In the theory of vibrations of a circular drum, the displacement of the drumhead can be expressed in terms of pure harmonic modes,
where 
are polar coordinates, 
, t is time, m is a positive integer, c is a material parameter, and 
is a Bessel function of the first kind. The quantity 
is a resonant frequency and is a positive root of the equation
are polar coordinates, , t is time, m is a positive integer, c is a material parameter, and is a Bessel function of the first kind. The quantity is a resonant frequency and is a positive root of the equation
which states that the drumhead is clamped around the rim. Tabulating approximations to the zeros of Bessel functions has occupied countless mathematician-hours throughout the centuries.
Too bad they didn't have MATLAB back in the day. Bessel functions are built in.
J3 = @(x) besselj(3,x);
fplot(J3,[0 20]), grid on
xlabel('x'), ylabel('J_3(x)') % ignore this line
title('Bessel function') % ignore this line
From the graph we see roots near 6, 10, 13, 16, and 19. We use fzero to find these roots accurately.
omega = [];
for guess = [6,10,13,16,19]
omega = [omega;fzero(J3,guess)];
end
omega
hold on, plot(omega,J3(omega),'.')
title('Bessel roots') % ignore this line
