# Measuring the tension of a drumhead

I'm working on an experiment to see how the tension of a drumhead impacts the frequency of its sound, but I'm not sure as to how I could measure this. I found this forum thread from 2012, which suggests several things. First of all, it suggests using a DrumDial, but it's very expensive and doesn't actually measure tension, but rather how far in the needle's able to push into the drum. The forum thread suggests using the following formula:

The variables are the radius a, the mass density p, and the velocity of the transverse waves c. I guess this is the speed of sound in the drum surface material. So if you know the fundamental frequency w1, you should be able to substitute into the equations given, rearrange, and find T, as c is given by rearranging the second equation.

a * w = c * 2.40483

T = c² * p

However, the source cited for this formula seems to be a website that no longer exists, so I can't find the reasoning behind it.

Any suggestions would be greatly appreciated! :)

As Chemomechanics has pointed out and as discussed in some detail here, the equations are related to the solution of the wave equation on a circular membrane. The solution corresponding to the fundamental on a membrane of radius $$a$$ is $$u(r, t) = A\cos\left(\frac{cz_1}{a}t+\phi\right)J_0\left(\frac{z_1}{a}r\right)$$ where $$A$$ and $$\phi$$ are constants that depend on initial conditions, $$J_0$$ is the zeroth-order Bessel function of the first kind, $$z_1\approx 2.40483$$ is its first zero (the smallest value of $$x$$ for which $$J_0(x)=0$$), $$c=\sqrt{T/\rho}$$ is the wave speed in the membrane, $$T$$ is the tension in the membrane (in $$\text{N/m}$$), and $$\rho$$ is the mass density of the membrane (in $$\text{kg/m}^2$$). The fundamental angular frequency is $$\omega_1 = \frac{cz_1}{a} = \sqrt{\frac T \rho}\frac{z_1}{a}.$$
I stress that this is the angular frequency, related to the frequency $$f_1$$ (in Hz) via $$\omega_1 = 2\pi f_1$$.