Rayleigh’s principle for one-way wave propagation asserts that the average kinetic energy in the wave must be equal to the average potential energy, i.e., 
Here, u is the horizontal particle velocity, v the vertical particle velocity, and p the pressure. This energy conservation formula can be used to determine a “natural” reference wavenumber k0 for propagation in any of the parabolic approximations to the Helmholtz equation.
a. Derive an approximate expression for k0 in terms of integrals of field quantities satisfying the standard parabolic
b. For a single mode propagating in an ideal, pressure-release waveguide show that the “natural” wavenumber found in (a) equals the modal eigenvalue.
c. Discuss the implications of multi-mode propagation for the choice of a reference wavenumber, particularly in lossy environments with mode stripping.
d. Consider next the alternative PE form given by Derive the approximate expression for k0 and show that for single-mode propagation in an ideal, pressure-release waveguide the “natural” wavenumber now equals the water wavenumber.
