Relation between Frequency and Overtone in Circular Membrane

Question

How does the frequency change along the radius of a circular membrane such as a snare drum (with respect to overtones). If I strike the drum in the center, I will get a frequency. Now how will this frequency change if I move say 7 cm from the center of the drum?

Answer 1

For the reasons I explain below, this is not a full answer to the question, but just a quick summary of what can be said in general about this problem.

The spectral composition (overtones) of the vibrations of a circular membrane is a relatively easy problem to solve (at least conceptually). However, you should be aware that the solution to this problem does not provide a solution to the more complicated problem of the frequencies of a real snare drum. In this case, additional elements enter into play (the presence of a second resonating membrane, the lower skin, the air inside the drum, the skin and lateral surface materials, the sizes of the drum, the form and kind of the beater, just to mention the most relevant) making a quantitative answer to your question impossible without additional data. Even with all the missing details, a realistic calculation would require a non-trivial numerical solution of the relevant equations.

As far as a simple circular membrane is concerned, the starting point is to determine all the vibrational modes, i.e. the possible standing waves of the membrane. It is an exercise of solution of partial differential equations (details can be found in this Wikipedia page ). The general solution in polar (cylindrical coordinates) can be written as a linear combination of the following set of normal modes (I am using the same notation as in the above mentioned Wikipedia page): $$ u_{mn}(r,\theta,t)=(A_{nm}\cos c \lambda_{mn}t + B_{nm}\sin c \lambda_{mn}t) J_m(\lambda_{mn}r) \left( C_m \cos m \theta + D_m \sin m \theta \right), $$ where $c$ is the phase speed of the elastic vibrations of the membrane, $m=0,1,\dots, n=1,2,\dots$. $\lambda_{mn}=\alpha_{mn}/a$, where $\alpha_{mn}$ is the $n-$th positive root of $J_m$, and $a$ is the radius of the membrane. The value of the constants depends on the initial perturbation of the membrane.

A particular motion of the membrane can be obtained by requiring that the sum of all these modes $u(r,\theta,t)=\sum_{mn}u_{mn}(r,\theta,t)$, describes the initial displacement and speed of displacement of the membrane $$ \begin{align} u(r,\theta,t=0)&=u_0(r,\theta)\\ \dot u(r,\theta,t=0)&=v_0(r,\theta) \end{align} $$ Typically, vibrations as a consequence of hitting the membrane in a point can be represented via an initially undistorted membrane ($u_0=0$) and the initial velocity concentrated around a small angular and radial region of the membrane. Notice, that a central hit would excite only axisymmetric modes, i.e. modes with $m=0$.