Consider an annular membrane fixed along its outer edge where the inner edge is attached to a mass $M$ that is allowed to move along the vertical direction only. The membrane has a uniform tension per unit length $P$ and mass per unit area $.$ The outer and inner radii are a and $b,$ respectively. By considering the transverse vibration only, show that the characteristic $($or frequency$)$ equation is given by

$\frac{J_0(a)Y_0(b)-J_0(b)Y_0(a)}{J_0(a)Y_1(b)-J_1(b)Y_0(a)}+\frac{2bP}{M{c}^2}=0$

where $=c$ and $c=\sqrt[\phantom{2}]{\frac{P}{}}.$

Hint: You may need to know that

$\frac{d}{dx}(J_0(x))=-J_1(x),$ and $,\frac{d}{dx}(Y_0(x))=-Y_1(x)$

The downward generated force from the membrane to the mass $M$ at $r=b$ is

$-P(2b)\frac{\text{d}\text{e}\text{l}\text{w}(b,,t)}{\text{d}\text{e}\text{l}\text{r}}$

Figure $2$: Annular membrane fixed at the outer edge and with a mass attached at the inner edge.