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A “two time scale” asymptotic expansion procedure describing the modulation of a propagating simple wave governed by a system of non-linear partial differential equations is applied to the deflection waves of non-linear elastic strings. Rapid deflection signals propagating into a general slowly varying disturbance are modulated. In addition, they themselves affect the equations for that disturbance. The two effects are separated naturally when, to prevent the cumulative growth inherent in most “high frequency” procedures, an averaging technique is introduced. The interaction of two deflection waves is given as a specific example.

The stability ofL 4 and the motion aboutL 4 in the restricted problem of three bodies is investigated when there is three-to-one commensurability between the long and short periods of motion, that is, when the mass ratio μ has the value μ=μ0.013516.... The two time scale method is used (1) to show thatL 4 is an unstable equilibrium point when μ=μ3, (2) to determine for what initial conditions periodic orbits occur when μ≈μ3, (3) to determine the stability of the periodic orbits, and (4) to investigate the boundedness of the motions aboutL 4 when μ≈μ3.