Conservation of energy for non-dissipative water waves

In this paper Reynolds transport theory is applied to obtain energy conservation equation (ECE). Depth integrated ECE is derived considering a control volume extended over water depth. It is demonstrated that mild slope equation (MSE) is derivable from depth integrated ECE for linear waves. Two equations are derived by separating real and imaginary parts of depth integrated ECE for multidirectional wave fields that can properly deal with reflecting waves. Geometrical optics equations, derived for non-reflecting waves, can be obtained from aforementioned equations as well. Depth integrated ECE is averaged over a wave period for monochromatic waves. The result is similar in form to energy transport equation (ETE) governing spectral wave models. The terms of energy density and energy flux, obtained here, are more accurate expressions, comparing to those of ETE. It should be mentioned that ETE is also obtained from ECE, the general form derived here. The strength of newly derived equations in dealing with reflective wave fields has been illustrated by simple numerical computations. In this research MSE has been derived by the new presented method. Besides, coupled equations for multidirectional waves as well as time averaged depth integrated ECE have also been obtained through mentioned new approach.