harmonic The Period of a simple harmonic motion is the time which Simple elapses from any instant until the moving point again moves in motion. the same direction through the same position. The Phase of a simple harmonic motion at any instant is the fraction of the whole period which has elapsed since the moving point last passed through its middle position in the positive direction. harmonic mechanism. 55. Those common kinds of mechanism, for producing recti- Simple lineal from circular motion, or vice versa, in which a crank motion in moving in a circle works in a straight slot belonging to a body which can only move in a straight line, fulfil strictly the definition of a simple harmonic motion in the part of which the motion is rectilineal, if the motion of the rotating part is uniform. It is also It is also approx The motion of the treadle in a spinning-wheel approximates to the same condition when the wheel moves uniformly; the approximation being the closer, the smaller is the angular motion of the treadle and of the connecting string. imated to more or less closely in the motion steam-engine connected, by any of the several methods in use, with the crank, provided always the rotatory motion of the crank be uniform. of the piston of a in S. H. 56. The velocity of a point executing a simple harmonic Velocity motion is a simple harmonic function of the time, a quarter of motion. a period earlier in phase than the displacement, and having its maximum value equal to the velocity in the circular motion by which the given function is defined. For, in the fig. of § 53, if V be the velocity in the circle, it may be represented by OQ in a direction perpendicular to its own, and therefore by OP and PQ in directions perpendicular to those lines. That is, the velocity of P in the simple harmonic V 0Q motion is PQ; which, when P is at O, becomes V. tion in S. H. 57. The acceleration of a point executing a simple harmonic Acceleramotion is at any time simply proportional to the displacement motion. from the middle point, but in opposite direction, or always towards the middle point. Its maximum value is that with which a velocity equal to that of the circular motion would Accelera. tion in S. H. motion. be acquired in the time in which an arc equal to the radius is described. V1 For, in the fig. of § 53, the acceleration of Q (by § 35, a) is Qo along QO. Supposing, for a moment, QO to represent the magnitude of this acceleration, we may resolve it in QP, PO. The acceleration of P is therefore represented on the same scale by V PO V2 PO. Its magnitude is therefore = PO, which is proportional to PO, and has at A its maximum value, acceleration under which the velocity V would be acquired in the time 20 as stated. Let a be the amplitude, e the epoch, and T the period, of a simple harmonic motion. Then if s be the displacement from middle position at time t, we have Lastly, for the maximum value of the acceleration, T T' T Composi tion of one line. 2π T 2π where, it may be remarked, is the time of describing an arc equal to radius in the relative circular motion. 58. Any two simple harmonic motions in one line, and of S. H. M. in one period, give, when compounded, a single simple harmonic motion; of the same period; of amplitude equal to the diagonal of a parallelogram described on lengths equal to their amplitudes measured on lines meeting at an angle equal to their difference S R A P tion of one line. of epochs; and of epoch differing from their epochs by angles Composiequal to those which this diagonal makes with the two sides of 8. H. M. in the parallelogram. Let P and P' be two points executing simple harmonic motions of one period, and in one line B'BCAA'. Let Q and Q be the uniformly moving points in the relative circles. On CQ and CQ' describe a parallelogram SQCQ; and through S draw SR perpendicular to B'A' produced. We have obviously P'R=CP (being projections of the equal and parallel lines Q'S, CQ, on CR). Hence CR=CP+CP'; and therefore the point R executes the resultant of the motions P and P. But CS, the diagonal of the parallelogram, is constant, and therefore the resultant motion is simple harmonic, of amplitude CS, and of epoch exceeding that of the motion of P, and falling short of that of the motion of P', by the angles QCS and SCQ respectively. B B' This geometrical construction has been usefully applied by the tidal committee of the British Association for a mechanical tideindicator (compare § 60, below). An arm C'Q' turning round C carries an arm Q'S turning round Q'. Toothed wheels, one of them fixed with its axis through C, and the others pivoted on a framework carried by CQ, are so arranged that Q'S turns very approximately at the rate of once round in 12 mean lunar hours, if CQ' be turned uniformly at the rate of once round in 12 mean solar hours. Days and half-days are marked by a counter geared to CQ. The distance of S from a fixed line through C shows the deviation from mean sea-level due to the sum of mean solar and mean lunar tides for the time of day and year marked by CQ and the counter. An analytical proof of the same proposition is useful, being as follows: 2πt 2πέ 59. The construction described in the preceding section exhibits the resultant of two simple harmonic motions, whether of the same period or not. Only, if they are not of the same period, the diagonal of the parallelogram will not be constant, but will diminish from a maximum value, the sum of the component amplitudes, which it has at the instant when the phases of the component motions agree; to a minimum, the difference of those amplitudes, which is its value when the phases differ by half a period. Its direction, which always must be nearer to the greater than to the less of the two radii constituting the sides. of the parallelogram, will oscillate on each side of the greater radius to a maximum deviation amounting on either side to the angle whose sine is the less radius divided by the greater, and reached when the less radius deviates more than this by a quarter circumference from the greater. The full period of this oscillation is the time in which either radius gains a full turn on the other. The resultant motion is therefore not simple harmonic, but is, as it were, simple harmonic with periodically increasing and diminishing amplitude, and with periodical acceleration and retardation of phase. This view is particularly appropriate for the case in which the periods of the two component motions are nearly equal, but the amplitude of one of them much greater than that of the other. To express the resultant motion, let s be the displacement at time t; and let a be the greater of the two component halfamplitudes. The maximum value of tan in the last of these equations is metrical methods indicated above (§ 58) lead to this conclusion by the following very simple construction. To find the time and the amount of the maximum acceleration or retardation of phase, let CA be the greater half-amplitude. From A as centre, with the less half-amplitude as radius, describe a circle. CB touching this circle is the generating radius of the most deviated resultant. Hence CBA is a right angle; and sin BCA= = AB Composition of S. H. M. in one line. composition in one line. 60. A most interesting application of this case of the com- Examples of position of harmonic motions is to the lunar and solar tides; of S. H. M. which, except in tidal rivers, or long channels, or deep bays, follow each very nearly the simple harmonic law, and produce, as the actual result, a variation of level equal to the sum of variations that would be produced by the two causes separately. The amount of the lunar equilibrium-tide (§ 812) is about 2.1 times that of the solar. Hence, if the actual tides conformed to the equilibrium theory, the spring tides would be 31, and the neap tides only 11, each reckoned in terms of the solar tide; and at spring and neap tides the hour of high water is that of the lunar tide alone. The greatest deviation of the actual tide from the phases (high, low, or mean water) of the lunar tide alone, would be about 95 of a lunar hour, that is, 98 of a solar hour (being the same part of 12 lunar hours that 28° 26', or the 1 angle whose sine is is of 360°). This maximum deviation 21, 2.1 would be in advance or in arrear according as the crown of the solar tide precedes or follows the crown of the lunar tide; and it would be exactly reached when the interval of phase between |