Ex-plosion-Genevated Water Waves 



3,4 Main Features of the Mathematical Models 



[Based on Eqs. (13) and (14)] , typical examples of various 

 models for initial surface deformations "HoCro) are given in Table I. 

 The first case, a parabolic water crater, is that proposed by Kranzer 

 and Keller [i 959] . 



The general features of traveling wave trains given by the 

 equations presented in Table I are: 



a. The waves travel radially from the explosion, 



b. The leading free sur fac e disturbance or leading wave 

 travels at velocity yj gD . 



c. At a given location, the frequency of individual waves 



increases monotomically, 



d. The amplitudes of individual waves (cosine function in 

 Eq, (12)) are modiilated into groups of successively 

 smaller amplitude by the slowly varying Bessel function 

 J^(kr^) in Eq. (14). 



e. The number of waves in a given group increases with 

 time or distance traveled, 



f. The length of a group increases linearly with time or 

 distance traveled, 



g. The frequency associated with a specific crest decreases 

 with time or distance traveled (equivalently, a given 

 crest moves forward within a group). 



h. The frequency associated with the maximum amplitude 

 of a given group is constant, 



i. The maximum height of a given group decreases as the 

 inverse of time or distance traveled. 



j. The maximum height of successive groups passing a 

 given point decreases with time. 



These features are partly illustrated in Fig. 4, which shows a com- 

 puted wave train at three different locations. The general decay of 

 wave height with distance is the result of both radial dispersion and 

 circular spreading. This radial dispersion is characterized by a 

 general increase in the wave length of individual waves with distance. 



Wave crests occur when cos (cot - kr) = 1 (Eq. (13)), and 

 crest order numbers are given by cot - kr = ■rr(2n- 1) where n is an 

 integer. It Is also interesting to note that in the case of deep water, 

 the trajectories of individual waves in the r - t plane are defined 

 by parabolae: 7r(2n-l ) = gt /4r, whosQ consecutive arrival times at 

 any point r will be in the ratios t:t/^fZ:t/^j5 , etc. Similarly, at any 

 instant of time t, the consecutive crest radii will have the ratios 

 r:r/3:r/5, etc. 



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