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 *λ*_{i}, then the eigenvalue equation tells us

…

Here, we now define ; with this we then have:

, *k*=1,2,…,*n*; or, using our value for *λ*_{i},

.

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 *i*th fundamental mode. Note that these are sine waves of decreasing wavelength: the *i*th 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.

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Tags: Coupled Oscillator, Discrete Model, Friday Physics, Longitudinal Wave, physics, Standing Wave

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