SECT. 5] WIND WAVES 665 



fraction of their length. It appears from hydrodynamical theory that the height 

 of waves cannot exceed about one-seventh of their length (Michell, 1893; 

 Wilton, 1913). At this limiting stage the crest of the wave develops a sharp 

 ridge with water sloping at 30'^ to the horizontal on either side (Lamb, 1945, 

 Art. 250). Further energy added to the wave would cause water to spill out of 

 the sharp crest and the "white caps" seen under a strong wind probably 

 develop in this way. In practice it is unusual to meet waves the height of which 

 exceeds one-tenth of their wavelength (Sverdrup and Munk, 1947). 



Precise discussions of waves in terms of a velocity potential can be found 

 elsewhere (Lamb, 1945, chap. IX; Stoker, 1957). Here it is sufficient to point 

 out an elementary argument which may assist the reader in recovering the 

 formulae relating the period wavelength and velocity of deep-water waves. 

 Everywhere on the undulating surface the water experiences horizontal and 

 vertical accelerations and the water surface arranges itself always at right 

 angles to the "false vertical", the combination of these accelerations and the 

 force of gravity. Now if it is true that each surface particle moves on a circular 

 orbit of diameter H (the w ave height crest to trough) in a periodic time T, the 

 acceleration towards the centre is "Itt'^HJT'^. When the particle is halfway up its 

 orbit (as point A in Fig. 1) this acceleration is purely horizontal and it combines 

 with gravity to give a false vertical inclined to the true vertical by an angle 

 whose tangent is 



But if the wave is sufficiently low to have a profile that is nearly sinusoidal with 

 height H and wavelength L, the water surface at a point half-way between 

 crest and trough is inclined to the horizontal by an angle whose tangent is ttHJL. 

 Equating these two expressions gives the wave equation, 



L = gT^l2n. (1) 



Since the velocity of advance C is necessarily equal to LjT, the equation can 

 also be written in the manner 



L = 2TTC^Ig (2) 



G = gTl^TT. (3) 



Numerical values for some typical ocean waves are listed in Table I. 



Table I 

 Typical Sea Waves 



Tj'pe Period, Wavelength, Velocity, Group velocity, 



sec m m/sec m/sec 



