ON THE THEORY OF HYDRAULICS. 219 



as a horizontal surface, and that the undulations, thus crossing each other, 

 'will proceed without any interruption, the motion of each particle being 

 always the sum or difference of the motions belonging to the separate 

 series. 



Supposing two equal and similar series of waves to meet each other in 

 such a canal, in opposite directions, the point in which their similar parts 

 meet must be free from all horizontal motion, so that any fixed obstacle in 

 an upright position would have the same effect on the motions of the fluid 

 on either side as the opposition of a similar series ; and this effect con- 

 stitutes the reflection of a series of waves, which is easily observed, when 

 they strike against a steep wall or bank ; and when this reflection is 

 sufficiently regular, it is easy to show that the combination of the direct 

 with the reflected motions must constitute a vibration of such a nature, 

 that the whole surface is divided into portions which appear to vibrate 

 alternately upwards and downwards, without any progressive motion, 

 while the points which separate the portions remain always in their natural 

 level. (Plate XX. Fig. 263.) 



But those series of waves which are usually observable in any broad 

 surface, and which constitute a number of concentric circles, are usually 

 reflected in such a manner as to appear to diverge after reflection from a 

 centre beyond the surface which reflects them, and to be subject to all those 

 laws, which are more commonly noticed in the phenomena of reflected light ; 

 but as these laws are of more practical importance in their application to 

 optics, than to hydraulics, it is unnecessary at present to examine their 

 consequences in detail. It may, however, be easily understood, that a new 

 series of waves, proceeding from a centre at the same distance behind the 

 reflecting surface as the centre of the original series is before it, would pro- 

 duce precisely the same effect as a fixed obstacle ; consequently the law of 

 reflection at equal angles is a very simple inference from this mode of rea- 

 soning. (Plate XX. Fig. 264.) 



When a series of waves proceeds in an equable canal, it is obvious that 

 the centre of gravity of the whole fluid neither rises nor falls; from this 

 analogy, as well as from the general application of the law of ascending 

 force, it is probable that in all cases of the propagation of waves, the place 

 of the centre of gravity remains unaltered ; so that when a circular wave 

 spreads further and further from its centre, its height is not diminished in 

 the same ratio as its diameter is increased, but the square of its height 

 only varies in this proportion ; that is, a wave which is a yard in diameter, 

 and an inch high, will retain a height of half an inch, when its diameter is 

 increased to four yards. 



Many of the phenomena of waves may be very conveniently exhibited, 

 by means of a wide and shallow vessel, with a bottom of glass, surrounded 

 by sides inclined to the horizon, in order to avoid the confusion which would 

 arise from the continual reflections produced by perpendicular surfaces. 

 The waves may be excited by the vibrations of an elastic rod or wire, 

 loaded with a weight, by means of which its motions may be made more or 

 less rapid at pleasure ; and the form and progress of the waves may be 

 easily observed, by placing a light under the vessel, so that their shadows 



