So I'm trying to solve a problem here and I have been working on it all day, clearly i'm in need of some guidance.
I have a rod of length $L$ and cross section area $A$, Young's modulus $E$ and constant density $\rho$. Now I want to study longitudinal waves in this rod when its end is attached to a wall. So first i derive the wave equation as it is done in my book and I arrive with this
$$ u_{tt}-c^2u_{xx}=0\\ u(0,t)=u(L,t)=0 $$
where u(x,t) is the displacement in point $x$ at time $t$ and $c=\sqrt{\frac{E}{\rho}}$.
I'm pretty sure that this is correct since I have a few examples in my book that are stated like this. Now I want to detect the eigenfrequency so I separate the variables and go through all that (not so important for this problem) and arrive with
$$ \omega_n = \frac{\pi}{L}\sqrt{\frac{E}{\rho}}\,n \,\,, \, n=1,2,3,.. $$
and this again seems right, dimensions checks out and the book have some similar examples.
Ok so now the first part seems okay and I move on to the next problem, a numerical approximation of this. The idea is to use $N+1$ number of strings along with $N$ masses with mass $M$, and then put these together in a long chain. So this model will end up looking like a system
$$ X'' =\frac{k}{m}AX = \frac{k}{m} \left( \begin{array}{ccc} -2 & 1 & 0 \\ 1 & -2 & 1 \\ 0 & 1 & -2 \end{array} \right) X $$
where m is the mass of these little masses and k is the constant for the springs. Now A can get how big as we want (size depends on N), just expand with $-2$ and $1$ along the diagonals and zeros on the upper and lower triangle, like a band matrix. Now I'm given the mass $ m = \frac{\rho A L}{N+1}$ and I'm asked again to find the eigenfrequencies for this numerical problem in the case for $N= 25$ and compare it with my first model, then chose the constant $k$ so the frequencies for these two models agree. I'm bit lost here but I ran $E = eig(A)$ in MATLAB and took
$$ \omega_n = \sqrt{\frac{k}{m}E(n)} = \sqrt{\frac{k(N+1)}{\rho A L}E(n)} $$ and then I tried to compare this with my first model but nothing agrees, the eigenfrequencies for my first model clearly grows as $n$ gets bigger and all the values of $E(n)$ are negative. I changes the sign in my matrix and that fixes the problem for this but the numbers aren't even correct here. I have worked on this for 10 hours now, It would mean a lot if someone could take the time to give me some advice!
First, why this expression for $c$? If I recall correctly then $$ c=\sqrt{\frac{E}{\rho}}\,. $$
Second, the eigenvalues you found are the eigenvalues of the operator $$ -\frac{d^2}{dx^2}, $$ and the eigenvalues are $\omega_n^2$, it is easier to deal with squares, hence to relate everything you need to find the eigenvalues of $-A$ (you use $A$ in two different contexts).
Your expression for $m$ is correct.
Now try to figure out your $k$. And compare with the spoiler: