The projection of elliptic harmonic motion upon any plane is elliptic harmonic motion. For its projection upon two lines at right angles to each other in the given plane will be simple harmonic, and their resultant will (by § 28) be elliptic harmonic motion.

31. The acceleration in elliptic harmonic motion is always directed towards the centre of the ellipse, and proportional to the distance from the centre.

For it is the resultant of the two accelerations μ. x 0, o (Fig. 13), along the principal axes (x and y being

the projections of P upon these axes), and these evidently compound into μ. PO.

Conversely, if a point P move in a plane curve round a point o with an acceleration always represented by p. PO, the path will be an ellipse of which o is the centre, and will be harmonically described. For if x and y (Fig. 14) are the projections of P on two rectangular axes through o, the accelerations of x and y will be μ. x 0, p. yo, so that the motions of x and y will be simple bar

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RESULTANT OF SEVERAL S.H. MOTIONS. monic; and the motion of P, being their resultant, will (by § 28) be elliptic harmonic motion.

32. The resultant of any number of s.H. motions along any lines whatever, if all have the same period, is elliptic harmonic motion. For the given motions can be projected upon three rectangular axes, and each of the projections will be simple harmonic. Those which are along the same axis will compound into one s.II. motion. The required resultant will therefore be obtained by compounding three S.H. motions of the same period along three lines at right angles, and we may regard o as the central point of each of the three motions, so that, ox, o y, oz being the three displacements from the origin o, the accelerations will be μ. x 0, μ. v O, μ. ZO. The resultant of these will be μ. ? O, P being the point whose projections are x, y, Z. But the motion of P is the resultant motion which we are seeking. Let a plane be drawn through the tangent to the path of P at a given moment, and also through o. lie in this plane, and will be the same as that of a free particle attracted towards o with a force varying as the distance. From the second part of the preceding section it follows that the motion will be elliptic harmonic.

The whole path of P will

33. We shall now investigate the amplitude of the S.H. motion which results from the composition of two S.H. motions in the same line.

Let A and B (Fig. 15) be the two points which travel uniformly round the auxiliary circles of the two compo

nents, c the point which travels uniformly in the auxiliary circle of the resultant. Then O A, O B, O C, are the three amplitudes, and o c, being the diagonal of a parallelogram of which O A, O B are the sides, may have any value intermediate between their sum and difference, according

FIG. 15.

to the magnitude of the angle A O B, which is the difference of phase (or difference of epoch) of the two components. When this angle is zero they have the same phase, and the resultant amplitude is the sum of the given amplitudes. When it is 180° they have oppo

site phases, and the resultant amplitude is the difference of the given amplitudes. In this case if the given amplitudes are equal the two components destroy each other, and the resultant is absolute rest.

The formula for the square of the amplitude of the resultant is evidently

o c2 =

O A2 + O B2 + 2 O A. O B COS A O B,

A O B being the difference of phase of the two components, and O A, O B their amplitudes.

34. If the two component S.H. motions have not rigorously the same period, as hitherto supposed, the angular velocities of A and B in the two auxiliary circles will not be rigorously equal, and the angle AO B will change at a constant rate. We shall suppose this rate to

SLIGHT INEQUALITY OF PERIODS.

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be slow in comparison with the angular velocities themselves, so that the angle A O B will only undergo a small change in each revolution. Then the path of c in one revolution will be nearly circular, and its velocity in this path sensibly uniform, so that the projection of its motion upon a given line will be very approximately simple harmonic; but the radius of the circle will gradually alter in successive revolutions, taking all values intermediate between the sum and difference of O A, O B. Remembering that the radius of the circle is the amplitude of the resultant, we see that the resultant of two S.H. motions of slightly unequal periods (both having the same line of motion) may be described as a S.H. motion with amplitude varying between the sum and the difference of the two given amplitudes.

35. The variation of the square of the amplitude is simple harmonic. For if we drop a perpendicular C D on OA or o A produced, we have

o c2 = 0 A2 + A c2 ± 2 0 A. A D,

where all the quantities on the second side are constant except A D. The variation of o c2 is therefore the variation of 2 0 A. A D, and is proportional to A D. But the motion of c relative to o A is uniform circular motion round A, and the motion of D along the line o A is therefore simple harmonic motion with A as central point.

It follows that the mean value of the term ± 2 0 A. A D is zero; and therefore the mean value of o c2 is o A2+0 B2,

or the mean value of the square of the resultant amplitude is the sum of the squares of the component amplitudes.

36. These principles explain the throbbing character of the sound which is produced by the combination of two sounds differing slightly in pitch. The drum of the ear vibrating under their joint influence performs vibrations whose amplitudes vary from the sum to the difference of the amplitudes due to the two separate sounds. If the separate effects of the two sounds were exactly equal and were simple harmonic, there would be momentary silence at the instant when the phases became opposite. Two 'stopped' organ-pipes mounted side by side on the same wind-chest, and tuned as nearly as possible to unison, will often maintain this opposition of phase (and almost complete extinction of sound) for a considerable time.

The alternations of loudness produced by the cause here explained are called beats. Each beat indicates that one of the two sources has gained a complete vibration upon the other; and hence, if the number of vibrations made by one source is known, the number made by the other can be found by adding or subtracting the number of beats.

37. They also explain the phenomena of spring and neap tides.

Speaking broadly, the variation of tidal level at a given place is the sum of two S.H. variations, one depending on the moon and the other on the sun, the