236 



INCIPIENT LEAKAGE IN A SURFACE DUCT 



puted therefrom were found to be in fair agreement 

 with the exact values, as is shown in Table 1. 



The physical nature of the duplicity of solutions 

 seems to be as follows. The solutions approaching a 

 limiting characteristic value for large duct height g 

 correspond to the case where the ground sinks to 

 great depths; the other solution corresponds, of 

 course, to the limiting case when the height of joint 

 rises to infinity. 



The relative importance of the two types of solu- 

 tion will depend on the ranges and heights considered. 

 At sufficiently great ranges the solutions with the 

 smaller value of A m will predominate, but the greater 

 the height considered the farther must one recede 

 from the source before the initial advantage of the 

 limiting solution due to a greater height gain is 

 overcome by the stronger horizontal attenuation. 

 The greater height gain of the limiting solutions at 

 high elevations is illustrated in Figures 4 and 5. In 

 these figures, the height-gain functions for the limit- 

 ing solutions are drawn in solid lines, those for the 

 Gamow solutions in dashed lines; and the unit of 

 height is the duct height. It should also be pointed 

 out that the normalization condition applied was 



1000 



Ui{z)dz = g , 



(13) 



so that, if a comparison of height-gain functions of 

 solutions of the same class for different values of g 

 is desired, the plotted values should be divided 

 by Vg~. 



100 



10 



0.1 



0.01 



3 



Figure 5. Height-gain functions of the second mode for 

 a bilinear M curve, s = — 1. z = height above ground 

 in natural units, g = height of joint in natural units. 

 Ui (z) = normalized wave function. 



