# 7. Kern E. Kenyon, Roll Wave Theory

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Volume 11: Pages 531-540, 1998

# Roll Wave Theory

Kern E. Kenyon

4632 North Lane, Del Mar, California 920144134 U.S.A.

When a thin layer of water flows down a smooth solid surface of arbitrary but constant slope angle, it is surprising that waves are almost always observed to be present instead of a layer of uniform thickness. The crests move down the slope at constant speed while maintaining their shape. These are the “roll” waves of mathematical hydraulics. A physical model is adapted from Einstein's gravity wave method [Naturwissenschaften 4, 509 (1916)] by including frictional dissipation and gravitational driving down the slope, and is used to explain the existence of roll waves and to calculate their phase speed. The computed phase speed greatly exceeds the speed of the current upon which the wave rides and practically always exceeds the phase speed that a shallowwater gravity wave would have if it propagated down the slope under a reduced gravity consistent with the given slope angle (and in the absence of a current). Increases in phase speed occur with increases in slope angle, mean layer thickness, and wave amplitude, and with decreases in the viscosity coefficient. Quite large speeds are predicted for fluids of low viscosity, and the possible application of roll wave theory to explain the observed fast turbidity currents at the bottom of the ocean is discussed. A uniform layer is predicted to occur only when a certain nondimensional number, called here the roll number R1, equals one. By definition R1 = gh3f(α)/v2, where g is the acceleration of gravity, h is the mean layer thickness, v is the laminar viscosity coefficient, and f(α) is a given nondimensional (trigonometric) function of the slope angle α. The average downslope mass flux is conjectured to be a minimum for a layer of constant thickness and to increase as the superimposed wave amplitude increases. It appears difficult to calculate the wavelength and shape with the present approach, because finiteamplitude effects must be considered, but a possible way to approach these computations is outlined.

Received: November 6, 1997; Published online: December 15, 2008