292 Theory of the Tides 



that, when the period of the force corresponds to the period of the free 

 wave, there is resonance. This is the case for the sun, when the depth of the 

 canal is 22 km or 13 nm, and for the moon when the depth is 205 km or 

 12 5 nm. In the actual case of the earth we have 



ff ° = jl h = in A 



a 2 n 2 R R R' 



For the oceans hjR is about ',„ at its maximum, and a will always be larger 

 than a so that the tides of the ocean will be inverted at the equator. The 

 tides which are caused by the disturbing bodies are not large; their amplitude 

 for a depth of 1300 ft (4000 m) is only 6 cm. For greater depths the tide 

 becomes larger, but the tides remain inverted until the critical depth of 22 km 

 or 13 nm; for depths beyond this limit the tides become direct and approximate 

 more and more values of the equilibrium theory. 



If the canal is not in the equator, but parallel to a circle of latitude <p, 

 whereas the orbit of the disturbing body remains in the equatorial plane, then 



COS & — COS (p cos {nt + A + £ ) 

 and we have 



Hcos 2 w 1 



2 1— /? 2 cos 2 </> 



cos2(w* + A+e). (IX. 13 a) 



If c < nR, the tides will be direct or inverted, depending on whether 

 cos^ ^ c/nR. Therefore, it is possible that in higher latitudes the tides are 

 opposite to the tides in lower latitudes. 



If the moon be not in the plane of the equator, but has the pole dis- 

 tance A, we have to use for cos# the equation (VIII. 12) in which % = 90°— </> 



1 c 2 H 



V =^ 1 2D2 • 2 sin2/sin2^cos(/2/ + A + e) 



2 c 2 — fi 2 R 2 sm 2 x 



+\ 2 C IT • , sin 2 xsin 2 Jcos2(Atf + A + f ) . (\XA3b) 

 2 c 2 — n 2 R 2 sm 2 x 



It is then seen that the first term is a diurnal tide of period 2n/n which 

 is added to the semi-diurnal tide. The diurnal tide vanishes, when the de- 

 clination of the disturbing body is zero, i.e. when the moon crosses the 

 equator, i.e. twice a month. The amplitude of the semi-diurnal tide with 

 a period n/n, however, is smaller than before in the ratio of sinM:l. 



In the case of a canal coinciding with a meridian we should consider the fact that the undis- 

 turbed figure of the free surface is one of relative equilibrium under gravity and centrifugal force, 

 and is therefore not exactly circular. We will assume by anticipation that in a narrow canal the 

 disturbances are sensibly the same as if the earth were at rest, and the disturbing body were to 

 revolve round it with the proper relative motion. If the moon be supposed to move in the plane 

 of the equator cos# = costf>cos(/z + /e) if nt+e is the hour angle from the meridian of the canal. 



