232 



CHARACTERISTIC VALUES FOR THE BILINEAR M CURVE 



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 some months ago. 

 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, 

 and if placed high enough it not only can reflect to 

 enormous distances but also becomes extremely 

 effective. It was decided not to give this effect too 

 much concern as all these calculations are made on 

 the basis of horizontal stratification. Doubtless all 

 sorts of small departures from a smooth curve occur 

 at various rather large heights, but they do not occur 

 perfectly stratified over areas of hundreds of square 

 miles, and only such perfectly stratified departures 

 could cause embarrassing results. Accordingly it was 

 decided that such fluctuations as occur probably 

 cause fading or fluctuation but do not cause the 

 particularly troublesome effect mentioned, because 

 they are local disturbances which are not stratified 

 over large areas. 



