Chapter 4 Coastal Problems 4.1 Resonance RESONANCEIN NARROWWATER BODIES One-dimensional approach - For narrow water bodies the one-dimensional approach has been used extensively to determine natural free oscillations. There is a great amount of literature dealing not only with natural water bodies but also idealized situations such as rectangular basins. A few advances (preceded by a brief introduction) will be selected. See also the excellent books on oceanography by Sverdrup et al. (1942), Proudman (1953), Defant (196 l), and Ippen (1966). Merian (1828) developed a theory for free longitudinal oscillations in a nonro- tating rectangular basin and gave the following formula for the period, Tn, of then th mode. 1 T =-- 2L “da where L is the length of the basin, D is the depth of the basin assumed to be uniform, and g is gravity. This formula holds when the bay is closed at both ends. On the other hand, for a rectangular bay of uniform depth, closed at one end and open at the other, the period of the first longitudinal mode is given by: (4.2) For this mode, there is a node at the mouth of the bay and an antinode at the head. In this situation, the length of the bay is + the length of the standing wave that is excited. The Merian formula gives a quick and rough estimation of the period of a bay. It can also be used to provide initial trial values for more accurate numerical calculations. Thorade (1931) stated that if the width of the bay is about the length, then a correction for the period could be applied by increasing the period (given by the Merian formula) by 10%. If the width is equal to the length then the period has to be increased by 32%. The Merian formula appears to consistently overestimate the periods compared to those determined by the more accurate numerical methods (Table 4.1). An improvement over the Merian formula is the Defant method described by Neumann and Pierson (1966) and Defant (196 1). For electrical analog methods to determine the free oscillations of one-dimensional systems see Defant (1961, vol. 2). 145 TABLE4. I. Computed period, T,, by numerical integration, of the first longitudinal mode of the five Great Lakes compared to periods, T,, determined from Merian formula. (Rockwell 1966) ~- Lake Superior 7.19 9.45 Michigan 8.83 10.53 Huron 6.49 9.77 Erie 14.08 16.76 Ontario 4.9 1 5.85 SYSTEMSWITH BRANCHES Defant (196 1) used the concept of electrical networks to describe procedures dealing with a system with branches (each assumed to have uniform depth). Occa- sionally the method of characteristics has been used to determine the characteristic oscillations of a basin with a branch (for example, see Horikawa and Nishimura 1968). However, these authors assumed uniform depth and a water body of regular shape. For this reason, although they give a rough estimate of the periods, these techniques cannot be used for accurate determination of the periods. Rao (1968) determined numerically free oscillations of the Bay of Fundy taking into account its two branches, Chignecto Bay and Minas Basin. Henry and Murty (1972) deter- mined the resonance modes for a system with five branches. Rao (1968) also applied corrections for rotational and frictional effects in one-dimensional systems based on the work by Platzman and Rao (1964). Murty and Boilard (1970) used essentially the same technique to determine the free oscillations of an inlet on the west coast of Canada. RESONANCEIN MULTIBRANCHEDINLETS Henry and Murty (1972) determined the resonant periods for a multi-branched inlet system on the west coast of Canada. Figure 4.1 shows the location of the Rivers Inlet complex treated in this study. Figure 4.2 schematically shows the five branches of this system with the grid scheme. The finite-difference forms of the continuity equation and the momentum equation are for a variable grid size : (4.3) and where M is the volume transport through a vertical cross section with area A and 146 QUEEN CHARLOlTE SOUND 0 SOkm FIG.4. I. Canadian west coast inlets. '2I-- Branch 4-_-- 17 -- BranchSt I ++#I --6 -Confluence 2 Branch 3-- FIG.4.2. Rivers Inlet on the west coast of Canada, map and schematic. 