Find the vibrations of a circular drum head of radius 1m and density $2 kg/m^3$, given a tension of $8N$, and an initial velocity of 0.0 and an initial displacement of $f(r)=2r^2-2$ (in m). Recall that $c^2=T/\rho$. Please show work.

$V = \sqrt{F/\rho} =\sqrt{8/2} =2 m/s$

$\lambda = C*T$

the wave has its maximum in the center of drum head and minimum at the exterior of drum head. It means the fundamental resonance is for $R =\lambda/4$.

Next resonances are for $R=\lambda/4 +\lambda/2$

$R = k*\lambda/4$ with $k =1,3,5,….$

$\lambda = 4*R/k = 4/k$ with $k=1,3,5,….$

$T = \lambda/C = 4/k*C=4/2k =2/k$

$\omega = 2*pi/T = 2*pi / (2/k) =\pi*k$ where $k=1,3,5,…$

the equation of wave is

$d^2x / d t^2 +1/\omega^2 *x =0$

$x = A*sin(\omega*t+\phi) = A*sin(\pi*k*t +\phi)$

$v = A*\pi*k*cos(\pi*k*t+\phi)$

$x(0) =2r^2-2 =2*0-2 =-2 m$

$v(0) =0$

$v(0) = A*\pi*k*cos(\phi) =0$

$x(0) =A*sin(\phi) =-2$

$cotg(\phi) =V(0)/X(0) = 0$ means $\phi = \pi/2$

$A*sin(\pi/2) =-2$ means $A=-2$

the equation of vibration in the center of the drum is

$X(t) = -2*sin(\pi*k*t +\pi/2)$ with $k=1,3,5,….$