WILSON— AEROPLANE ENCOUNTERING GUSTS. 223 



the pilot would have to react so quickly — the constants of integra- 

 tion might turn out, as I have just suggested, such that the motion 

 during the first quarter minute was not far different from that in 

 the case of the simple gust. This was what was found to happen in 

 the case of the periodic gust above treated (§9). The amplitude of 

 the vertical motion so far as the particular solution was concerned 

 turned out to be about 5.3/, but the constants of integration were 

 such as to postpone the major effect of the particular solution until 

 30 or 40 seconds had elapsed. If we have a damped harmonic gust 

 and such a postponement were operative, the damping would become 

 effective and the gust might turn out to have at no time an effect 

 much in excess of the maximum effect of a single gust of the form 



Infinitely Sharp Gusts. 



21, In my previous paper I discussed gusts /(i — e-^*) rising 

 from zero to / with various degrees of sharpness depending on the 

 value of r — the larger ;-_, the sharper the gust. An infinitely sharp 

 gust would be one for which r was indefinitely large. Such a gust 

 would represent an absolute discontinuity in the velocity of the wind. 

 This is impossible, though it represents a state of aerial motion 

 which may be nearly approached. Moreover, the infinitely sharp 

 gust could not strike the machine all over at once, and hence the 

 theoretical effect of such a gust on the assumption that the machine 

 is instantaneously immersed must differ from the actual effect upon 

 a machine running into a discontinuity in the wind velocity. 



For this reason one may well limit his considerations to finite 

 gusts with a value of r not greater than 5, say, as I did. Neverthe- 

 less if the calculation of the effect of an infinitely sharp gust is 

 simpler than for a finite gust and if the limiting motion derived for 

 such a gust is not appreciably different from that for a sharp gust 

 of reasonable sharpness, the discussion of the limiting case will be 

 justified. 



22. Consider first the longitudinal motion and a head-on gust 

 Mi = /(i — ^"'"0, r enormously large. According to the symbolic 

 method D= — r must be substituted to find the particular solution 

 for e'^K As, however, A is of the fourth degree in D and all the 



