HOMEWORK #3

1. Pre-dating Grant-Madsen, a popular approach to estimate initiation of motion criterion under waves was the formula proposed by Eagleson et al. (1961) –

where D is the grain size of interest, n is the kinematic viscosity, is the deep-water wave height, R is the submerged specific gravity of the sediment, g is gravity, T is the wave period and

where k is the wavenumber and h is the still water depth.

The criterion was based upon SMALL laboratory experiments of normally incident waves on a gently sloping beach.

Knowing that , show that the Eagleson et al. criterion may be expressed as Shields criterion (i.e., ) and comment upon its general applicability to natural environments.

Hint: Simplify the model using the expressions you are given and rules of hyperbolic functions. Then use the expressions given in class for the various relationships between t_m and u_bm - only one will produce a dimensionally consistent result.

2. Using a combination of the Shields curve in your notes and Grant-Madsen, calculate the maximum orbital velocity u_bm necessary to initiate motion of quartz sediment of diameters D = 5, 0.7 and 0.1 mm – for both a 10 sec and 5 sec wave. How does u_bm compare to the maximum velocity in a river U_r, assuming the ‘rule of thumb’ . Discuss (briefly) the reason for the differences.