Vibrating beam - physical interpretation

[Unco]

Homework Statement

This question arises from partial differential equations work, but concerns introductory-level physics.

The interpretation is of solutions representing the displacement (a function, u(x, t), of position and time) of a vibrating bar, fixed but hinged at each end.

My question #1:
I have found the ratio of the "energy" (the square coefficient of the eigenfunction, and independent of time), for the first three harmonics for three different initial displacements. The bar has length L and these initial displacements are (a) x(L-x), (b) x^2(L-x), (c) x^3(L-x), so correspond to hitting the bar in the center, and progressively off center.

Some definitions - pitch of a note: the frequency of the fundamental harmonic; tone: frequency of higher harmonics; purest tones are those of lower energy in higher harmonics.

For (a), even harmonics give zero displacement and hence zero energy. Odd harmonics for (a) are the same as those of (b). Energy of harmonics, even or odd, of (c) are higher than both (a) and (b).

My question is, which is considered to produce the purest tone, for playing a xylophone, for instance: having no even harmonics (which means no n=2 harmonic but also fewer higher - say, n=10 - harmonics) as in (a), or having similarly low energy harmonics in (b) which doesn't miss out on n=2, etc. I take it they're both better than (c).

---
My question #2:
Also relating to a vibrating beam, I have found (following instruction) the ratio of the frequency (coefficient of t in a cosine term) of the second harmonic to the fundamental to be greater for a tuning fork than that of a vibrating string. Is it better, therefore, to use a tuning fork to tune a musical instrumental rather than a vibrating string because the pitch of a note is more marked/distinguishable to the ear for the tuning fork?

Apologies for any incorrect physical terminology. "Energy" is described to us in inverted commas, but I hope the description has some recognisable physical element to it, anyway.

 

[member 553083]

Homework Equations N/AThe Attempt at a SolutionMy attempt at a solution #1:A higher energy in higher harmonics would produce a less pure sound, so I think (a) would produce the purest tone for playing a xylophone.My attempt at a solution #2:Yes, it is better to use a tuning fork to tune a musical instrument rather than a vibrating string because the pitch of a note is more marked/distinguishable to the ear for the tuning fork.