The experimental investigation was extended to the long-term evolution of 

 an initially uniform wave train by Lake, et al. (1977). When the average 

 value of ak exceeded 0.1, the onset and development of BF sideband insta- 

 bilities were observed in early stages of evolution, followed by a spread 

 of spectral energy to many frequencies in addition to the carrier and side- 

 band frequencies. Envelope solitons in the "disintegrated" wave train were 

 observed to be consistent with soliton solutions of the nonlinear Schrodinger 

 equation. However, Lake, et al. presented clear evidence that envelope soli- 

 tons are not the ultimate state of wave-train evolution. They observed that 

 the highly modulated train, characterized by a broad spectrum, actually 

 demodulated and very nearly returned to its initial state as a uniform wave 

 train. A small decrease in carrier frequency was observed in some cases. 

 They suggested that the ultimate evolution of a train of steep waves is a 

 series of periodically recurring states of modulation and demodulation, not a 

 series of stable envelope solitons. They used numerical techniques to solve 

 the nonlinear Schrodinger equation with periodic initial conditions. It was 

 demonstrated that periodically recurring modulated-demodulated states are 

 characteristic of solutions to the nonlinear Schrodinger equation. 



Lake and Yuen (1978) extended the investigation further to include wind- 

 generated laboratory waves. They presented strong evidence that a broad 

 wind-wave spectrum is better represented as a coherent collection of bound 

 frequency components than as independent components . They proposed a non- 

 linear wind-wave model consisting of a single dominant wave frequency with all 

 other spectral energy bound to the dominant wave. It was proposed that free 

 wave energy exists, but the existence is primarily in very short waves gener- 

 ated by local winds and wave breaking, representing a negligible fraction of 

 the total energy. This model is obviously a drastic change from the long-held 

 conception of a spectrum as a random collection of free wave components. 



Both Lake, et al. (1977) and Lake and Yuen (1978) reported an important 

 characteristic which cannot be accounted for by the nonlinear Schrodinger 

 equation. When the waves are sufficiently steep, the BF sidebands develop and 

 strong modulation followed by demodulation occurs, but the carrier frequency 

 of the demodulated wave train becomes the frequency of the low-frequency side- 

 band in the original train. Lake and Yuen indicated that discrete shifts such 

 as these may be the primary mechanism by which energy in a developing sea is 

 transferred to lower frequency. Thus the nonlinear Schrodinger equation fails 

 to model a crucial characteristic of developing wind waves. 



Modulation frequency was investigated in laboratory wind waves by Lake and 

 Yuen (1978). Identifiable modulations were present with modulation periods on 

 the order of equation (3) but they were generally somewhat longer. 



The modulation period given by Benjamin and Feir (eq. 3) applies strictly 

 to small values of wave steepness, ak. An extensive theoretical investi- 

 gation in which the instabilities of finite-amplitude deepwater waves were 

 calculated over a large range of wave steepness was reported by Longuet- 

 Higgins (1978a, 1978b). Two types of instability were identified — super- 

 harmonic instabilities, which are associated with wave breaking, and BF-type 

 subharmonic instabilities. However, as wave steepness increased beyond about 

 0.346, the BF-type instabilities became stable. A new type of subharmonic 

 instability with a very high growth rate appeared at a wave steepness of 

 about 0.41. The practical significance of this theoretical instability is 

 uncertain. 



17 



