TIDES AND CURRENTS IN BOSTON HARBOR 103 



of the mixed type is at a minimum, the tide at such times resembling the semi- 

 diurnal type. It is the characteristics of the predominating tide that determine 

 the type of tide at any given place. With the aid of harmonic constants the type 

 of tide may be defined by definite ratios of the semidiurnal to the diurnal 

 constituents. 



Type of tide is intimately associated with diurnal inequality and hence depends 

 on the relation of the semidiurnal to the diurnal tides; and it is the variation in 

 this relation that makes possible the various forms of the mixed type of tide. 



HARMONIC CONSTANTS 



Since the tide is periodic in character, it may be regarded as the resultant of 

 a number of simple harmonic movements. In other words, if h be the height of 

 the tide, reckoned from sea level, then for any time t, we may write h = A cos 

 lat + a)4-B cos {bt+l3)+ ... In the above formula each term represents 

 a constituent of the tide which is defined by its amplitude or semirange. A, B, 

 etc., by an angular speed, a, b, etc., and by an angle of constant value, a, /3, etc., 

 which determines the relation of time of maximum height to the time of begin- 

 ning of observation. 



We may also regard the matter from another viewpoint and suppose the moon 

 and sun as tide-producing bodies to be replaced by a number of hypothetical 

 tide-producing bodies, each of which moves around the earth in the plane of the 

 Equator in a circular orbit with the earth as center. With the further assump- 

 tion that each of these hypothetical tide-producing bodies gives rise to a simple 

 tide, the high water of which occurs a certain number of hours after its upper 

 meridian passage and the low water the same number of hours after its lower 

 meridian passage, the oscillation produced by each of these simple tides may be 

 written in the form h = A cos (at + a) as above. The great advantage of so 

 regarding the tide is that it permits the complicated movements of sun and 

 moon relative to the earth to be replaced by a number of simple movements. 



Each of the simple tides into which the tide of nature is resolved is called a 

 component tide, or simply a component. The amplitudes or semiranges of the 

 component tides, together with the angles which determine the relation of the 

 high water of each of these component tides to some definite time origin and 

 which are known as the epochs, constitute the harmonic constants. 



The periods of revolution of the hypothetical tidal bodies or the speeds of the 

 various component tides are computed from astronomical data and depend only 

 on the relative movements of sun, moon, and earth. These periods being inde- 

 pendent of local conditions are therefore the same for all places on the surface 

 of the earth; what remains to be determined for the various simple constituent 

 tides is their epochs and amplitudes which vary from place to place according 

 to the type, time, and range of the tide. The mathematical process by which 

 these epochs and amplitudes are disentangled from tidal observations is known 

 as the harmonic analysis. 



The number of simple constituent tides is theoretically large, but most of 

 them are of such small magnitude that they may for all practical purposes be 

 disregarded. In the prediction of tides it is necessary to take account of 20 to 

 30, but the characteristics of the tide at any place may be determined easily 

 from the 5 principal ones. 



It is obvious that the principal lunar tidal component will be one which gives 

 two high and two low waters in a tidal day of 24 hours and 50 minutes, or more 



2X 360° 

 exactly in 24.84 hours. Its speed per solar hour, therefore, is 04 84 =^^°-^^' 



This component has been given the symbol M2. Likewise, the principal solar 

 tidal component is one that gives two high and two low waters in a solar day of 



2X360° 

 24 hours. Its angular speed per hour is therefore — 04 — =30°. 00. The symbol 



for this principal solar component is S2. 



Since the moon's distance from the earth is not constant, being less than the 

 average at perigee and greater at apogee, the period from one perigee to another 

 being on the average 27.55 days, we must introduce another hypothetical tidal 

 body, so that at perigee its high water will correspond with the M2 high water, 

 and at apogee its low water will correspond with the M2 high water. In other 

 words, the tidal component which is to take account of the moon's perigean 

 movement must, in a period of 13.78 days, lose 180° on M2, or at the rate of 



^^y^=13°.0Q per day. Its hourly speed, therefore, is 28°.98- '^^=28°.44. 



This component has been given the symbol N2. 



