i.e. executed in the line of the wave's advance. The wave, therefore, no longer presents itself to the eye as exclusively a condition of alternate protuberance of outline in opposite directions, but mainly as a state of more tightly, or less tightly, packed ears.

Fig 13.

B

The

annexed figure gives a rough idea how this takes place. The wind is supposed to be moving from left to right and to have just reached the ear A. Its neighbours to the right are still undisturbed. The stalk of C has just swung back into its erect position. The ears about B are closer to, and those about C further apart from, each other than is the case with those on which the wind has not yet acted.

After this illustration, it will be easy to conceive a kind of wave-motion in which there is no longer any movement transverse to the direction in which the wave is advancing.

15. Let a series of particles, originally at rest in equidistant positions along a straight line, as in that at the head of Fig. 14, be executing equal

periodic vibrations in that line, in such a manner that each is the same fixed amount further back in

its path than is its neighbour to the left, and therefore exactly as much more forward than is its neighbour to the right.

(0) shows the condition of the row of particles at the moment when that on the extreme left is beginning its swing from left to right, which, in accordance with the direction of the wave's advance as indicated by an arrow in the figure, we may call its forward swing. The equidistant vertical straight lines fix the extent of vibration for each separate particle. The constant amount of relative backwardness, or retardation' as it is technically called, between successive particles is, as in Figs. 5 and 7, one-eighth of the period of a complete vibration. Thus, proceeding from left to right along the line (0), we have the first particle beginning a forward swing, the second, third, fourth and fifth particles entering respectively on the fourth, third, second and first quarters of a backward swing, and the sixth, seventh, eighth and ninth particles on the fourth, third, second, and first quarters of a forward swing.

Since the ninth particle is just beginning forward swing, its situation is exactly the same as that of the first. Beyond this point, therefore, we have only repetition of the state of things between the first and ninth particles. The row (0) is therefore made up of a series of groups, or cycles, of the same number of particles arranged in the same manner throughout. Two such cycles, included by

the large brackets A and B, are shown in (0). Each cycle is divided by the sub-brackets a, a and b, b′ into two parts. In a and b the distances between successive particles are less than, and in a' and b' greater than, the corresponding distances when the particles occupied their undisturbed positions. The cycles correspond to complete waves on the surface of water, the shortened and elongated portions of each cycle answering to the crest and trough of which each water-wave consists.

(1) shows the state of the row of particles when an interval of time equal to one-eighth of the period of a complete particle-vibration has elapsed from the moment shown in (0). The wave A has meantime advanced by one-eighth of its own wave-length into the position pointed to by the dotted lines.

The following rows (2), (3), (4), &c., show the state of things when two-eighths, three-eighths, foureighths, &c., of a vibration-period has elapsed since (0). In each successive row the wave A is further advanced by one-eighth of its wave-length.

In (8), a whole vibration-period has elapsed since (0). Accordingly each particle has performed one complete vibration, and returned to the position which it held in (0). The wave A, meanwhile, has travelled constantly forward so as to be, in (8), where B was in (0), i.e. to have advanced one

whole wave-length.

Hence, while any particle per

forms a complete vibration the wave advances one wave-length. Accordingly, for waves due to longitudinal vibrations, i.e. such as are executed in the line of wave-advance, a proposition holds good corresponding to that established, in § 9 p. 21, for waves due to vibrations performed perpendicularly to the direction of wave-motion.

16. In the waves shown in Fig. 14, the particles in the bracket a are mutually equidistant, as are also those in the bracket b. This is due to the fact that, in the case there represented, the vibrating particles move uniformly, i.e. with equal velocity, throughout their paths. If we take other modes of vibration, we shall find that this equidistance no longer exists. Fig. 15 shows three distinct modes of vibration with the wave resulting from each, on the plan of Fig. 14. The extent of vibration and length of wave are the same in the three cases.

In (I) the particles move quickest at the middle, and slowest at the ends of their paths; in (II) fastest at the ends and slowest in the middle; in (III) slowest at the left end, and fastest at the right.

The shortest distance separating any two particles contained in a is, in (I), that between 7 and 8; in (II), that between 8 and 9; in (III), that between