VALUES FOR FIRST MODE FOR BILINEAR M CURVE 171 
in W except at great distances. Some other mode, 
which dees not suffer from the depth-loss effect, 
although it may have greater attenuation, will be 
the important mode at smaller distances. 
The conclusions for positive s cannot be expected 
to apply unless the lower part of the M curve is 
really sensibly straight over a considerable part of 
its length. For negative s (trapping) this requirement 
is not so important. 
It was mentioned that other models had been 
employed by various investigators in calculating 
field strength in the presence of a duct. The British 
used an index distribution given essentially by Y = 
(z — 2z"/m), where m lies between zero and unity. 
When m = % the problem could be treated by a 
phase integral method, which Booker had done. The 
differential analyzers at Manchester and Cambridge 
had been used to obtain the characteristic values for 
other values of m. The linear variation of index had 
been studied by Hartree and Pearcey. In this case 
of linear exponential variation Y = z + Ae-s:, where 
A and B are adjustable parameters. This model offers 
the advantage that the index is an analytic function 
of z and also that the modification term approaches 
zero with increasing height. 
An alternative method (Langer’s) for joining the 
two parts of an otherwise bilinear M profile was 
brought up. This method gives a solution in terms 
of Bessel functions and solves the difficulty perfectly 
for joining two straight lines. 
It was inquired whether, in case of positive s it 
had been ascertained that for large g there were no 
roots of the secular equation corresponding to a 
linear M curve having the slope of the lower segment. 
There was the possibility that the root found might 
be one of a possible pair and that there might be 
another solution of the wave equation for positive s 
which had not yet been discovered. 
The author replied that the roots varied continu- 
ously as g varied and that the investigation had 
dealt with the root obtained when starting with the 
first standard value for g = 0. What happened with 
increasing g when the start was made from some 
other standard value of g = 0 was not known defi- 
nitely, but the effects were believed to be peculiar. 
It is expected that there may be some values lying 
fairly near the s squared value for the imaginary 
part. They are not considered to lie close to the s 
squared value for the real part, as they would for 
the simple assumption previously mentioned—that 
when the joint is very high the upper segment can 
be forgotten and the curve can be assumed to be a 
single line all the way. This is believed incorrect, 
because when the result is derived by taking only 
the first terms in the asymptotic expansions, com- 
puting a small correction from the next terms in the 
asymptotic expansions produces terms which are 
infinite compared to the first terms. This means that 
the value s squared times D is an impossible one. 
It may well be that there are results with s squared 
times A plus some different value of B rather than 
simply s squared times B, but these have not been 
investigated. This does not occur for the first mode, 
which is all that this report covers, but it may 
happen that some other mode goes over to that 
value. Any mode which does so would probably not 
suffer from depth-loss effect and would be the 
important mode close in when there was a thick 
layer with positive slope. 
The need was pointed out for stressing the differ- 
ence between “completely trapped’? modes and 
“leaky” modes. With completely trapped modes the 
field decreases exponentially with height, and the 
power carried by each mode is finite, but with leaky 
modes the field increases exponentially with height, 
and the power carried by each mode is infinite. This 
means that completely trapped modes may exist 
separately, but leaky modes may not. The expan- 
sions of fields in terms of leaky modes are thus 
essentially mathematical and from physical considera- 
tions it is no longer possible to anticipate that these 
expansions would be convergent; the question of 
convergence has to be settled formally. The reac- 
tions of trapped and leaky modes to small perturba- 
tions are quite different. The former are relatively 
insensitive and the latter are very sensitive. In 
considering the field at a certain distance from the 
transmitter, it must be ascertained whether the 
relevant modes are affected by changes in the 
dielectric constant at heights large compared with 
this distance; if this is the case particular care must 
be taken in proving the sum to be still the same, 
since even a perfectly reflecting layer at such great 
heights can have little effect on the field in the 
region of interest. 
It was noted that these remarks pertained to a 
phenomenon which had greatly puzzled the investi- 
gators for several months. The trouble occasioned 
by the concept that infinite energy is carried by a 
mode does exist. This means that the formula in 
terms of modes is valid only if all those modes are 
summed that make any appreciable contribution. It 
becomes extremely difficult to carry out the summa- 
tion when there are numerous modes, as they begin 
to cancel each other more and more with progress 
into that region. This occurs in leaving the diffraction 
region to which this work is meant to apply and in 
approaching the optical region. The question of what 
a small departure from the shape of the curve at 
great heights does is something which was very 
troublesome during studies made of the first mode 
There is no doubt that a small departure from a 
smooth shape of the M curve has an enormous 
effect on the results if it occurs at a great height. If 
the departure is located high enough it need not 
amount to more than a millionth of an M unit to 
spoil the calculation completely. That is because it 
is a reflecting layer similar to the Heaviside layer, 
