Deep-sea Tides, and the Effect of Tidal Friction. 167 



3. And first as to the geometrical characteristics of a fluid 

 wave uniformly propagated westward at any rate (a). 



Taking the earth's radius as unity, and any point on the 

 equator as the origin, let co measure the longitude westward of 

 any other point, and let k be the mean depth of the water and 

 y the '' small elevation' of the surface above the level at co at a 

 definite moment of time. 



For the wave to be propagated with a persistent form and at 

 the rate a, the height at co must in a time 8t change from y to 

 the value which y now has at a distance a$t behind it, or co — ctht 

 from the origin ; that is, 



. \ ^+&c. = --^-ao7+&c., 

 at da) 



or 



ti~ *di W 



And if this relation exist everywhere between the differential 

 coefficients, the condition will be fulfilled for finite intervals. 



Not only the heights, but every other measure or mark of 

 disturbance must be propagated onwards at the same rate, if 

 the wave is to have a permanent character ; so that if v be the 

 average forward velocity of the particles in a vertical section at 

 co, we must have 



dv dv 



di^^dco • ( B ) 



Now there is a necessary connexion between the changes in v 

 and y, known as the equation of continuity. 

 j The increase or decrease of height of surface between two 

 near points (co and o>-f Bay) is caused by the excess or defect of 

 the influx -and efflux of w r ater on either side, and is proportional 

 to that bulk of water divided by the horizontal section Sco. The 

 influx from the east is proportional to the area of the vertical 

 transverse section, or tc-\-y, and to the average forward velocity 

 v, or to kv nearly ; and the efflux westward is similarly propor- 

 tional to 



/c(v-\- jr-S&H-&C.). 



Therefore the decrease in bulk is 

 and the rate of fall 



K-r- Sco + &c, 

 dco 



dy dv 

 dt dco 



