- #1
Miles123K
- 57
- 2
- Homework Statement
- The problem description is as follows. I am expected to find the frequency of the standing wave apparently.
- Relevant Equations
- F = ma
First I worked out the dispersion relations, which is pretty easy:
##M \ddot x_j = K x_{j-1} + K x_{j+1} - 2K x_j -mg \frac {x_j} {l} ## (All t-derivatives)
We know ##x_j## will be in the form ##Ae^{ijka}e^{-i\omega t}##
so the above becomes:
## -\omega^2M = K (e^{-ika}+e^{ika}-2)-\frac {g} {l}##
Use trig identities to simplify and we get:
## \omega^2 = \frac {4K} {M} sin^2(\frac {ka} {2}) + \frac {g} {l}##
Now I think I am supposed to consider the forces on Block 0, so:
##F_{left} + F_{right} = F_0 = M \ddot x_0##
## K x_{j-1} + K x_{j+1} - 2K x_j = M \ddot x_0 = -M \omega^2 x_0## I think this is the subsystem mentioned in the hint?
In the question, it mentioned considering a normal mode of the form:
##A(x) = A_0 e^{-k \left| x \right |}##
So, I just plugged that into the above equation and got something like:
##2Ke^{-ka} = 2K - m \omega^2##
##e^{-ka} = 1 - \frac {m \omega^2} {2K}##
Log the above:
##ka = - ln(1-\frac {m \omega^2} {2K})##
However, I am unsure whether this is the correct solution. Do I just plug this ##ka## into the dispersion relations we got earlier?
Could someone check my answer?