# Relation between Frequency and Overtone in Circular Membrane

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?

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 $$m=0$$.