an analytical expression for the families of wave spectra has been 

 derived. These spectra have been compared with those proposed by others 

 and were found to be in good agreement with available data. In addition, 

 a theory for waves of finite height has been derived, and will be 

 presented in a paper before the Coastal Engineering Conference in August 

 1961. This theory is an exact one, and may be extended to any order. 

 It is represented by a summation in harmonic series, each term of which is 

 in an expanded form. The terms of the series when expanded result in an 

 approximation of the exact theory and this approximation is identical to 

 Stokes' wave theory extended to the same order. The theory represents 

 irrotational-divergence less flow. 



(g) Equilibrium Profile and Model Scale Effects Studies. 



Testing was continued in a small tank utilizing low specific 

 gravity material (crushed coal) to study the effect of scale on movable 

 bed models under wave action. Profiles derived from these tests bear 

 basic resemblances to the profiles obtained with large (up to 5-1/2 foot) 

 waves on a sand beach, from which the small-scale tests were modeled. 

 However, certain differences still occur. One possible explanation for 

 some of these is the wide range of specific gravities of individual coal 

 grains, even though the average specific gravity is modeled according to 

 the settling velocity of the beach sands tested. 



(h) Pubble-Mound Stability. 



Large-scale (7.5 to 1) tests on stability of rubble-mound structures 

 under wave action were continued to spot check results of the small-scale 

 test program at the Waterways Experiment Station. These tests involve a 

 1 on 1-1/2 slope rubble breakwater using approximately 1-foot diameter, 

 160-pound stone. Tests to determine the design wave (i.e., the height 

 of wave which will just initiate damage after 1-1/2 hours of wave attack) 

 have been completed, as have tests with damaging waves of heights 

 approximately 1.5 times the design height. These latter tests were run 

 to check the rate and amount of damage caused by waves higher than the 

 design wave. Final analysis of the data and interpretation of results 

 is awaiting precise calibration of the wave tank and determination of 

 wave heights at the structure location without the structure in place. 

 Essentially all tests made to date have involved non-breaking waves, 

 or waves which, if breaking, were not fully breaking. As a distinct 

 possibility exists that somewhat greater damage may be caused by fully 

 breaking waves having the same incident deep water height as non- 

 breaking waves, a 1 on 10 concrete beach slope is being installed in 

 front of the breakwater to permit generation of a fully breaking wave 

 on the breakwater base. T^sts with this breaking wave condition will 

 be repeated for the same wave generator settings as were used previously 

 to obtain the design wave. 



39 



