Frequency of drum As picture blow, there are two drums, one is round drumhead , and another is elliptical drumhead. If the drumhead has same tension, I mean they are same tight , what is relation of their frequency ?



 A: Equation 6.7 in Vibration of Membranes gives the frequency of waves on a rectangular membrane as
$f=\frac{v}{2\pi}\sqrt{(\frac{m}{a})^2+(\frac{n}{b})^2}$
where $v=\sqrt{\frac{T}{\rho}}$ is the speed of waves on the membrane. which depends on density $\rho$ and tension $T.$ Here $a>b$ are the lengths of side of the rectangle and $m,n$ are the numbers of anti-nodes in each direction.
From this you can see that even for the fundamental mode $m=n=1$ the frequency is different for a square $(a=b)$ than for a rectangle $(a \ne b)$. In this case, if the area $A=ab$ is constant then we can write
$f=\frac{v}{2\pi}\sqrt{(\frac{b}{A})^2+(\frac{1}{b})^2}$
which is a minimum for the square $(b^2=A)$.  
The equivalent result for elliptical membranes is more complex and can only be expressed numerically - see Eigenfrequencies of an Elliptic Membrane.
In Appendix A an expansion is given for the fundamental mode $(m=n=1)$ in terms of $x=1-\frac{b}{a}$ which has a maximum error of $0.0002$ :
$2\pi \frac{bf}{v}=2.4048-1.1924x+0.1768x^2+0.3923x^3-0.2107x^4$
where $2b$ is the minor axis of the ellipse. You could use this to compare the frequencies for the circle and ellipse.
In Appendix B the authors state that $f/\sqrt{\frac{1}{a^2}+\frac{1}{b^2}}$ is approximately constant for eccentricities up to about $0.65$, noting that this applies also for rectangles of any size. Unfortunately they do not state whether the area or the major axis is constant, nor whether this result applies only for the fundamental mode or for all modes.
As for rectangular membranes, the lowest frequency for any mode of an ellipse of constant area is found when the ellipse is a circle. As you note, for a given area the circle has the lowest frequency of any shape of membrane. 
