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Drums
When you hit a drum the membrane forms standing waves
There are "special" modes of vibration, wherein all points oscillate with the *same frequency*
There are actually an infinite # of such modes (called eigenmodes), for a given surface!
I've plotted the first 2
pic.twitter.com/oH1drV9NTy
Consider a drum head of radius R.
Its motion is governed by the 𝒘𝒂𝒗𝒆 𝒆𝒒𝒖𝒂𝒕𝒊𝒐𝒏
Imposing the edges of the drum are fixed tells us
the "special" frequencies
The solution u_n(r,t) contains J₀(kr): the zeroth Bessel function
α_n is the nᵗʰ root of J₀(x)pic.twitter.com/8A7Twn4Zxi
Wave Speed
The wave speed 𝑣 is the speed at which the waves will propagate along the surface
𝑣² is proportional to the tension (per unit length) across the drumhead
𝑣² is inversely proportional to the mass (per unit area) of the drumpic.twitter.com/InGdrw8BqX
What about the tension of the drum head?
Take an elliptic drum, and you get Mathieu functions, a fundamentally different kind of beast.
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