28 Lord Kelvin on the 



and £0(32, 1, i) the diagram shows that they are quite im- 

 perceptible at the end of period 4, and begin to be considerable 

 at the end of period 8, which would be the exact time of 

 arrival if there was a definite " group- velocity " equal to 

 half the wave-velocity. The largeness of S^(32, 1, t), approxi- 

 mately uniform throughout the first four periods, is explained 

 in § 141. Its gradual augmentation through periods 5, 6, 

 7, 8, depends on the wave propagation of disturbances from 

 the origin, as shown for S^(32, 1, t) and £d>(32, 1, t) in the 

 second and fourth curves. 



§ 143. The £0(0, 1, t) curve of fig. 38 may be compared 

 with the curve of the same designation in fig. 3G. Thev 

 differ because of a quarter period difference in the phase of 

 commencement of the disturbing pressure, which commences 

 suddenly at its maximum for all the curves of fig. 30, and 

 commences at zero for all the curves of fig. 38. If the 

 S^, S^ curves for initiating pressure commencing at zero 

 were drawn, they would differ from the first and third curves 

 of fig. 36 in being at the commencement tangential to the 

 line of abscissas, instead of being inclined to it in the positive 

 direction, as shown in fig. 36. The f curves are all initially 

 tangential to the line of abscissas, but the tangency is only 

 of the first order in fig. 36,, while it is of the second order in 

 fig. 38. 



§ 144. The third and fourth curves * of fig. 38 show the 

 whole history for the points, ,v = 0, and a- = \, of the surface 

 displacement expressed by the formulas 



fyC*. 1, t)=- f^ sin K< -<?)/>(.>', 1, ?) (191), 



which expresses the surface displacement due to surface- 

 pressure expressed by 



ITO, 1, t)=-miwtf(js, 1, 0) . . (192). 



The fifth curve of fig. 38 shows the history, after period 3, 

 to almost half a period after period 9, of the disturbance at 

 the place ^ = 32. The disturbance has not yet become sinu- 

 soidal, but w^ould certainly become almost exactly sinusoidal 

 after a few more periods. 



§ 145. In fig. 39, two sets of five curves show, for case </> 

 and case \jr, the periodically varying water-surface on each 

 side of the. middle, at any long enough time after the be- 

 ginning of the motion, to give a regular regime of sinusoidal 



* The scale of ordinates of the third, fourth, and fifth curves of fig. £8 

 is double that of the first and second, indicated on the figure. 



