THEORY OF SIMPLE WAVES 



321 



Fix, n) = Fix, 0) + ivdFix, 0)/by) (37) 



AoTj then amounts to the second term of expression (37). 

 Substituting expression (18) for t], taking its derivative 

 with respect to i/, and letting y = 



Sin = 2i7ra-/\) cos' /v(.r - ct) (38) 



The total correction is therefore 



A(77/X) = (AiTj + A,7/)/X 



= T C^y [cos-^ kix - ct) - sin'^ kix - ct)] 



= TT C'V cos 2k ix - ct) (39) 



After letting t = this is seen t(j be identical with tlie 

 second term of Stoke's equation (34). 



In a great majority of potential-flow problems con- 

 nected with waves, the work is based not on the wave 

 height itself, but on the velocity potential of the wave 

 motion. The primary purpose of the foregoing discus- 

 sion has been to demonstrate that iLwrk based on the 

 wave-velocity potential is correct icithin a second-order ap- 

 proximation in the potential gravity ivave theory. While a 

 simple harmonic wave is usually referred to in the intro- 

 ductory text of any such work, the results apply in reality 

 to very nearly trochoidal wa\'e form. 



3.2 Maximum Wave Height. As the wave height in- 

 creases, and the terms of higher order become significant, 

 the crest of the wave becomes sharper and sharper, until 

 finally it takes the form of a sharp cusp. The value of 

 the minimum included angle at the cusp was derived by 

 Stokes on the following basis: A uniform velocity — c is 

 imposed on the water, so that the waves become sta- 

 tionary in space. At the cusp the velocity must be zero, 

 and in close vicinity to it the water flow corresponds to 

 the one occurring in a re-entrant angle. Using polar co- 

 ordinates and the notation indicated in Fig. 3, the velocity 



ne 



w/2. 



(42) 



Fig. 3. Sketch used in connection 

 with Section 3,2 



potential of such a flow is (Lamb, 1-D, par, G3, p. G8 and 

 p. 418) 



(j) = Ar" sin nO (40) 



The velocity normal to the bounding surface (i.e., at 

 e = ±a) 



-id<t>/rdd) = = «.4r"-i cos nd (41) 



and therefore 



The velocity along the surface is 



U = -id4>/dr) = -nAr"-^ sin nd at 6 = ±a (43) 



Since the water motion in the wave is solely due to 

 gravity and inertial forces, and since the velocity is zero 

 at the cusp, the velocity at any vertical distance r cos d 

 below the cusp is 



U = (2grcose)'/' (44) 



B}' equating the exponents of r, in — 1) in equation (43) 

 and V2 ill equation (44), n is evaluated as 3/2. From 

 equation (42) it then follows that max 6 = a = 7r/3; 

 therefore the total included angle 2a is 120 deg. 



Michell (1-1893) and Havelock (1-1918) found that 

 the corresponding limiting wave height from trough to 

 crest is O.I'fX (expre.s.sed to two decimal points). Such a 

 limiting height is approached in experiments only with a 

 perfect wave form. Any irregularity of the wave 

 causes a prematiu'c breaking of crests. 



3.3 Mass Transport. The volume of w^ater flowing 

 between trhe free surface and a depth y below it is ex- 

 pressed by the stream function - 



^ = --ace'"' cos kix - ct) (45) 



where to the second order of approximation 



y = yo + ae""' cos kix — ct) (4(3) 



Substituting equation (46) into (45) and retaining only 

 the flrst two terms of a serial expansion of the ex- 

 ponential, 



\p = —ace"'"' [1 -\- kae""" cos kix — ct)] cos A;(.r - ct) 



(47) 



The mean rate of flow for the water layer contained 

 between the surface and the depth yo can be oljtained by 

 integration of \p over the cycle period T. Tli(> mean rate 

 of flow is independent of the position .r, and without loss 

 (jf generality can be evaluated at x = 0. Noting fm'ther 

 that onh' terms containing cosine squared in the integral 

 contribute to the value of the integral, the mean value of 

 ^, designated here by ip, is 



1 r^ 



i/' = - - ka-ce-""' cos- 2Tvt/T dt 



T Jo 



(48) 



= -^/ika.^ce-""" 



The mean velocity ii of the water flow at the depth yo 

 is obtained by partial dift'erentiation 



u = -diA/dj/o = k'-a^ce-""" (49) 



wiiich is identical with the expression derived by Lamb 

 (1-D, p. 419, equation 16). 



As an example, a wave 600 ft long and 30 ft high (a = 

 15) can be considered. Its celerity is 55.4 fps. The sur- 

 face layer of water is shown by eciuation (49) to drift at 



^ Table 1 in this Appemlix. 



