Surface Wave Along a Locally Reacting Cylinder

→ See also Mechel, Vol. I, Ch. 11 (1989)

The topic here is a surface wave along a cylinder, not around a cylinder. The cylinder has a diameter 2a and is locally reacting at its surface with the normalised radial impedance W= Z/Z${}_{{0}}$ = 1/(Z${}_{{0}}$G) ($G=$ admittance). The wave is supposed to have an axial symmetry. It is formulated as

$$\begin{array}[]{@{}l}p(r,z)=P_{0}\cdot K_{0}(\Gamma _{r}r)\cdot e^{{-\Gamma _{z}z}}\quad;\quad\Gamma _{r}^{2}+\Gamma _{z}^{2}=-k_{0}^{2}\,,\\ Z_{0}v_{r}(r,z)=\displaystyle\frac{j}{k_{0}}grad_{r}p=\displaystyle\frac{-j\,\Gamma _{r}}{k_{0}}\, P_{0}\cdot K_{1}(\Gamma _{r}r)\cdot e^{{-\Gamma _{z}z}}\\ \end{array}$$

(1)

with the modified Bessel function K${}_{{0}}$(z) of the second kind of zero order. The boundary condition at $r=a$ leads to the characteristic equation for $\Gamma$ ${}_{{r}}$a:

$$\Gamma _{r}a\cdot\frac{K_{1}(\Gamma _{r}a)}{K_{0}(\Gamma _{r}a)}=-j\, k_{0}a\cdot Z_{0}G=-j\cdot U.$$...