})^{N+1} > 1. Thus, for the case c < −2 eigenfrequencies do not exist too.
These supplementary investigations can be summarized as follows: The possible eigenfrequencies of the vibrating chain lie between 0 and \sqrt{2T/ma}:
0 < |ω| <\sqrt{\frac{2T}{ma}}. (7.39)