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).

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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.

 

[Nate810]

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.

 

[m2003]

Homework Equations
The physical interpretation of a vibrating beam involves the concept of resonance and harmonics. Resonance occurs when the frequency of an external force applied to a system matches the natural frequency of the system, resulting in a large amplitude of vibration. In the case of a vibrating beam, the natural frequency is determined by the length, density, and tension of the beam.

The Solution
The vibrating beam can be thought of as a musical instrument, where the different harmonics correspond to different notes. The fundamental frequency (first harmonic) is the lowest note produced by the beam, and each subsequent harmonic produces a higher note. In terms of the physical interpretation, the harmonics represent the different modes of vibration of the beam.

In regards to the question #1, the purest tone would be produced by having no even harmonics, as in case (a). This is because even harmonics are not present in the natural vibration of the beam and can only be produced by external forces. Therefore, having no even harmonics would result in a more "natural" sounding tone.

For question #2, the tuning fork may be better for tuning instruments because it produces a more distinguishable pitch. This is because the tuning fork has a fixed frequency, while the frequency of a vibrating string can vary depending on factors such as tension and length. However, the choice of which to use for tuning would ultimately depend on the preference of the person tuning the instrument.