MECHANICS 687 Analytically the resultant motion is expressed by X = a COS (tat + e) + a COS (cat + e ) , = (a cos e + a cos e ) cos at - (a sin e + a sin e ) sin coi , provided that P cos Q = a cos e + a cos e , and PsinQ = asitie + a sme . These expressions give for the amplitude of the resultant P /(acos e + a eose ) 2 + (asm e + a sin e ) 2 , = V(a 2 + 2aa cos (e - e ) + a" 2 . This may be put in cither of the forms e - e ) or + a ) 2 - 4aa siii 2 ^(e - e ) or J(a - a ) from which the above conclusions follow at once. Also, for the epoch of the resultant, we have a sin + a sine tan Q = -, -, . a cose + a cose When e and e are both positive and less than |ir, this is obviously intermediate in value to tan e and tan e . When the periods of the components are not exactly equal, the simple artifice which follows enables us still to apply the same method of composition. We have now x = a cos ( wt + e) + a cos (w t + f ), = acos(w< + e) + a cos(<+e + (co - u)t) . Hence the above values of P and Q will still satisfy the conditions if we write 4 ( - w)t instead of e . Thus we may treat the two components as being of the same period, but make the epoch of one of them steadily increase with an angular velocity equal to the difference of the angular velocities in the generating circles of the components. The triangle OPQ will no longer preserve its form ; it will pass continuously through all the various forms which we have sesn would be given to it by various differences of phase in the com ponent simple harmonic motions. The time in which it returns to a former value is evidently 27r/( -), which is greater the more nearly equal are the periods of the components. 50. One of the best examples of the principles we have just discussed is furnished by the tides. If there were but one tide-producing body, we should have (approxi mately) a simple harmonic rise and fall of the sea-level at any given place twice over in the course of about twenty- four hours, and the phase would depend simply upon the distance of the tide-producing body from the meridian (whether above or below the pole). The joint effect of the sun and moon is practically the resultant of the effects which they would separately produce. Hence, when these bodies are in conjunction or in opposition (i.e., at new or at full moon), the whole rise of the tide is the sum of the solar and lunar tides ; and we have what are called " spring tides." When the moon is in quadrature, the amplitude of Spring the tidal rise or fall is the excess of the lunar over the nd nea P solar tide, for it is low water as regards the sun when it is high water as regards the moon. In intermediate positions the effect lies between these extremes, but the joint high- tide lies nearer to the crest of the lunar than to that of the solar tide. In the first and third quarters of the moon, high tide is earlier than the high tide due to the moon alone ; in the second and fourth later. This is what is called " priming" and " lagging " of the tides, and is seen at once Priming to follow from the construction given above. Had the and lag- lunar and solar tides been of equal amplitude, spring tides 8 m o- would have been of double the altitude of either, and there would have been no tide at all at the time of neap. The mode in which we have treated this special case is an illustration of the general method (above described) of combining simple harmonic motions in which the periods are slightly different. 57. What we have said of the tide-waves holds of Compo- course of all waves in which the separate disturbances are sition of so small that the joint effect is found by superposing the n t j^ e separate effects. Thus when, at sea, two series of waves gene . of equal length meefe at any place, the resultant is still a rally, set of waves of the same length, but the altitudes and phases of the components determine those of the resultant. When crest meets crest, we have waves of the sum of the original amplitudes ; when crest meets trough, the difference. lu the latter case we have still water when the amplitudes of the components are equal. What is called a "jabble," where, for a short time, a portion of a stormy sea is almost calm, and after a little it is violently agitated, is the result of a number of " cross seas." 58. If we now. consider the instantaneous form of a Common section of the surface, instead of the successive displacements pheno- of one portion of it, we can easily account for a striking mena - phenomenon which is very frequently observed on a shelv ing beach. We often notice that every ninth or tenth wave or so is higher than those immediately before or after it. This is the result of superposition of two or more sets of waves in which the distance from crest to crest is different in the different sets. In the joint system we have, represented as in 53, phenomena akin to the spring and neap tides, and the priming and lagging of the tides. Fig. 17 shows part of the result when the amplitudes are equal, and the wave-lengths as 15 to 17. It gives also a rough approximation to the whole result when the lengths are as 7 to 8 or as 8 to 9 Jompo- ntion of nore
- han two
simple larmo- uic mo tions of Two simple harmo nic mo- .tions at right ingles. 59. To compound any number of simple harmonic motions, of equal periods, in one line, we may obviously take them two by two, and apply the preceding process over and over again till we have as final resultant another simple harmonic motion of the common period. Or thus: x = 2 cos (cat + e) = cos o>S( cos e) - sin co2( sin e) = Pcos (wt + Q) , where P cos Q = 2( cose), P sin Q = 2(a sine). When the separate periods are not equal, and not even nearly equal, it is only in special cases that any simplifica tion can be effected by analytical processes. But this is not much to be regretted, because for most purposes a graphic method is sufficiently accurate, and it can always be easily carried out. 60. We must now consider the composition of simple harmonic motions in directions at right angles to each other ; but for the present we confine ourselves to the case in which their periods are equal. In this case we A 17. know that the acceleration is in the same ratio to the displacement in each of the two rectangular directions. Hence by the general theorem rt a of 50 the motion is elliptic, with uniform description of areas about the centre. To analyse this, suppose, at starting, that their amplitudes also are equal. Let OA, OB (fig. 18) represent the two rectangular directions. With centre O, and radius equal tothe common ampli tude, describe a circle. Let AOE, EOF represent the epochs of the two components (the corresponding circular motion being supposed positive for each), then obviously EOF exceeds by a right angle the difference between the B Fig. 18.
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