Growth of a Train of Waves. 29 



vibration as far as two or three wave-lengths on each side of! 

 the middle. The third curve in each case is a curve of sines. 

 The first curve represents the surface at the beginning of a 

 period from ir to (i + l)T. The fifth curve, being the first 

 curve inverted, represents the water-surface at the middle of 

 the period. The other two curves may be described as 

 components of the first and third, according to the following 

 formula : 



£0, l,*)=Psinft>£-QcosG>* . . (193), 



where P= -A cos 2irx/X .... (1'91), 



and Q is a continuous transcendental function of x, having 

 equal values for ±x, expressed by (195) for positive or 

 negative values of <r, exceeding a wave-length. 



For x positive, Q— — A sin 2irx/X ; for x negative, Q= + A sin 2irx/X (195), 



where A denotes the semi-amplitude of the vibration, at any 

 time long enough after the beginning, and place far enough 

 from the middle of the disturbance, to have very approxi- 

 mately sinusoidal motion. The determination of the trans- 

 cendental function Q. and the calculation of A, for both P 

 and Q, will be virtually worked out in § 151 below. 



§ 146. We have now an exceedingly interesting and 

 suggestive analysis of the circumstances represented in 

 fio\ 39. Consider separately the two motions corresponding 

 to Psintwtf alone, and to — Q cos cot alone. The motion 

 Psin«£, if at any instant given from x = -co to #=-f oo 5 

 would continue for ever, as an infinite series of standing- 

 waves, without any surface-pressure. Hence our application 

 of surface-pressure is only required for the Q-motion : and 

 if this motion be at any instant given from x= — oo to 

 #=-j-3o, it will go on for ever, provided the pressure 



—cos G)£^ (jc, lj 0) is applied and kept applied to the 



surface. 



§ 117. The plan of § 11(3 may be generalised as follows : — 

 Displace the water according to the formula (193) with P 

 omitted, and with Q any arbitrary function of x for moder- 

 ately great positive or negative values of x, gradually changing 

 into the formula (195) for positive and negative values 

 outside any arbitrarily chosen length MOX (MO not 

 necessarily equal to ON). Find mathematically the sinu- 

 soidally varying surface-pressure, F(a?) cos cot, required to 

 cause the motion to continue according to this law. Super- 

 impose, upon the motion thus guided by surface-pressure, 



