Le Mehaute 



^-tJ^Ty^) (20) 



in which it is understood that k is the root of Eq, (16) 



It can be seen from Eq. (20) that for any fixed value of r, 

 the least nonzero value of k for which dA/dk = is independent 

 of r; this means that the maximum of the first wave envelope (where 

 dA/dk = 0) is associated with a constant value of k (and, therefore, 

 wavelength and period) throughout its propagation. This constant 

 value of k at the first envelope maximum, k^^^^ depends only on 

 the nature of the source disturbance 110(^0) through the factor "nCk). 



Evaluating A at k^g^^, we can write 



1/2 



A,no,f = { ^(k)( _dv/dk ) } Lk " C°^«t^^* 



(21) 



for a particular source deformation 'no(ro) , which means that the 

 amplitude of the maximum waves is inversely proportional to r. 



Before we can proceed with the quantification of the theoretical 

 model, we must select an appropriate form of ilo(rQ). The two con- 

 straints on this choice are, first, that the resulting wave envelope 

 shape be sufficiently similar to observed shapes that some manipu- 

 lation of numerical coefficients will give an accurate fit; and, second, 

 that the Hankel transform of 'nQ(ro) be within our power to obtain in 

 a closed form. 



In addition, it would be nice -- although it is not really neces- 

 sary --to have 110(^0) intuitively resemble the effective surface 

 deformation due to an explosion. For all these reasons, we are led 

 to try simple polynomials in r for Tlo(rQ) with crater-like shapes. 



Of the three forms which have been used in practical work 

 (Table I) , the last has been tentatively established as most suitable: 









> R 



where "Homax ^s a coefficient which, for the sake of simplicity, will 

 be written as "Ho in the following. 



The wave amplitude is then given by 



84 



