Wave generation capability has evolved from the time when researchers were completely at the mercy of a limited technology to the present where the capability of wave generators and computers exceeds the current understanding of wave dynamic processes." ...Ed Funke and Etienne Mansard (1987) ' This paper was the source of the quotation opening this chapter. 2 English translation by M. Pilch. Physical Models and Laboratory Techniques in Coastal Engineering Downloaded from www.worldscientific.com by NANYANG TECHNOLOGICAL UNIVERSITY on 08/25/15. For personal use only. 7.2. TWO-DIMENSIONAL GOVERNING EQUATIONS Physical Models and Laboratory Techniques in Coastal Engineering Downloaded from www.worldscientific.com by NANYANG TECHNOLOGICAL UNIVERSITY on 08/25/15. For personal use only. CHAPTER 7. LABORATORY WAVE GENERATION Multiplying both sides of Eqn. 7.49 by cosh[ki(h + z)], integrating the equation between z = -h and z = 0, and arranging gives A -USo , f °h f (z) cosh[kl (h + z)] dz (7.50) 2k1 f °h cosh2 [kl (h + z)] dz All of the summation terms have gone to zero because of orthogonality considerations given by the Sturm-Liouville theory (Dean and Dalrymple 1984). Similarly, multiplying Eqn. 7.49 by cos[k3, (z + h)] and performing the integration causes the progressive wave term to go to zero yieldingThe final step is to formulate the solution for the first-order wavemaker problem. The sea surface elevation in the wave tank is found by substituting the velocity potential given by Eqn. 7.46 into the first-order dynamic free surface boundary condition given by Eqn. 7.22 and evaluating the boundary condition at z = 0, i.e.,
171(x,t) = oA cosh( klh)cos( klx -at) + 9+ sin(at ) E Cn ek i n x cos ( k3. h) (7.52) n =1 9Far from the wave board all of the summation terms will disappear, and to first order , the sea surface will be represented as the progressive wave solution 171(x,t ) = 2 cos ( klx -at ) (7.53)where H is the wave height . Equating Eqns . 7.52 and 7. 53 and discarding the series terms gives the basic wavemaker relationship H = 2o-A cosh(klh) (7.54) 9where A is given by Eqn . 7.50. Solution for a specific type of wavemaker is found by substituting for f ( z) in Eqn . 7.50 and solving the integrals. Physical Models and Laboratory Techniques in Coastal Engineering Downloaded from www.worldscientific.com by NANYANG TECHNOLOGICAL UNIVERSITY on 08/25/15. For personal use only. P. _ a tanh kh l smh kh + 1 ( 1 -cosh kh) ( pghSo l kh (si.nh 2kh + 2kh l L kh ] z T J Physical Models and Laboratory Techniques in Coastal Engineering Downloaded from www.worldscientific.com by NANYANG TECHNOLOGICAL UNIVERSITY on 08/25/15. For personal use only.