Diffraction of light by two parallel superposed supersonic waves, one being then-th harmonic of the other: A critical study of the methods leading to approximate solutions in finite form
The system of difference-differential equations for the amplitudes of the light diffracted by two parallel superposed supersonic waves, consisting of a fundamental tone and itsn-th harmonic, is treated by two approximative methods giving solutions in finite form:
The method of successive approximations (SA method) and the method of theN-th order approximation (NOA method). The SA method reduces the system of difference-differential equations to a series of differential equations for each of the amplitudes; the integration of each differential equation becomes possible through the knowledge of the foregoing ones. In the NOA method the system of difference-differential equations is replaced by a system of 2N+1 simultaneous ordinary linear differential equations, the characteristic equation of which has only purely imaginary roots. Explicit calculations are made when the fundamental tone is accompanied by the second or the third harmonic; for the NOA method the approximationsN=1 andN=2 are considered. For values of the parameterρ=λ2/μ0μ1λ*2 large with respect to unity both methods lead to nearly the same results. If, however,ρ is of the order unity, the intensities of the diffracted orders obtained by the SA method are only valid for values ofζ=2πμ1z/λ small with respect to unity. For the same order of magnitude ofρ, the NOA method leads to more acceptable results; for values ofan=μn /μ1 (n=2, 3) of the order 10−1 there exists only a small difference between the intensities obtained in the first and in the second approximation; those differences become more appreciable for growing values ofan.