396 



SCIENCE. 



[N. S. Vol. IX. No. 220. 



here indicated by stationary particles half 

 a wave-length apart, while the antinodes 

 vibrate 9". In all positions the form of the 

 compound harmonic curve is at all times a 

 simple sinusoid, but its mode of motion as 

 compared with the same curve while both 

 components are direct is totally different. 



Again, if the first pair of cams are in the 

 same null phase (pointers away from each 

 other) the first particle is a node, suc- 

 ceeded by four other nodes one-half wave- 

 length apart, and the wave is initially at 

 maximum amplitude. If the first pair of 

 cams are in opposite null phases (±0) the 

 initial harmonic curve is linear, the first of 

 four nodes one-fourth wave-length ahead, 

 etc. 



Reflection. — The first of these cases cor- 

 responds to reflection from a denser, the 

 second to reflection from a rarer, medium 

 at the origin. It is worth while to examine 

 the interpretation of both cases* for trans- 

 verse waves first, and thereafter, §26, to sim- 

 ilarly treat longitudinal waves. 



If the direction of a wave is reversed, 

 particles without displacement (±0) are 

 changed half a period in phase (becoming 

 =P 0) ; particles at maxima or minima {±m) 

 are not changed in phase at all, while 

 the phases of intermediate particles are 

 changed in the corresponding harmonic ra- 

 tio. This may be tested at once by sup- 

 posing the full wave, Fig. 12, to advance 

 first in direction d, thereafter in direction r, 

 when the particles vibrating in the line aa 

 will respectively rise and fall, thus passing 

 between opposed phases ; etc. 



The transverse wave advances through a 

 given medium at rest, with the zero of dis- 

 placement (±0) in the wave front, so under- 

 stood. Hence to reverse the direction of a 

 wave is to reverse thephaseof the wave front. 



If the transverse wave encounters a denser 

 medium this implies that the particles 

 therein situated are capable of reacting with 



*Waves as here considered are essentially steady. 



forces in excess of those corresponding to the 

 original medium. If the medium is quite 

 impermeable (as when the wave on an 

 elastic cord meets the peg) the reaction is 

 exactly equal and opposite to the action. 

 Thus if a wave advances toward the dense 

 medium with a crest or group of pulls up- 

 ward the medium itself must at every in- 

 stant react with equal pulls downward. 

 This reaction, which in its succession is 

 bound to be rythmic like the impinging 

 wave, is the impulse of the reflected wave, 

 which must all be returned into the first 

 medium (i. e., be reversed in direction) if 

 none can enter the new medium. 



ISTow, let the particle in the wall aa (Fig. 

 12) be in the zero of phase {+ 0). The 

 direct wave advancing, as shown by d, is in 

 the act of increasing the displacement. It 

 is developing an increasing pull up. The 

 reflected wave (prolonged) r is simultane- 

 ously in the act of developing the counter 

 pull down ; it is, in like degree, tending to 

 decrease displacement : but, though the 

 phases impressed by the direct and reflected 

 wave are thus initially quite opposite, both 

 waves d and r momentarily constitute con- 

 tiguous parts of the same harmonic curve. 

 If this curve separates at aa, with the parts 

 d and r moving with equal velocity in op- 

 posite directions the conditfon for action 

 equilibrated by reaction at aa is maintained 

 throughout all time. 



The explanation is essentially the same 

 if the reaction is not complete (permeable 

 dense medium). In this case the amplitude 

 of r will be smaller, other conditions re- 

 maining the same. 



Hence in the machine the pointers are to 

 be set for equal and opposite displacements 

 at the origin, beginning with the null 

 phases of each component wave — the case 

 of Fig. 12, where if d and r were moving 

 in the same direction, or the pulleys not 

 cross-belted, the two components would 

 meet in the same place. 



