TIDES AND CURRENTS IN BOSTON HARBOR 111 



It is to be noted that there is no necessary relationship between the velocity 

 of the tidal current at any place and the rate of advance of the tide at that place. 

 In other words, if the rate of advance of the tide is known we can not from 

 that alone infer the velocity of the current, nor vice versa. The rate of advance 

 of the tide in any given body of water depends on the type of tidal movement. 

 In a progressive wave the tide moves approximately in accordance with the 

 formula r='\Jg(l in which r is the rate of advance of the tide, g the acceleration 

 of gravity, and d the depth of the waterway. In stationary-wave movement, 

 since high or low water occurs at very nearly the same time over a considerable 

 area, the rate of advance is theoretically very great; but actually there is always 

 some progression present, and this reduces the theoretical velocity considerably. 



The velocity of the current, or the actual speed with which the particles of 

 water are moving past, any fixed point, depends on the volume of water that 

 must pass the given point and the cross-section of the channel at that point. 

 The velocity of the current is thus independent of the rate of advance of the 

 tide. 



DISTANCE TRAVELED BY A PARTICLE IN A TIDAL CYCLE 



In a rectillinear current the distance traveled by the water particles or by 

 any object floating in the water is obviously equal .to the product of the time 

 by the average velocity during this interval of time. To determine the average 

 velocity of the tidal current for any desired interval several methods may be 

 used. 



If the curve of the tidal current has been plotted, the average velocity may 

 be derived as the mean of a number of measurements of the velocity made at 

 frequent intervals on the curve; as, for example, every 10 or 15 minutes. From 

 the current curve the average velocity may also be determined by deriving the 

 mean ordinate of the curve by use of the planimeter. For a full tidal cycle of 

 flood or ebb, however, since the current curve generally approximates the cosine 

 curve, the simplest method consists in making use of the well-known ratio of 

 the mean ordinate of the cosine curve to the maximum ordinate which is 2-i-w, 

 or 0.6366. 



The latter method has another advantage in that the velocity of the tidal 

 current is almost invariably specified by its velocity at the time of stre^th, 

 which corresponds to the maximum ordinate of the cosine curve; hence, the 

 average velocity of the tidal current for a flood or ebb cycle is given imme- 

 diately as the product of the strength of the current by 0.6366. And though 

 this method is only approximate, since the curve of the current may deviate 

 more or less from the cosine curve, in general the results will be sufficiently 

 accurate for all practical purposes. For a normal flood or ebb period of 6.2 

 hours the distance a tidal current with a velocity at strength of 1 knot will 

 carry a floating object is, in nautical miles, 0.6366X6.2 = 3.95, or 24,000 feet. 



DURATION OF SLACK 



In the change of direction of flow from flood to ebb, and vice versa, the tidal 

 current goes through a period of slack water or zero velocity. Obviously, this 

 period of slack is but momentary, and graphically it is represented by the 

 instant when the current curve cuts the zero line of velocities. For a brief 

 period each side of slack water, however, the current is very weak, and in 

 ordinary usage "slack water" denotes not only the instant of zero velocity 

 but also the period of weak current. The question is therefore frequently 

 raised, How long does slack water last? 



To give slack water in its ordinary usage a definite meaning, we may define 

 it to be the period during which the velocity of the current is less than one- 

 tenth of a knot. Velocities less than one-tenth of a knot may generally be dis- 

 regarded for practical purposes, and such velocities are, moreover, difficult to 

 measure either with float or with current meter. For any given current it is 

 now a simple matter to determine the duration of slack water, the current curve 

 furnishing a ready means for this determination. 



In general, regarding the current curve as approximately a sine or cosine 

 curve, the duration of slack water is a function of the strength of current — the 

 stronger the current the less the duration of slack — and from the equation of the 

 sine curve we may easily compute the duration of slack water for currents of 

 various strengths. For the normal flood or ebb cycle of 6** 12.6"> we may write 

 the equation of the current curve y = A sin OASSlt, in wl^ch A is the velocity of 

 the current in knots at time of strength, 0.4831 the angular velocity in degrees 

 per minute, and t is the time in minutes from the instant of zero velocity. 



