The second index tells us how many zero crossings the field distribution has in the radial direction. So if we fix all other parameters and just move radially outwards, how many zero crossings the field variation would see is essentially given by the second index. If we don't have any zero crossing then we have , if we have one zero crossing then , and so on.
6.
For the characteristic equation (12a) becomes
(13a)
So either of the brackets could be equal to zero.
(a)
If we take first bracket equal to zero, the equation gives the characteristic equation of the transverse electric mode.
Similarly, the second bracket equal to zero, gives the characteristic equation of the transverse magnetic mode.
(b)
Since represents transverse electric and transverse magnetic modes, the and modes have field
distributions which are essentially circularly symmetric.
If then we always get a field distribution which is hybrid. So the transverse electric and transverse magnetic fields have only radial variation, and they do not have any variation in the direction.
(c)
If we take the first bracket equal to zero, we get
________ (2-13b)
This is the characteristic equation for mode.
We get multiple solutions for this equation because function is an oscillatory Function.
Using recurrence relation for the Bessel function, we have and .
, if we have one zero crossing then 
, and so on. 
the characteristic equation (12a) becomes 
(13a) 
represents transverse electric and transverse magnetic modes, the 
and 
modes have field
then we always get a field distribution which is hybrid. So the transverse electric and transverse magnetic fields have only radial variation, and they do not have any variation in the 
direction. 
________ (2-13b) 
mode. 
function is an oscillatory Function. 
and 
. 
