274 Theory of the Tides 



the amplitude varying with declination, A = lHcos 2 <pcos 2 d then we can 

 consider this variation for a short time to be constant for a point on the 

 earth. If, we introduce the sidereal time and the right ascension a instead 

 of the hour angle of the disturbing body, (see p. 265) we obtain for the com- 

 bined semi-diurnal lunar and solar tide, the equlibrium tide 



^ 3 = Acos2(d-a)^A'cos2(6-a') (IX.2) 



in which the quantities marked with a prime refer to the sun. The super- 

 position of these two waves can be imagined to give a single wave with variable 

 amplitude and phase. If we put: d — a = 6 — a + (a — a) 



% = ^ 3 cos2(0-a + a 3 ) (IX.3) 



in which 



A s = } {A 2j rA' 2j r2AA'cos2(a—a')} and tan2a 3 = -r^—r, ^ — - — 77 . 



A+A cos2(a — a ) 



The amplitude A^ will have a maximum when a = a and a = a' +180°. 

 This happens at the conjuction and opposition of moon and sun, e.g. at the 

 time of the syzygies or at full and new moon. These are the spring tides. At 

 the same time, a 3 = 0, i.e. the spring tides appear at the moment 6 = a 

 and a+ 180°, which is at the upper and lower culmination of the moon, which 

 for full and new moon coincides with midnight and noon. 



The minimum of A 3 occurs when a— a = 90° and 270°, which is at the 

 time of the quadratures (first and last quarter of the moon respectively). 

 ;/ 3 then becomes smallest; these are the neap tides. In this case too a 3 = 0, 

 and ?] 3 is reached when 6 = a, i.e. again at the upper and lower culmination 

 of the moon, which at this time occurs approximately at 6 h and 18 h. At any 

 other time a 3 is different from zero and high water does not occur at the 

 moon's transit, but this phase displacement is not very large (semi-monthly 

 inequality in the time of occurrence). 



The combined action of the principal lunar and solar tides as seen by an observer at the equator 

 is illustrated in Fig. 118 drawn by Bidlinghanger (1908). The upper part shows the conditions 

 at full and new moon. At noon both bodies are in the zenith, the water level is H times as high 

 as for the luna* tide alone (spring tide). The next figure shows the position for the first and the last 

 quarter. At 6 a.m., when the sun rises, the moon is at the zenith, lunar and solar tide oppose each 

 other (neap tide). The following figures show halfway between spring and neap tide, neap and 

 spring tide respectively. In the first case, the moon is in the 1st and 5th octant and reaches its 

 highest point at 9h, whereas the sun and its tide are still rising. High water is retarded by about 

 an hour against the culmination of the moon. In the second case (3rd and 7th octant) the moon 

 goes through the meridian at 3 a.m., whereas the nadir flood of the sun passed three hours earlier. 

 Hence, high water appears approximately an hour before culmination of the moon. 



The daily inequality in height and time can be easily understood from the illustrations of 

 Figs. 119 and 120. If the disturbing body is not upon the equator, the axis of rotation of the spheroid 

 does not coincide any more with the axis of the earth (see Fig. 117). In Fig. 1 19 are shown isohypses 

 of the spheroid for both hemispheres, when the disturbing body (moon) stands in 28°N. latitude 

 in the zenith above the centre meridian. The corresponding point then lies on the other hemisphere 



