File talk:Drum vibration mode02.gif

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Correction of the relative frequencies

[edit]

The three pictures File:Drum_vibration_mode01.gif, File:Drum_vibration_mode02.gif, and File:Drum_vibration_mode03.gif consist of 19 frames with a duration 100 ms for each frame. In particular when showing them next to each other (see e.g. File:Membrane-normal-modes.gif) this leads to the wrong impression that they all had the same period, resp. the same frequency.

I will follow the argumentation of Vorlesung zur "Physik der Musikinstrumente" (in German) (PDF) 86-91 (5 November 2007). Retrieved on 9 August 2017.. From those notes, it is clear that the frequencies should differ, and that they can be derived as follows.

The pictures show a circular membrane with fixed boundary. With areal mass density , surface tension , and radius , the equations of motion in radial coordinates are given by

leading to solutions

with the mth Bessel function and

The need to vanish on the boundary (i.e. for ). Hence, the are given as being the nth zero of the mth Bessel function , respectively.

The solutions depicted in the three gif files are the (0,1), (0,2), and (0,3) modes: no azimuthal dependency indicates modes. The first gif indicates a zero only at the boundary, the second has one ring-shaped node, the third has two nodes.

The first zeroes of are 2.405, 5.520, 8.654, 11.791, ... leading to relative frequencies of , , and .

I will modify the gif files in the following way: in File:Drum_vibration_mode01.gif, I will set the duration of each frame to 230 ms. In File:Drum_vibration_mode02.gif, I keep the 100 ms. In File:Drum_vibration_mode03.gif, I will shorten the period to 64 ms.

The original gif files have been created using Matlab. They are in the public domain. I informed the original author User:Oleg Alexandrov on his Commons talk page and he did not contradict my planned modification of his pictures. The gif files were modified using the gif-x tool downloadable from the psynetic web site.

--Dogbert66 (talk) 00:09, 9 August 2017 (UTC)[reply]