1904.] Train of Waves emitted by a Hertzian Oscillator. 83 



when ct<r. Figs. 1 3 of Plate 6, represent parts of the curves, of 

 which is ordinate and r is abscissa, at instants which are the ends of 

 the second, third, and thirteenth periods from the beginning of the 

 vibrations.* The parts of the curves in figs. 1 and 2, which are very 

 near the oscillator, are omitted, and, in the three curves, the parts 

 which lie beyond the advancing wave-fronts are indistinguishable from 

 the axis of abscissae. In figs. 1 and 2, as originally drawn, A was 

 given the value 0*1, and v and e have the same values as in the 

 previous discussion. In fig. 3 A was given the value ]. The 

 curve in fig. 3 has been drawn for values of r between r = 10 A and 

 r = 13 A. In each case the terminal point of the curve towards the 

 right represents the value of at the advancing wave-front at the 

 instant in question. Near the oscillator, the maxima and minima 

 values of diminish as the distance of them from the oscillator 

 increases, as is shown in figs. 1 and 2. This is due to the pre- 

 ponderance of the factor 1/r 3 when r is small. When the front of the 

 train of waves has travelled over as few as three wave-lengths, this 

 tendency is already checked by the tendency of the factor e vr ^ to 

 increase with r, as is seen in fig. 2, where the last minimum is almost 

 exactly equal to the previous maximum. When the front of the 

 train of waves has travelled over a larger number of wave-lengths, the 

 maxima and minima near the front exceed those at a little distance 

 behind the front, as is shown in fig. 3, where there is a regular 

 increase in the maxima and minima values as the front of the train 

 of waves is approached. A comparison of figs. 1 and 2 with each 

 other shows the diminution of the maxima and minima at the same 

 places as time goes on. This is due to the damping of the oscillations 

 by radiation. The same comparison shows also that the maxima and 

 minima near the front of the train of waves do not suffer diminution 

 to the same extent, and the same thing is shown by comparing 

 fig. 3 with these, allowance being made for the difference of scale. 

 In fact, the disturbance at the front of the wave-train suffers diminu- 

 tion through spherical divergence only, for the factor e-"< c *~ r )/ A has the 

 value unity at the front of the waves, and, when r is at all large, the 

 value of at the front is very nearly equal to 



- A sin (27T6/A) (47r 2 + i> 2 )/AV, 

 so that it is very nearly proportional to r~ l . 



* In the arithmetical work which is requisite for tracing these curves and in 

 some of the remaining arithmetical work of the paper, I had the good fortune to 

 secure the collaboration of Mr. J. W. Sharpe, formerly Fellow of Gonville and 

 Caius College, Cambridge, who made the necessary calculations. The paper is 

 much more complete than it would have been without his help. 



