Whitham’s variational method is formulated so as to apply to weak second-order resonant interactions among waves whose amplitudes and phase angles vary slowly with position and time. The method is applied in detail to capillary-gravity wave interactions. An internal gravity waves problem is also discussed briefly. The method leads to new and substantial simplifications of the interaction equations. This makes possible the proof of local conservation of total mean wave energy and momentum laws. These, together with another integral of the motion, are found to be of central importance in classifying and characterizing the slow modulations of planewave-like form. Such a classification is given in detail for all initial values of phase angles and relative amplitudes. All progressive uniform waves in the capillary range are found to be unstable with perturbation growth rates which can be of first order in the wave slopes. In this formulation amplitude dependent first-order corrections of classical frequency and/or wave-number arise for all waves participating in a resonance. A few predictions which could be verified by simple experiments are made.

The stability of progressive internal waves of modes 1 and 3, propagating down a long tank filled with a linearly stratified salt water solution, is studied theoretically and experimentally. Examination of the spectra of the waves shows when a1 > 10−2, where a is the wave amplitude and l is the vertical wavenumber, that single internal waves excite waves of several resonant triads, where the excited waves belong to that set of triads with the largest theoretical growth rates. For example, a wave of mode 3 with a non-dimensional frequency around 0.66 excites waves of the following triads: (5,8,3), (6,9,3), (8,11,3), (9,12,3) and (10,13,3), where the integers are mode numbers. The spontaneous appearance of these naturally excited triads greatly complicates attempts to isolate and study preselected wave interactions. In one case, when waves of mode 1 and 3 with al > 10−2 were generated simultaneously while tuned to the (1,3,4,7) multiple resonance, the fastest growing wave was neither a wave of mode 4 located at the difference frequency nor a wave of mode 7 at the sum frequency, but rather a wave of mode 9 located at a frequency slightly above that of the 4-wave.

Successful synthesis of a 1,2-dioxetane was not accomplished until 1968 when the synthesis of trimethyl-1,2-dioxetane was reported.1 Since that time many 1,2-dioxetanes have been synthesized and characterized.2 Four synthetic methods have been utilized in their production: (1) closure of a /3-halohydroperoxide with base, (2) addition of singlet oxygen to enol ethers, enamines, and alkenes with unreactive allylic hydrogens, (3) assisted closure of a /3-halohydroperoxide with silver ion, and (4) 9,10-dicyanoanthracene (DCA) photooxidations.3We now report a fifth method which we have used for the synthesis of adamantylideneadamantane dioxetane. A solution consisting of 50 mL of methylene chloride, 1.71 g of tetra-n-butyl ammonium perchlorate, and 104.5 mg of adamantylideneadamantane (1) was subjected to a constant potential electrolysis (1.6 V vs. SCE) at a platinum gauze electrode. The electrolysis was conducted while the solution was continually agitated by bubbling oxygen vigorously through the cell and until current passage through the cell ceased. The methylene chloride was then

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