Figure 3. Three-dimensional gravity-wave making due to an obstacle O. Waves generated t seconds 

 ago are shown. OP = Vi, PQ = cf, since < OOP = sin 1 (c/V), < OOP must be a right angle; 

 so locus of O is a semicircle. Locus of R (mid-point of PQ) is therefore also a semicircle. Similarly 

 waves generated i' seconds ago lie on semicircle through P', and the different semicircles envelop 

 a wedge of total angle 39°, within which all waves are found. Waves on boundary of wedge 

 are set as shown. 



sin _1 (c/F) to the stream. At a time / after they have been emitted they might be 

 expected to be at a distance ct from the centre of the disturbance, which however has 

 drifted a distance Vt downstream. In this case their crests would lie along a line through 

 the disturbance, like a "Mach line" in supersonic aerodynamics. All the waves emitted 

 at the same instant would be on their Mach lines, and the whole lot would lie on the 

 semicircle through the obstacle O and the point P to which the water passing the 

 obstacle at the instant of emission has drifted. However, this neglects the fact that 

 each wave group proceeds at the group velocity, and has travelled a distance Ut, not 

 ct, since it was emitted. As a consequence the waves have not got as far as the semi- 

 circle. For stream speeds which are well sub-critical all the waves produced are 

 essentially "deep-water" waves with U zz Vi c, and then the waves lie on an inner semi- 

 circle of half the diameter. Every wave, produced at any time, lies on one such semi- 

 circle, and all the semicircles lie lie in a wedge of total angle 2 sin -1 ^) = 39°; a 

 characteristic pattern is formed by the specially strong waves at the boundary of the 

 wedge, which are set at an angle of 55° to the stream. However, as the stream speed 

 V approaches \/ (gh) this semi-circle gets elongated at the front, and the wedge expands. 

 At supercritical speeds the wedge contracts again because only waves with c < \/ (gh) 

 are present, and it is then a genuine Mach wedge of semi-angle sin _1 ( \/(gh)/V). 



