SECT. 5] ANALYSIS AND STATISTICS 569 



Chapter 21 for the effect of capillarity, neglected here for simplicity.) There is 

 also the kinematic surface condition 



^ + ^ + ^^ + ^^=0, , = ^. (3) 



dt dz dx dx dy 8y ' 



If the wave amplitude is small enough for the second-order terms in (2) and 

 (3) to be ignored, then the following periodic solution is well known : 



, - ag cosh k{z + h) . , _ i ■ n 



(p = vrji, — ^^^ y"^'^ ^*-*^ " + y^ ^'^^ V — at+ e), (4) 



(J cosn fCih 



where 6 is an arbitrary direction of propagation, e an arbitrary phase angle, 

 and the circular frequency a ( — 27T/period) is related to an arbitrary wave 

 number k ( = 27r/wavelength) by the equation 



CT^ = gk tanh kh. (5) 



For practical purposes (5) is usually replaced by the simpler asymptotic form 

 a" — gk when the depth is greater than about a quarter wavelength; alter- 

 natively, if the depth is very small compared with the wavelength, we have the 

 non-dispersive limit, (CT/^")= phase velocity = -y/grA. 

 From (4) we obtain the surface elevation 





z = 



which is a sinusoid of amplitude a with infinitely long crests parallel to the line 

 y = xtaind, and the first-order horizontal and vertical velocity perturbations 

 and pressure perturbations given by 



_^, _^, _^, and - ^ 

 dx dy dz 8i 



respectively. 



Though this periodic solution is often used for rough calculations on sea 

 waves, it is clearly unrealistic on account of its infinitely long crests and con- 

 stancy of amplitude and frequency. A generalization of (4), also much used in 

 analytical work, follows from Fourier's Integral Theorem, by means of a double 

 integral over (0<^<oo, 0^^^277), with assigned amplitude and phase func- 

 tions a{k, 6), €{k, 6). This integral form has the advantage that it can be fitted 

 to almost any instantaneous wave pattern defined in ( — oo<a;<oo, 

 — CO <y < oo), even one which is virtually zero outside a finite zone, and the 

 subsequent wave motion deduced. Application of this sort of analysis to 

 models of sea waves with idealised initial conditions (e.g. Pierson, 1955) does 

 yield some results confirmed qualitatively by observations. For example, the 

 independent travel of wave-fronts from a storm centre with group velocity 

 dGJdk, causing the spectrum at a distance from the storm to increase slowly in 

 frequency (Barber and Ursell, 1948) ; also that a small group of waves initially 

 confined to a few wavelengths spreads into an ever-widening group with an 



