WATER DEPTH 131 



length were only about 40 feet (ratio of 20: 1), at the 3.6-foot line, 

 if its initial length were about 200 feet, and at the 5.3-foot line if its 

 initial length were about 500 feet, assuming that the condition of the 

 wind and the slope of the bottom were such that the relationship holds. 

 Allowance must therefore be made for the shapes of the waves offshore ; 

 longer waves will break at greater depths than will shorter ones, if of 

 equal heights originally. If this factor is taken into account, and if 

 it is assumed that the ratio between depth of water and height of 

 breaker falls between about 1.3 to 1, and 2 to 1, as it usually does under 

 ordinary conditions of wind and weather, waves of different heights 

 initially, but of different degrees of steepness, may be expected to break 

 at approximately the depths given in table 32. 



We must call attention here to the fact that the ratio of length to 

 height for breakers at the instant of breaking, as deduced from the 

 theoretical relationship illustrated in Hgure 21 for waves advancing 

 over an evenly shoaling bottom, is uniformly greater than the sup- 

 posedly critical 7/1 (or than the inverse ratio of 0.14) unless they are 

 as steep as that initially. And this discrepancy increases, the greater 

 the original length of the wave is relative to its height, before it be- 

 gins to feel the bottom in its advance shoreward. Thus, a wave 10 

 feet high originally should, theoretically, be about 7.4 times as long 

 as high when it broke if it were 82 feet long (period 4 seconds) orig- 

 inally, about 10.6 times as long as high if its original length were 184 

 feet (period 8 seconds), about 14 tjmes as long as high at breaking if 

 its original length were 328 feet (period 12 seconds), and 24 times as 

 long as high if it were 738 feet long initially ( period 12 seconds) . And 

 we believe, from our own observations, that breakers commonly are 

 considerably more than 7 times as long from crest to crest as they are 

 high, although the very event of breaking makes it obvious that their 

 crests have steepened to the point of instability. 



The reason that waves do not reach the supposedly critical 7/1 

 ratio before they break is that the decrease in the length of a wave, 

 that accompanies its advance over a shoaling bottom, involves an alter- 

 ation of another sort in its profile, as has long been known, by which 

 the troughs become so much longer and flatter but the crests so much 

 shorter, that breakers caused by an old swell often appear as abrupt 

 ridges separated by stretches of nearly level water. The result of 

 this alteration is that the crests, acting independently of each other, 

 steepen to the angle of instability while the lengths, from crest to 

 crest, are still many times more than 7 times as great as the heights 

 of the latter. It is only when the surf is the product of storm seas, 

 steep enough to be breaking already, that this general rule does not 

 apply in greater or less degree. 



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