EVALUATION AND DEVELOPMENT OF WATER WAVE THEORIES FOR 

 ENGINEERING APPLICATION 



by 

 R. G. Dean 



1. INTRODUCTION 



The following were the primary goals of the research reported: (l)for given wave 

 conditions, to estabUsh a rational basis for selection of one of the numerous available 

 progressive -water-wave theories and (2) to tabulate the most appropriate wave theory or 

 theories in a form convenient for preliminary design use. The main emphasis has been an 

 attempt to assist the engineer in his selection and apphcation of wave theories in marine 

 design problems. The research has proceeded in several distinct phases which are described 

 briefly below. 



An early phase of the research was related to evaluating the analytical validity of 

 water-wave theories; that is, the degree to which the various available theories satisfy the 

 equations constituting the mathematical formulation. The results of this phase, first 

 published in September, 1968 (Dean, 1968a), estabhshed that, of the eight theories included 

 in the study, the Stream-function fifth-order provided the best fit over a wide range of wave 

 conditions. For very shallow water waves, the Airy and first-order Cnoidal theories provided 

 the best fit. However, because the Stream-function theory can be extended to quite high 

 orders, it was expected that it would provide the best fit, even for most shallow water wave 

 conditions. Based on the results of this phase, the following phases concentrated on further 

 exploration and development of the Stream-function theory for engineering apphcation. 



The second phase represented an examination of near -breaking wave conditions using the 

 Stream-function theory (Dean, 1968b). This problem is compUcated because breaking 

 conditions represent a mathematical as well as a hydrodynamic instability, and therefore the 

 computational aspects are not straightforward. The results of this study indicated that of 

 the two stability criteria, the kinematic criterion rather than the dynamic criterion governs 

 at breaking. It was also found that near breaking the pressure distribution was hydrostatic 

 rather than characterized by a zero pressure gradient as predicted by some other studies. 

 The complexities of the numerical computations led to an attempt to estabhsh the breaking 

 index for only three relative water depths (shallow, transitional zone and deep). It was 

 found that for shallow and deepwater waves, the breaking heights estabhshed from the 

 Stream-function wave theory were up to 28 percent higher than those established earher by 

 other investigations. For transitional depth conditions, however, the breaking heights 

 determined in the study agreed well with those of earlier investigations. 



The third phase of the investigation (Dean and LeMehaute, 1970) was related to the 

 "experimental validity of water wave theories" as compared to "analytical validity." The 

 motivation of this phase was the recent pubhcation of a fairly comprehensive set of 

 measurements of water particle velocities for shallow water waves and comparison with a 

 number of wave theories by LeMehaute, Divoky, and Lin (1968); a comparison with the 



