A closed-form waveguide invariant β for a Pekeris waveguide is derived. It is based on the modal Wentzel-Kramers-Brillouin (WKB) dispersion equation and implicit differentiation, in conjunction with the concept of the "effective boundary depth," ΔH(θ), where θ is the propagation angle. First, an explicit formula for β(m,n) between mode pairs is obtained assuming an ideal waveguide of the effective waveguide depth, H+ΔH(θ), and provides an excellent agreement with the reference value for the Pekeris waveguide of depth H obtained using the normal mode program kraken. Then, a closed-form expression for a group of adjacent modes is derived: β=(H+ΔH(θ))/(H/ cos θ-ΔH(θ)), which can be approximated by β=cos θ as ΔH(θ)/H≪1, the analytical expression for an ideal waveguide.
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