File talk:Drum vibration mode03.gif

Correction of the relative frequencies

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 $\sigma$, surface tension $T$, and radius $R$, the equations of motion in radial coordinates are given by

$\sigma {\frac {\partial ^{2}z}{\partial t^{2}}}=T\left({\frac {1}{r}}{\frac {\partial }{\partial r}}(r{\frac {\partial z}{\partial r}})+{\frac {1}{r^{2}}}{\frac {\partial ^{2}z}{\partial \varphi ^{2}}}\right)\ ,$

leading to solutions

$z_{mn}=Ae^{i\omega _{mn}t}e^{im\varphi }J_{m}(\xi _{mn}{\frac {r}{R}})\$

with $J_{m}$ the mth Bessel function and

$\omega _{mn}={\frac {\xi _{mn}}{R}}{\sqrt {\frac {T}{\sigma }}}\ .$

The $z_{mn}$ need to vanish on the boundary (i.e. for $r=R$). Hence, the $\xi _{mn}$ are given as being the nth zero of the mth Bessel function $J_{m}$, respectively.

The solutions depicted in the three gif files are the (0,1), (0,2), and (0,3) modes: no azimuthal dependency indicates $m=0$ 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 $J_{0}$ are 2.405, 5.520, 8.654, 11.791, ... leading to relative frequencies of $f_{01}=1.000$, $f_{02}=2.295$, and $f_{03}=3.598$.

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)