THEORY OF SIMPLE WAVES 



323 



iipijroaches the front, while it.s former place in the group 

 is oeeupietl in suece.s.sion by other waves which ha\-e 

 come forward from the rear. 



The simplest analytical representation of such a gnnip 

 is obtained by the superposition of two systems of waves 

 of the same amplitude, and of nearly, but not ((uite, the 

 same wave length. The correspontling etiuation of the 

 free surface will be of the form 



7) = a sin (/.'.r — wt) + a sin (/,'.r 

 ,(A- - A-'),r - ' .(c. 

 sin |V2(/>-+ A-').*- 



1 1 

 I . 



Lo't) 



where co is written for the circular fre(|uency kc = '2Trc/\. 

 If l\ k' be very nearly etjual, the cosine in this expres- 

 sion varies very slowly with .r, so that the wave profile at 

 any instant has the form of a cur\-e of sines in which the 

 amplitude alternates gradually between the values and 

 2a. The surface, therefore, presents the appearance of 

 a series of groups of wa\'es separated at ecjual interx'als liy 

 bands of nearly smooth water. Since the distance lie- 

 tween the centers of two successive groups is 2iv/{k — k') , 

 and the time occupied l)y the system in shifting through 

 this space is 27r(a) — w'), the group-\-el<icilv ((', say) is 

 = ((^ - co')/(fc - k') 



I' 



r/a- dk 



m) 



This result hokls for any case of waxes tra\'eling 

 through a uniform medium. In application to waves on 

 water of depth h, 



c = 



g tanh kh\/' 



and therefore, for the group x'clocity, 



U = d{kc)/dk = V2 c 1 + ^--— (63) 



2kh 

 ■<nih 



The ratio which this liears to the wave celerity c in- 

 creases as kh diminishes, Iwing 1/2 when the depth is 

 very great, and unity when it is \-ery small, compared 

 with the wave length. 



It is observed that the group velocity is identical with 

 the rate of transmission of the energy in waves. Indeed, 

 an isolated group of waves cannot advance into still 

 water unless its energy is transmitted at the velocity of 

 the group. 



4.4 Damping of Waves. The theory of waves dis- 

 cu.s.sed up to now neglects the viscosity and assumes a 

 perfect or in viscid fluid. The waves described by this 

 theory represent therefore a "conservative system"; 

 i.e., there is no loss or gain of energy, and the total energy 

 contained in the sj'stem remains constant. Under 

 "energy" here is meant the sum of the potential (gravity) 

 and kinetic energies. Loss of energy means conversion 

 of it into the energy of molecular motion; i.e., heat. In 

 an actual fluid po.s.sessing a small viscosity, as is the case 

 with water, the e(|uations resulting from the potential 

 theory are still found to describe the flow correctly 



(Lainii, i-D, Art. ;U(), pages (i2.3-()25), but there is a 

 certain small rate of dissipation of energy due to internal 

 friction resulting from the distortion of fluid elements. 

 The wa\-es, therefore, cannot propagate without change 

 of form, but propagate with gradually and very slowly 

 decreasing amplitude. Ki|uation (.18) for the total 

 enei-gy per unit area of sea surface can be rewritten as 



E = ^/■2pkc-a- 

 and the rate of energy loss expressed as 

 d/dt{ \'2pkc-a-) 



(64) 



(().5) 



I'rom a consideration of tlie work ilone by \'iscous 

 forces (Lamb, l-I), p. 624) the rate of energy dissijiation 

 is 



-2m/.''c-o- (()6) 



where n is tlie viscosity of water. By e(|uating (6.")) and 

 (66) the rate of decrease of wa\-e am))lifu(le is 



da 

 dt 



= -2vk'a 



(67) 



or 



flllC 



-•2,kn 



(68) 



where Oo is the initial wave amplitude at t = (L a the 

 amplitude at the time /., and v is the kinematic \'iscosity 

 = fi/ p. The time neces.sary for the wave amplitude to 

 l)e I'educed to 1/c of its original amplitude, callecl the 

 modulus of decay, is 



t = X=/8!' 



(69) 



Since k is s(|uarcd in the ex]3onential, the short waves 

 with large k are damped rapidly, while the long waves 

 with a small k are attenuat(>d extremely slowly. Table 4 

 (taken from O'Brien and Mason, 1-G, page 42) gives the 

 length of the wave travel in which the amplitude is re- 

 duced to 1/c of its initial value. 



Table 4 



\^'ave 



period, 



sec 



1 







10 



Wave 



length, 



ft 



5.1 



128 



512 



Wave 



celerity, 



fps 



5.1 



25,6 



51 2 



Wave 

 travel, 

 miles 



32.5 



102000 



3250000 



The foregoing theoretical computations are conlirmed 

 ciualitatively by observing waves in nature. The small 

 ripple is foimd to disappear in a few minutes, while the 

 long swells are found to travel hundreds or thousands of 

 miles without appreciable attenuation. This slow rate 

 of damping of long swells makes it po.ssible to forecast 

 the surf conditions, which often have their origin in 

 a far distant storm. The rate of amplitude decay just 

 given is entirely due to internal friction. In the actual 

 .sea there is additional and apparently larger dissipation 

 of energy due to turbulence and to the bi-eaking of steep 

 waves. 



