76 Prof. J. D. Everett's Elementary Investigations 



8. The actual horizontal acceleration of a particle of the 

 water at any moment is the algebraic sum of two horizontal 

 components — one due to the moon and equal (as just shown) 

 to §M sin 20, and the other due to the slope of the surface. 

 The magnitude of the latter component is g multiplied by the 

 tangent of the slope ; and its direction is always towards the 

 lower side. The particle vibrates under the joint action of 

 these two accelerations, each of which is a simple-harmonic 

 function of the time with a period of half a lunar day. Hence, 

 by the reasoning of section 2, they attain their maxima 

 together. The maxima of the component due to the moon 

 are at the points 45° distant from the line of centres ; these 

 must therefore be the points of steepest slope ; and as the 

 points of steepest slope in a simple-harmonic wave are mid- 

 way between crest and trough, there will be either a crest or 

 a trough exactly under the moon. The reasoning of section 5 

 shows that, if the actual period be less than the period of free 

 vibration, the two components conspire; but if greater, they are 

 in opposition. If they conspire, the slope for 45° on each side 

 of the point under the moon must be towards this point ; that 

 is to say, there must be low water under the moon. If they 

 are in opposition, the slope must be the other way, and there 

 must be high water under the moon. 



The periods of the free and forced vibrations are inversely 

 as the velocities of free and forced waves, since a wave of 

 either kind travels half round the equator in one period. 

 Hence, if the forced wave travels faster than the free wave, there 

 is loiv water, and if slower, high water, under the moon*. 



9. We can now calculate the height of the tide, if the ratio 

 of the actual period T to the free period T\ be known. 



Let z denote the height of any point of the surface of the 

 water measured from mean level at time t, at a point of the 

 canal distant w in the forward direction from a fixed point of 

 reference, V the actual velocity of the wave, X the wave- 

 length, which is irr (where r denotes the earth's radius), and 

 2h the difference between high and low water. Then, by the 

 standard formula for simple-harmonic waves, we have 

 9 f> 



z ?= h cos -^- (Vt — x) =s h cos - (Yt — x) t 



The acceleration due to the slope is 



dz 2h . 2, v , . 



* On cpmparing the reasoning of this section with that employed by- 

 Mr. Abbott in a paper on the same subject (Phil. Mag. January 1870), it 

 will be seen that Mr. Abbott has neglected the acceleration due to slope. 



