J.J. Stoker - Water waves. The mathematical theory with applications (796980), страница 80
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However, in order to be effective as a reflector of waves such a floatingstructure would probably have to be built with a fairly large dimension in the direction of travel of the incoming waves. As a consequenceof the length of the structure, it would be bent like a beam under theaction of the waves and hence could not in general be treated withaccuracy as a rigid body in determining its effectiveness as a barrier.This brings with it the possibility that the structure might be bentso much that the stresses set up would be a limiting feature in thedesign. The specifications (as suggested by Carr) for a beam havinga width of one foot (parallel to the wave crest, that is) were:Weight: 85 pounds2/'ftMomentof inertia (of area of cross-section): 0.2Modulus of elasticity: 437 X 10 7 pounds I ft 24/2.The depthSimple harmonic progressing waves having periods of 8 and of 15 sees, were to be considered,and these correspond to wave lengths of 287 and 539 feet, and to circular frequencies a of 785 x 10~ 3 and 418 x 10~ 3 cycles per second,of the wateristaken as 40feet.respectively.
The problem is to determine the reflecting power of thebeam under these circumstances when the length of the beam is varied.we assume a wave train to come from the right handbeam and that it is partly transmitted under the beam toIn other words,side of theand partly reflected back to the right hand side.of the reflected wave and the amoftheamplitudeR/Bof the incoming wave is a measure of the effectiveness ofhandtheleftTheratioBsideRplitudethe beam as a breakwater.Before discussing the case of beams of finite length it is interestingand worthwhile to consider semi-infinite beams first. Since the calculations are easier than for beams of finite length it was found possibleto consider a larger range of values of the parameters than was givenWATER WAVES446above.
The results are summarized in the following tables (takenfrom[F.5]):TABLE AXa(ft)l\ W (pounds)IE^^-}(ft)2\secj539287225150113In Table\0.418850.200.785850.201.0850.201.5850.202.0850.20437437437437437XXXXXhR/B(ft)Jft10 710 740400.1910 7400.2310 7400.3210 7400.430.14A the beam design data are as given above. At the two speci-fied circular frequencies of 0.418 and 0.785 one sees that the floatingbeam is quite ineffective as a breakwater since the reflected wave hasan amplitude of less than 1/5 of the amplitude of the incoming wave,even for the higher frequency (and hence shorter wave length), whichof the incoming energy is reflected back.
Atmeans that less than 4henceandsmaller wave lengths, the breakwaterhigher frequencies,is more effective, as one would expect. However the approximatetheory used to calculate the reflection coefficient R/B can be expected%to be accurate onlyifthe ratio Xjh of wave length to depth is suffifor the case A287 ft. (a =- .785) the re-and evenciently large,flection coefficient of value 0.19maybe in error by perhaps 10% oris only about 7, and the errors for the shorter wavebewouldgreater. Calculations for still other values of thelengthsshownin Table B.
The only change as compared withareparametersmoresince A/ATABLE BW5390.4182870.785384384EIhRIB0.204377400.510.20437 X 10 7400.75X10the first two rows of Table A is that the weight per foot of the beamhas been increased by a factor of more than 4 from a value of 852pounds]ft* to a value of 384 pounds]'ft The result is a decided increase.in the effectiveness of the breakwater, especially at the shorter wave2length, since more than half (i.e. (.75) ) of the incoming energy wouldbe reflected back.
However, this beneficial effect is coupled with adecided disadvantage, since quadrupling the weight of the beamLONG WAVES IN SHALLOW WATERwould causeit447to sink deeper in the water in like proportion and henceis the same as themight make heavy anchorages necessary. Table CTABLECA except that the bending stiffness has beenafactorof10byby increasing the moment of inertia of thebeam cross-section from 0.2 // 4 to 2.0 // 4 Such an increase in stiffnessresults in a noticeable increase in the effectiveness of the breakwater,but by far not as great an increase as is achieved by multiplying theweight by a factor of four.
