Editing Wind chime (section)

== Mathematics of tubular wind chimes ==
[[File:Free-Free Euler-Bernoulli Beam animation.gif|thumb|The mode 1 (lowest frequency) vibration of a free Euler–Bernoulli beam of length 1]]

A wind chime constructed of a circular tube may be modelled as a freely vibrating [[Euler–Bernoulli beam theory#Example: free–free (unsupported) beam|Euler–Bernoulli beam]]<ref name="Hite">{{cite web |url=http://leehite.org/Chimes.htm |title=Say it with Chimes |last1=Hite |first1=Lee  |access-date=2016-08-01}}</ref> and the dominant frequency in cycles per second is given by:

:<math>
\nu_1 = \frac{\beta_1^2}{2\pi} \sqrt{\frac{EI}{\mu}} = \frac{22.3733}{2 \pi L^2}\sqrt{\frac{EI}{\mu}}
</math>

where ''L'' is the length of the tube, ''E'' is the [[Young's modulus]] for the tube material, ''I'' is the [[second moment of area]] of the tube, and &mu; is the mass per unit length of the tube. Young's modulus ''E'' is a constant for a given material. If the inner radius of the tube is ''r<sub>i</sub>'' and the outer radius is ''r<sub>o</sub>'', then the second moment of area for an axis perpendicular to the axis of the tube is:

:<math>
I=\frac{\pi}{4}(r_o^4-r_i^4)
</math>

The mass per unit length is:

:<math>
\mu=\pi \rho (r_o^2-r_i^2)
</math>

where &rho; is the density of the tube material. The frequency is then

:<math>
\nu_1=\frac{22.3733}{4\pi L^2}\sqrt{\frac{E}{2 \rho} }\,\sqrt{D^2+W^2}
</math>

where ''W=r<sub>o</sub>-r<sub>i</sub>'' is the wall thickness and ''D'' is the average diameter ''D=r<sub>o</sub>+r<sub>i</sub>''.
For sufficiently thin-walled tubes the ''W<sup>2</sup>'' term may be neglected, and for a given material, the main frequency is inversely proportional to ''L''<sup>2</sup> and proportional to the diameter ''D''.

For the main mode of vibration, there will be two nodes on the tube, where the tube is motionless during the vibration. These nodes will be located at a distance of 22.416% of the length of the tube from each end of the tube. If the tube is simply supported (not clamped) at one or both of these nodes, the tube will vibrate as if these supports did not exist. A wind chime will give the clearest and loudest tone when it is hung using one of these node points as the attachment point. These attachment points are also the same as used by other similar instruments such as the [[xylophone]] and [[glockenspiel]].