Improved explicit approximation of linear dispersion relationship for gravity waves

For water waves the transcendental dispersion relationship is solved by iterative methods when wave period and water depth are given and wavelength or wave number is required. A highly accurate explicit approximation to linear dispersion relationship is proposed based on Eckart's explicit relationship. While Eckart's expression is accurate to within 5%, the improved relationship has a maximum relative error of less than 0.05%. A simpler form of the relationship with 0.2% accuracy is also given.