179 



in depth, and of quite regular section, the published high water inter- 

 vals show that high water at Eagle, at the head of the fiord, is but 2 

 minutes later than at Halibut Bay, 55 miles down the channel and 

 not far from the entrance. In closed channels of more usual depths, 

 the frictional resistance is considerable if the length of the channel 

 is sufficient to set up any appreciable currents, and the tidal fluctu- 

 ations may more nearly resemble a wave of the progressive type. 



CEITICAL LENGTHS OF A CANAL WERE THE FLOW FRICTIONLESS; NODES 



IN A CLOSED CANAL 



346. Critical lengths oj a closed canal. — The formulas for the tides 

 and currents which would be produced in a closed canal by a com- 

 ponent of the tide at the entrance, were the flow frictionless (equations 

 (251) and (252)) shows that these would reach infinite proportions if 

 the length of the canal were such that 7 = x/2, 3x/2, 57r/2, etc., since 

 cos 7 would then become zero. Since 7=2x L/\ (equation 236) these 

 critical lengths occur when i=X/4, 3X/4, 5X/4, etc.; i. e., when the 

 length, L, of the canal is one-quarter, three-quarters, etc., of the 

 wave length, X, of the component. 



It is apparent that if a closed canal is not of great length, the cur- 

 rents set up by the filling and emptying of the tidal prism are 

 moderate, and the slopes produced by a small increase in the tidal 

 ranges in the canal are sufficient to check the momentum of the mov- 

 ing water. As the length of a canal increases, the increase in the tidal 

 ranges in the canal further accentuates the currents, and if these 

 were not restrained by frictional resistance, they would reach infinite 

 proportions if the canal had the critical length of one quarter of the 

 wave length of the component. If the length of the canal exceeds 

 this critical length, the currents at the head of the canal are in the 

 opposite direction to those at the entrance, and the inomentum of 

 the water is correspondingly controlled. The currents then are finite 

 until the length of the canal reaches the second critical length; and so on. 



347. Critical lengths of a connecting canal. — Unless the amplitudes 

 and timing of the component of the tide at the entrances to a con- 

 necting canal are such as to produce a simple progressive (or retro- 

 gressive) wave, a critical length for the component is reached when 

 sin 7 = 0, and hence when L = %\, X, 1)^X, etc. The condition of flow 

 in each half of the canal at the first of these critical lengths is hke that 

 in a closed canal of one c[uarter of the wave length of the component. 

 The water entering through both entrances would pile up in the 

 canal without limit, were there no frictional resistance. Wlien the 

 length of the canal is equal to the wave length of the component, the 

 positive and negative currents exactly balance each other, and the 

 net work done in the acceleration and deceleration of the current 

 becomes zero, so that frictional resistance would alone limit the flow. 



