Suppose we have a large system of weights connected with springs. This oscillating system has periodic boundary condition and with the help of Fourier we are able to describe displacement of $j$-th weight as
$x_j(t) = \sum_k A_k e^{i\omega_k t} e^{i2\pi k j /N}$
$i$ in imaginary unit. Index $j$ tells us which weight and index $k$ which mode of oscillation. They both run from 0 to N-1.
Initial conditions say $x_j =0$ for all $j$, $\dot{x}_0 =v_0$ and $\dot{x}_0 =0$ for $j\neq 0$.
Coefficients $A_k$ are complex.
Frequency: $\omega_k=4\sqrt{m/k}\sin{\left(\frac{2\pi k}{N}\right)}$.
Say N=10. Then we can write down 10 equations for displacement ($x_0$,…$x_9$) and 10 equations for velocity ($\dot{x}_0$,…$\dot{x}_9$). Taking into consideration initial conditions and frequencies that can be computed, our only unknowns are coefficients $A_0$ to $A_9$. 10 coefficients mean 20 unknowns, since they have real and imaginary part.
So 20 equations and 20 unknowns sounds ok. I have put it all in matrix form and it turns out that due to $\omega_0=0$ I always get one column full of zeroes. How do I deal with this zero value frequency? My other question is: are negative frequencies different from positive? If I don't consider sign, I only get 3 different frequencies for system of 10 weights (and for n=100 I get around 25 different frequencies). Isn't the number of normal modes supposed to be equal to N?
Best Answer
In general terms your problem should be coming from a system of ODEs of the form $$\ddot{\mathbf x} + Q\mathbf x = 0,$$ where $Q$ is a matrix that has to satisfy to certain hypotheses. If you want this equation to describe oscillations, then $Q$ can be at most positive semi-definite (i.e. non-negative eigenvalues). If $Q$ is not diagonal already then you can find a change of coordinates $\mathbf x\mapsto\mathbf y$ where the ODEs take the form $$\ddot{\mathbf y} + \Omega^2\mathbf y = 0,$$ where $\Omega^2$ is a diagonal matrix whose entries are the squares of the natural frequences, so that signs don't really matter. Any zeroes appearing in $\Omega^2$ are some sort of degeneracies, telling that there are certain directions along which there are no oscillations and in that subspace, say $D\subset\mathbb R^n$, the equations look like $$\ddot{\mathbf y}|_D = 0,$$ which has the general solution $$\mathbf y|_D(t) = \mathbf u_0 + \mathbf v_0t.$$ If the system has to be confined somewhere, then the initial condition $\mathbf v_0=0$ must be enforced, otherwise the system will drift indefinitely along the (antitransformed) direction of $\mathbf v_0$ with speed related to the magnitude of this vector.