Kettledrum Membrane Modes

What is modelled

An ideal tensioned membrane fixed at the rim obeys the wave equation, and its free vibrations are products of a Bessel function in radius and a cosine in angle. A mallet stroke excites the modes in proportion to their shape at the strike point, so striking near the centre favours the circular modes and striking near the rim favours the ones with nodal diameters.

Equations

• Wave speed c = √(T/σ), mode frequency fmn = jmn c / (2πa) where Jm(jmn) = 0.
• Shape u = Jm(jmn r/a) cos(mθ); strike amplitude ∝ Jm(jmn r0/a) / (ω Jm+1(jmn)2).
• Air loading as extra moving mass β = κ ρair a / σ, so f' = f / √(1 + β).

Checks run in the page

• Bessel zeros found by bisection against the tabulated values.
• Orthogonality and normalisation of two circular modes by numerical integration.
• A leapfrog wave simulation of the radial equation against the formula for the fundamental.
• The same simulation with the air mass added to the skin.

Honest simplifications

The head is an ideal membrane without stiffness, the rim is perfectly rigid and air loading is a single empirical coefficient, not the full coupled radiation problem. A real timpani also has a bowl, a pitch-shifting kettle volume and non-uniform skin, so this is not a tuning guide. The animation is slowed by a large factor.

• Press Struck at the centre and see only circular modes ring, with the nodal-diameter modes silent.
• Press Near the rim to bring in the higher modes and a brighter spectrum.
• Compare Vacuum with Tuned kettledrum to see how much the air lowers the pitch.