MS Thesis R. Hager - Hawaii National Marine Renewable Energy ...

I iAg e ekz i( kx t)(30)cosh( k(z d))If the water depth, d, is finite, then the depth parameter, e kz , is rewritten as cosh( kd),satisfying the seabed boundary condition:IAg cosh( k(z d))sin( kx t) cosh( kd )To satisfy the free-surface boundary condition, the phase velocity is described by thedispersion relation:k g tanh( kd ) k 12C (31)Energy, however, does not travel at the phase velocity. Rather, energy travels at the groupvelocity, which is seen in a kinematic or dynamic view. The kinematic view is presentedbelow. A band of waves with similar lengths and traveling in a similar direction at a groupvelocity, C g , comprise a wave envelope:C g (32) kThen Eq. (32) converges to:d C 2kd 1 dk 2 sinh(2kd) C g (33)Figure 3.2 depicts the individual waves moving with a phase velocity traveling in a band witha group velocity.41