If the stiffness were to be made infinite(i.e. if the beam were made rigid) the reflection coefficient could bemade unity, and no wave motion would be transmitted. This isevidently true for a semi-infinite beam, but would not be true for afirsttwo rows of Tableincreased.rigidbody ofIn Tablebeamfinite length.Dstiffnessthe difference as compared with Table A is that thetaken to be zero.
This means that the surface of theisassumed to be covered by a distribution of inert particlesweighing 85 pounds per foot. (Such cases have been studied by Goldstein and Keller [G.I].) As we observe, there is practically no reflection and this is perhaps not surprising since the mass distribution perwaterisunit length has such a value that thebeamsinksdowninto the wateronly slightly.One might perhaps summarize the above results as follows: A verylong beam can be effective as a floating breakwater if it is stiff enough.However, a reasonable value for the stiffness (the value 0.2 givenabove) leads to an ineffective breakwater unless the weight of thebeamper square footweight of water.In practiceitisa fairly large multiple (say 8 or 10) of theseems unlikely that beams long enough to be consideredWATER WAVES448would be practicable as breakwaters.
(The term "longenough" might be interpreted to mean a sufficiently large multiple ofthe wave length, but since the wave lengths are of the order of 200feet or more the correctness of this statement seems obvious. ) It therefore is necessary to investigate the effectiveness of beams of finitelength. Such an investigation requires extremely tedious calculationsso much so that only a certain number of numerical cases have beentreated. These are summarized in the following tables.semi-infinitea=I.785,(ft)A=R/Ba=I.418,(ft)145.917.549.2287.9372.9A=539R/B.17196.9.53291.8.1398.5.75443.0.90145.9.10583.6.74656.2.62.33874.9.07450.4.32875.4.08583.6.12948.3.54656.3.13oo.14875.4.32oo.19196.9291.8In these tables the parameters have values the same as in the firsttwo rows of Table A, except that now lengths other than infinitelength are considered.
The most noticeable feature of the results givenin the tables is their irregularity and the fact that at certain lengthseven certain rather short lengths the beam proposed by Carrseems to be quite effective. For example, when the wave length isof the287 ft. a beam less than 50 ft. long reflects more than 80ateffectiveAofbeam443isalsolengthequallyincoming energy.ft.the longer wave length of 539 ft.*%* Itmight not be amiss to consider the physical reason why it is possible thatof finite length could be more effective as a breakwater than a beam ofinfinite length. Such a phenomenon comes about, of course, through multiplereflections that take place at the ends of the beam.
Apparently the phases someabeamtimes arrange themselves in the course of these complicated interactions in sucha way as to yield a small amplitude for the transmitted wave. That such a processmight well be sensitive to small changes in the parameters, as is noted in thediscussion, cannot be wondered at.LONG WAVES IN SHALLOW WATER449However, the maximum effectiveness of any such breakwateroccurs for a specific wave length within a certain range of wavelengths; thus the reflection of a given percentage of the incoming waveenergy would involve changing the length (or some other parameter) ofthe structure in accordance with changes in the wave length of theincoming waves.
Also, the reflection coefficient seems to be rathersensitive to changes in the parameters, particularly for the shorterstructures (a relatively slight change in length from an optimum value,or a slight change in frequency, leads to a sharp decrease in the reflection coefficient). It is also probableas was indicated earlier on thebasis of calculationsby Wells [W.10]that the shallow water approximation used here as a basis for the theory is not sufficiently accuratefor a floatingbeam whose lengthistoomuch less than the wave length.does seem possible to design floating breakwaters ofNevertheless,reasonable length which would be effective at a given wave length.itPerhaps it is not too far-fetched to imagine that sections could beadded to or taken away from the breakwater in accordance withchanging conditions.Another consequence of the theory which is also obvious ongeneral grounds is that there is always the chance of creating alarge standing wave between the shore and the breakwater because ofreflection from the shore, unless the waves break at the shore; thiseffect is perhaps not important if the main interest is in breakwatersoff beaches of not too large slope, since breaking at the shore line thenalways occurs.
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