In cylindrical polar coordinates ), the Hemholtz equation is

attachment 1 (cylind wave)

From above, bessel function is obtained to solve the function in terms of r. With proper boundary condition, equation below is obtained.

attachment 2 (bessel char)

I just need to get the roots k_{mn}for n=1,2,3... and m=0, with a=0.306, b=0.179

It can be done numerically but i don't know how,

In Matlab, i run this codes (it give same answer as in literature) but the paper could be wrong since it didnt tell how it was obtained. It suppose to be bessel function of order 0 (m=0). But in this codes, m is 4 and n is 2.34. In another paper, the root given by this code is same as for m=0, n=4. In codes below, from my understanding nu2 and nu3 suppose to be 0 since m is 0, but it doesnt give answer.

nu2=4; nu3=2.34;

z = (0:0.2:100)';

a = besselj(nu2,z);

da = diff(a);

b = bessely(nu2,z);

db = diff(b);

c = besselj(nu3,z);

dc = diff(c);

d = bessely(nu3,z);

dd = diff(d);

f=da.*dd;

g=db.*dc;

h=f-g;

plot(f);

hold all

plot(g);

hold all

plot(h);

hold all

axis([10 25 0 .01])