538 XL HYDRODYNAMICS. 



the opposite side we have their difference. Consequently, at any 

 point, the difference of two consecutive high waters is twice the 

 diurnal tide. This difference is generally small, showing that the 

 latter tide is small. It vanishes for points on the equator, and at 

 the times of the equinoctial tides. 



The third term, which vanishes for latitude 35 16', does not 

 depend on the moon's hour -angle, but only on its declination. This 

 declinational tide, depending on the square of sind, has a period of 

 one -half a lunar month. 



Beside the tides due to the moon, we must add those due to 

 the sun, for which the factor outside the brackets in 172) is some- 

 what less than one -half that due to the moon. The highest tides 

 therefore occur at those times .in the month when the sun and the 

 moon are on the meridian together, namely at new and full moon. 

 These are known as spring -tides. The lowest occur when the moon 

 is in quadrature with the sun, and the lunar and solar tides are in 

 opposition. These are known as neap-tides, and occording to this 

 theory would be only one -third the height of the spring -tides. The 

 greatest spring- tides would be those in which the moon was on the 

 equator, or the equinoctial spring- tides. Now it is found that, instead 

 of this, the high tides come about a day and a half later. Consequently, 

 although the equilibrium theory indicates to us the general nature 

 of the different tides to be expected, it does not give us an accurate 

 expression for their values. Roughly speaking we may say that the 

 tides act as if they were produced as described by the action of the 

 sun and moon, but that the time of arrival of the effects produced 

 was delayed. 



A correction was introduced into the equilibrium theory by 

 Lord Kelvin, to take account of the effect of the continents. For if 

 the height of the tide were given by the equation 171), removing 

 the various volumes of water in the space actually occupied by land 

 would subtract an amount of water now positive, now negative, so 

 that the condition of constant volume would not be fulfilled. In 

 order that it still may do so, the integral 169) is to be taken only 

 over those parts of the earth's surface covered by the sea, The 

 value of V is then not zero. The effect of this is to introduce at 

 each point on the earth's surface change of time of the arrival of 

 each tide, varying from point to point. The practical effect of this 

 correction is not large. 



20O. Tidal Waves in Canals. In the dynamical theory of 

 the tides, taking account of the inertia of the water, we have the 

 problem of the forced oscillations of the sea under periodic forces. 

 As a simple example illustrating this method we shall consider waves 



