A close approximation of wave dispersion relation for direct calculation of wavelength in any coastal water depth

The most commonly used wave dispersion relation, k0h=khtanh(kh)k0h=khtanh(kh), has been closely approximated as an explicit formula for direct calculation of wave number kk in any coastal water depth hh by using two different root-finding methods. One explicit formula derived with the Newton–Raphson method has a maximum relative error of only 0.01% in calculating kk in any water depth. This is the most accurate explicit one proposed so far and should be used to compute kk in wave-related models that require a large of number of wavelength calculations. The other explicit formula derived with the one-point iteration method has a maximum relative error of 0.1%, but is of a simple form and can be easily applied to compute kk in any water depth with a hand calculator. The validity of the wave dispersion relation under the coastal conditions is also investigated based on the comprehensive field data on wave pressure and orbital velocity.