Last week we found the frequencies for the fundamental modes for the motion of n identical masses, each of mass m, connected in a line between two fixed walls by n+1 springs of spring constant k. In particular, we found that the system of differential equations reduced to finding the eigenvalues and eigenvectors of the matrix
where . We showed how the eigenvalues of this matrix can be obtained from those of
which has the same eigenvectors as . Lastly, we recall that the eigenvalues of were given by , i=1,2,…,n.
Note that if we have an eigenvector with corresponding eigenvalue λi, then the eigenvalue equation tells us
Here, we now define ; with this we then have:
, k=1,2,…,n; or, using our value for λi,
Consider the functions , i=1,2,…,n. First, for all our i, we have . Secondly, we see that
Thus, if we let , k=1,2,…,n, then the vector is the eigenvector corresponding to the ith fundamental mode. Note that these are sine waves of decreasing wavelength: the ith mode has i-1 zeroes (nodes) in the interval 0<v<n+1, and thus i-1 changes of phase, as we observed previously (noting the nodes via changes in relative phase). Note that our horizontal displacement of the masses as a function of time and of equilibrium position of the masses is given by the equation of a standing wave; this is particularly noticeable when n is large and i is much smaller than n. So our n mass oscillator represents a discrete model that approaches a one-dimensional longitudinal standing wave, such as a sound wave bouncing between two perfectly fixed walls.