410 REPORTS ON THE STATE OF SCIENCE. — 1919. 



same period, like Osborne Reynolds' series of independent pendulums, so 

 that V varies as X, g=l, G=0; and the appearance is explained of the 

 stationary patches in the waves of the cornfield in a wind, compared by 

 Homer to the stage crowd of the public assembly in the market-place. 



Echelon Waves. 



4. These waves have been described and illustrated by W. Froude 

 {Trans. Listitution of Naval Architects, I.N. A., 1877), and an explanation 

 was given by Sir W. Thomson, reprinted in his Popular Lectures ; also 

 by R. E. Froude, on Ship Resistance {Greenock Philosoj^hical Society, 

 1894), and Hovgaard (I.N.A., 1909). 



They are seen stretching aft from the bow wave in a line of echelon, 

 and at sea on a calm day they can be watched from the deck of a steamer 

 till they disappear over the horizon, sifted out as pure waves with no 

 group velocity effect. On a small scale the wake of a duck will exhibit 

 the echelon waves quite clearly, or of a stick drawn through the water. 



The effect is explained by a combination of interference and group 

 velocity ; the following geometrical construction is submitted to replace 

 previous long analytical calculations. 



The bow wave is thrown off in a hump at in fig. 1, with the 

 crest OE at an angle 6 with the beam, and it is followed by the next 

 wave BC, thrown off at B on the side OB of the ship. 



Looking along BC, a hump is seen at C on the wave, and this is 

 followed by a series of equidistant humps on the line 00, showing 

 a series of waves following the ship in echelon, but with the crest 

 parallel to the bow wave at ; the problem is to calculate the angle ^ 

 which OC makes with the beam. 



The perpendicular OA on BC is the wave length A, between the 

 parallel crests OE, BC ; and representing the speed of tlie ship, K knots, 

 by the vector OK, and drawing KV perpendicular to AO produced, then 

 OV is the vector representing the wave velocity, V knots. 



The hump at C is produced by the interference of the wave A'C, of 

 slightly different length OA', \ + dX, and velocity OV, V + dV, at a small 

 angle AOA' ; and then in the figure 



(1) 



and drawing OF perpendicular to OA, 



i2^ AB ^ OE ^ A (TV BC ^ , _ A dV ^ G 



^ ' AC OF V d\ 'OF V dA V' 



G denoting the group velocity of the waves. 



When V varies as the qth power of A, 



(3) G = {l-q)Y, 



IA\ AB ABAC , n, r. n^ X clY 



(^) AIJ = AO AC = *"^ ^ '''' (^-^) = V JA = 2 



