# 9. Kern E. Kenyon, Force and Torque on a Wall from Reflected Surface Gravity Waves

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Volume 17: Pages 95-102, 2004

# Force and Torque on a Wall from Reflected Surface Gravity Waves

Kern E. Kenyon

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

An elementary derivation is given for the total dynamic force perpendicular to a rectangular section of a twodimensional vertical wall when a freely propagating surface gravity wave reflects from the wall. Starting assumptions are as follows; a plane wave of moderately small amplitude travels normal to the wall and is completely reflected without breaking; the water is infinitely deep and homogeneous with no current flowing and no wind blowing; the wall is rigid, impervious, and smooth and extends indefinitely horizontally and vertically. The dynamic force equals the time rate of change of momentum, and the linear momentum per unit volume of a surface gravity wave is the particle (fluid) density multiplied by the wellknown steady Stokes drift velocity, which points in the direction of wave propagation. During reflection of the wave by the wall, the linear momentum per unit volume within one wavelength reverses sign in one wave period, causing a steady force per unit vertical area on the wall at each mean depth. With increasing mean depth, the force per unit vertical area decreases exponentially, where the exponent equals twice the wavenumber times the mean depth. Integration of the force per unit vertical area over the entire surface of the wall gives the total steady normal force on the wall: (fluid density) × (acceleration of gravity) × (wave amplitude squared) × (horizontal length of wall). Exactly the same total force on the wall occurs if the mean depth of water is finite and constant. By Newton's third law (action equals reaction) the wall exerts an equal but opposite force on the wave. No data are found suitable for comparison with the proposed force formula, and other published theories, of which there are more than 25, have not been found to agree with it within a factor of two. Since surface gravity waves have angular momentum, related to the orbital particle motion, and since the angular momentum changes sign after reflection, the waves also exert a torque on the wall at all mean depths, within the depth of wave influence, that is directed horizontally and parallel to the wall. The total torque on the wall is calculated, and in general it is a function of the wavelength and the mean depth in addition to being proportional to the square of the amplitude.

Received: October 30, 2003; Published online: December 15, 2008