and as M = n 2 /c, as previously defined, this can be rewritten as 

 which reduces to 



2tt d* n . /R*\/n2 d* 



S 2 1 I S \( 2. s 



gT z 2irc \r*/\ R* 



(291) 



Lines of constant 2tt d| n 2 / (gT ) are also plotted in Figure 38. The 

 minimum period of the trapped waves is defined where a line of constant 

 2it dj n 2 /(gT 2 ) is tangent to the line where d*/ (S 2 Rg) is constant for 

 the given values of dj, S 2 , and R| . Solutions for equations (264) 

 and (265) at greater values of Rg/ip (at longer wave periods) define 

 trapped waves. Solutions for equations (264) and (265) for smaller 

 values of Rg/ri (at longer wave periods) define the damping zone dis- 

 cussed by Lozano and Meyer (1976) . 



Solutions of equations (264) and (265) for values of n 2 , defined 

 by equation (286) , define caustics at the inner limit of the trapped 

 wave zone near the shoreline (i.e., where rS ■* Rg) • These solutions 

 will give the maximum trapped wave periods, ^max' ^ ut tne solutions 

 tend to break down at this point as the parameter, U, defined by 

 equation (66) as U = (H/d)(L/d) 2 , becomes very large (U >> 1). However, 

 the minimum trapped wave period and the outer limit of the trapped wave 

 zone can be approximated using equations (264) and (265) . 



Figure 38 shows that, for varying values of ctj , the minimum trapped 

 wave period will increase as aj decreases. This is expected since 

 shorter period waves, at lower values of a ls tend to pass into deep 

 water and are not trapped. 



The theoretical solutions given by equations (264) and (265) are for 

 the case of a coastline approximated by a circular arc. In the limiting 

 case, this will approach a straight coastline. Solutions for irregular 

 coastlines must be obtained by numerical methods. The theoretical solu- 

 tions presented here can be used to verify numerical methods used for 

 more complex solutions for irregular coastlines. An example of a numeri- 

 cal solution is given by Houston, Carver, and Markle (1977) using a 

 finite-element numerical model developed by Chen and Mei (1974). 



*********** EXAMPLE PROBLEM 19 * * 



* * * * 



GIVEN : A curved section of coastline is convex, with a radius of curva- 

 ture R| = 100,000 meters (62.14 miles), and the depth at the toe of 

 the shoreline slope d* = 30 meters. A tsunami reflects from the shore- 

 line slope and refracts over a shelf where the bottom slope of the shelf 



128 



