An empirical relationship is presented for the incipient motionof bottom material under solitary waves. Two special casesof bottom material are considered: particles of arbitrary shape,and isolated sphere resting on top of a bed of tightly packedspheres.
The amount of motion in the bed of particles of arbitrary shape isshown to depend on a dimensionless shear stress, similar to the Shieldsparameter. The mean resistance coefficient used in estimating thisparameter is derived from considerations of energy dissipation, and isobtained from measurements of the attenuation of waves along a channel. Atheoretical expression for the mean resistance coefficient is developedfor the case of laminar flow from the linearized boundary layer equationsand is verified by experiments.
For the case of a single sphere resting on top of a bed of spheres,the analysis is based on the hypothesis that at incipient motion thehydrodynamic moments which tend to remove the sphere are equal to therestoring moment due to gravity which tends to keep it in its place. Itis shown that the estimation of the hydrodynamic forces, based on anapproach similar to the so-called "Morison's formula", in which the drag,lift, and inertia coefficients are independent of each other, is inaccurate.Alternatively, a single coefficient incorporating both drag,inertia, and lift effects is employed. Approximate values of this coefficientare described by an empirical relationship which is obtainedfrom the experimental results.
A review of existing theories of the solitary wave is presented andan experimental study is conducted in order to determine which theoryshould be used in the theoretical analysis of the incipient motion ofbottom material.
Experiments were conducted in the laboratory in order to determinethe mean resistance coefficient of the bottom under solitary waves, andin order to obtain a relationship defining the incipient motion ofbottom material. All the experiments were conducted in a wave tank40 m long, 110 cm wide with water depths varying from 7 cm to 42 cm.The mean resistance coefficient was obtained from measurements of theattenuation of waves along an 18 m section of the wave tank. Experimentswere conducted with a smooth bottom and with the bottom roughened witha layer of rock. The incipient motion of particles of arbitrary shapewas studied by measuring the amount of motion in a 91 cm x 50 cm sectioncovered with a 15.9 mm thick layer of material. The materials used haddifferent densities and mean diameters. The incipient motion of sphereswas observed for spheres of different diameters and densities placed ona bed of tightly packed spheres. The experiments were conducted withvarious water depths, and with wave height-to-water depth ratios varyingfrom small values up to that for breaking of the wave.
It was found that: (a) The theories of Boussinesq (1872) and McCowan(1891) describe the solitary wave fairly accurately. However, thedifferences between these theories are large when used to predict the forceswhich are exerted on objects on the bottom, and it was not established whichtheory describes these forces better. (b) The mean resistance coefficientfor a rough turbulent flow under solitary waves can be described asa function of Ds, h, and H, where Ds is the mean diameter of theroughness particles, h is the water depth, and H is the wave height.(c) Small errors in the determination of the dimensionless shear stressfor incipient motion of rocks result in large errors in the evaluationof the diameter of the rock required for incipient motion. However, itwas found that the empirical relationship for the incipient motion ofspheres can be used to determine the size of rock of arbitrary shape forincipient motion under a given wave, provided the angle of friction ofthe rock can be determined accurately.