147 surface width B; 9 is the water-level deviation from the equilibrium position; g is gravity; A x is the grid size at grid point, i; and u is the frequency of oscillation. Frictional and Coriolis effects are ignored. The conditions at confluence I are: and where superscript denotes the branch and subscript denotes the grid point in a given branch. At confluence 2 the following conditions have to be satisfied : and Branch 1 is open to the sea and has 39 grid points, branch 2 has 16, branch 3 has 5, branch 4 has 12, and branch 5 has 17. The boundary conditions of the problem are: M(l) IS arbitrary (4.10) and (4.1 1) A standard iteration technique described by Rao (1968) has been used to determine the resonance periods. (A) of Table 4.2 shows the periods determined by starting at sea and applying the final boundary condition at the head of branch 5. Although some modes are located with a few iterations, others took considerably more iterations. Let M’ and MZ denote the values of M!;I1at the two values of bracketing the zero-crossing at the end of the fine search with frequency increment, .A 0. Table 4.3 shows the zero-crossing behavior for a well-behaved and an ill- behaved mode for two different values of no. This table shows that for a well- behaved mode, M‘f? is a smooth function of u near the zero-crossing and the values remain small. For an ill-behaved mode the values of M’ and MZ remain large. The authors ascribed this behavior to accumulation of numerical errors and called the ill-behaved modes false modes (these could be normal modes for a part of the system, however) and denoted them by F. 148 TABLE4.2. Natural periods of Rivers Inlet computed by: (A) starting at sea and monitoring at head of branch 5, (B) starting at sea and monitoring at head of branch 2, (C) starting at head of branch 4 and monitoring at head of branch 2. Numerals denote genuine modes whereas the modes identified by F are spurious. (A) (B) (C) Mode No. Period No. Period No. Period No. (rnin) iterations (miti) iterations (min) iterations 88.99 2 88.99 88.99 3 - - 37.17 - 33.75 2 33.75 33.75 3 - - 27.50 30 25.08 3 25.08 25.08 5 - 24.52 34 22.04 52 - - 21.67 5 21.67 3 20.10 46 - - - 15.27 3 15.27 2 15.27 2 - ~ 14.33 25 I 11.36 1 1.36 2 11.36 3 - 9.95 313 - 9.86 9.86 7 9.86 3 (5 1 TABLE4.3. The values MI (denoted by MIand M2) at the two values of u bracketing the zero-crossing in fine searches with two different frequency increments Ao for a well-behaved mode (number 4 for case A in Table 4.2) and for all ill-behaved mode (number F:,4 for case A in Table 4.2). Values 01 M'15j (cni' s-' ) Well-behaved Well-behaved Ill-behaved Ill-behaved mode mode mode mode A(i = IO-'' nu = 10-12 Ao = IO-'' nu = M' -7.138 X IO' -3.225 X 10' -7.4b3 X IO" -1.135 X 1013 MZ 2.850 X 7.625 X l.cl18 X IO' 1.318 X 1012 Table 4.2 also shows the modes, starting at sea (branch 1) and applying the final boundary condition at the head of branch 2 (i.e. monitoring in branch 2). Whereas the true modes denoted by numerals stay the same (the reason for the disappearance of genuine mode 4 will be considered below) an entirely different set of false modes appears, shown as (B). (C) shows the results from starting at the head of branch 4 and monitoring at the head of branch 2. To understand this peculiar behavior of the modes, the nodal structures in terms of M and 7 were examined. Figure 4.3 shows the structures of modes 1 and 2 (both well behaved) and compares the well-behaved mode 3 with the false mode F:P. 149 MODE 1 MODE 2 41 Confluence 1 Confluence 2 r3 Confluence 1 Confluence 2 8 3- 3-1 -t ----___--__ 1 2- --I11-_ -. ...._ -.. 1- I 10 20 30 -.. 14 Grid Point Number aI21 /‘ ,f’ ‘\ -4 -2 - ‘,. -3 - 7- M ---______ -8 -4 - -4 MODE 3 MODE F:,, mfluence 2 Confluence 1 Confluence 2 5 4 t, -12 4 \ \ \\{41 -8 3 2 2 _i -4 1 h h 0 vL_ -1 -2 -3 -4 -5 - -12 FIG. 4.3. Structure of modes 1, 2, 3, and F:,4 in Rivers .Inlet on the west coast of Canada in terms of volume transport (broken line) and water level (solid line). An examination of the modal structures revealed the following important points: (1) when motion in the branch where the solution is begun is small compared to motion in the other branches, a false mode could occur, and (2) when the motion in both the starting and monitoring branches is small, a genuine mode could disappear. Standard iteration technique is not suitable to determine free oscillations of multibranched inlets because all permutations and combinations of starting and monitoring must be used to eliminate the false modes.