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Note: 1 lecture, §10.2 in [EP], exercises in §11.2 in [BD]

The eigenfunction series can arise even from higher order equations. Suppose we have an elastic beam (say made of steel). We will study the transversal vibrations of the beam. That is, assume the beam lies along the axis and let measure the displacement of the point on the beam at time . See Figure 5.2.

The equation that governs this setup is

for some constant .

Suppose the beam is of length 1 simply supported (hinged) at the ends. The beam is displaced by some function at time and then let go (initial velocity is 0). Then satisﬁes:

(5.5) |

Again we try and plug in to get or

We note that we want . Let us assume that . We can argue that we expect vibration and not exponential growth nor decay in the direction (there is no friction in our model for instance). Similarly will not occur.

Write , so that we do not need to write the fourth root all the time. For we get the equation . The general solution is

Now , . Hence, and , or . So we have

Also , and . This means that and . If , then and so . This means that otherwise is not an eigenvalue. Also must be an integer multiple of . Hence and (as ). We can take . So the eigenvalues are and corresponding eigenfunctions are .

Now . The general solution is . But and hence we must have and we can take to make for convenience. So our solutions are .

As eigenfunctions are just sines again, we can decompose the function on using the sine series. We ﬁnd numbers such that for we have

Then the solution to (5.5) is

The point is that is a solution that satisﬁes all the homogeneous conditions (that is, all conditions except the initial position). And since and , we have

So solves (5.5).

The natural (angular) frequencies of the system are . These frequencies are all integer multiples of the fundamental frequency , so we get a nice musical note. The exact frequencies and their amplitude are what musicians call the timbre of the note (outside of music it is called the spectrum).

The timbre of a beam is diﬀerent than for a vibrating string where we get “more” of the lower frequencies since we get all integer multiples, . For a steel beam we get only the square multiples . That is why when you hit a steel beam you hear a very pure sound. The sound of a xylophone or vibraphone is, therefore, very diﬀerent from a guitar or piano.

Example 5.2.1: Let us assume that . On we have (you know how to do this by now)

Hence, the solution to (5.5) with the given initial position is

Exercise 5.2.2: Suppose you have a beam of length 5 with free ends. Let be the transverse deviation of the beam at position on the beam (). You know that the constants are such that this satisﬁes the equation . Suppose you know that the initial shape of the beam is the graph of , and the initial velocity is uniformly equal to 2 (same for each ) in the positive direction. Set up the equation together with the boundary and initial conditions. Just set up, do not solve.

Exercise 5.2.3: Suppose you have a beam of length 5 with one end free and one end ﬁxed (the ﬁxed end is at ). Let be the longitudinal deviation of the beam at position on the beam (). You know that the constants are such that this satisﬁes the equation . Suppose you know that the initial displacement of the beam is , and the initial velocity is in the positive direction. Set up the equation together with the boundary and initial conditions. Just set up, do not solve.

Exercise 5.2.4: Suppose the beam is units long, everything else kept the same as in (5.5). What is the equation and the series solution?

Exercise 5.2.5: Suppose you have

That is, you have also an initial velocity. Find a series solution. Hint: Use the same idea as we did for the wave equation.

Exercise 5.2.101: Suppose you have a beam of length 1 with hinged ends. Let be the transverse deviation of the beam at position on the beam (). You know that the constants are such that this satisﬁes the equation . Suppose you know that the initial shape of the beam is the graph of , and the initial velocity is 0. Solve for .

Exercise 5.2.102: Suppose you have a beam of length 10 with two ﬁxed ends. Let be the transverse deviation of the beam at position on the beam (). You know that the constants are such that this satisﬁes the equation . Suppose you know that the initial shape of the beam is the graph of , and the initial velocity is uniformly equal to . Set up the equation together with the boundary and initial conditions. Just set up, do not solve.