本文於成功大學水工試驗所大型斷面水槽(300 m x 5 m x 5.2 m)進行一系列因不同波浪尖銳度、坡度和正規化波群頻率(主、副頻頻率差)所激發之長週波振盪。根據三種不同型態之非線性波列進行實驗分析，分別是初始均勻波列(單頻波)、調變波列(雙主頻波與三頻波)。實驗結果顯示，在同一坡度條件下，亞重力波之生成會隨著波浪非線性增強而減弱;此外也發現波浪調變所形成之頻率下移是一個重要的影響因子。再者，在固定之波浪條件下，此低頻成分波之強度會隨著坡度變緩而有一變小的趨勢。而實驗中所觀察到之長週波能量最大值則對應到正規化波群頻率 的位置。
實驗結果進一步闡述亞重力波之生成是由於非線性波群在淺化過程中局部激發所產生的，特別是在外碎波點處。為了瞭解亞重力波的物理機制，本文藉著小波及雙頻譜分析來探討波浪在演化過程中之傳遞特性;此外，在淺化過程中所觀察到的水動力機制也將一併進行討論。
入射強制長波在淺化過程中計算所得之成長率 ，其指數值介於0.25至2.5之間，也提供正規化坡度參數 判斷陡坡及緩坡條件之參考依據。在本文中也證實正規化坡度參數的改變會影響海岸線附近長波的反射率大小，故可得知海岸線附近之長波反射率大小與碎波水深及入射長波之波高有關。再者，入射波群在斜坡上之演化、長波與入射波群之位相差則是藉由自相關分析來探討不同位置所發生之時間延遲現象。本文也提出在內碎波帶由於長波碎波所導致的能量耗損，在相對緩坡的條件下較陡坡條件明顯。文中也指出，當短波之平均碎波點對應到自由駐波結構的節點位置時，淺灘動盪之生成能量最大。然而，當駐波結構不甚明顯及海岸線之溯升高度受到明顯壓抑時，將導致碎波點有向外海移動的趨勢。
在湧上區波動機制方面的探討則著重於坡度效應、入射波高及頻率差對於殘存之短週波能量及長週波振盪之間的關係。而反射自由長波之最大與最小的生成能量則是分析其縱斷面結構如何受湧上區溯升增幅大小的影響。經由實驗分析也證實了湧上區之溯升、溯降週期深受底床坡度、初始波浪尖銳度及頻率差影響。本文主要貢獻在於提供系統性、完整的實驗分析，俾使長波在複雜的湧上區波動存在情況下，其生成與耗散的機制得以明瞭。

A series of experiments were conducted in a super wave flume (300m x 5m x 5.2m) to examine the low-frequency motion induced by different incident wave steepness, sloping gradients and normalized group frequency (sideband space). Three types of waves including initial uniform wave train (mono-chromatic wave) and modulated wave train (bi- and tri-chromatic waves) are utilized for incident wave conditions. From the experimental results, it is found that for a given slope gradient the infra-gravity wave component decreases as wave nonlinearity increases and frequency downshift is a predominant factor. Furthermore, the magnitude of low-frequency components decreases with the decrease of sloping gradients for a given initial wave conditions. In addition, the maximum value of low-frequency motion is found to be close to the normalized group frequency, .
Since the wave steepness and bottom slope have been validated to dominate the generating mechanism of infra-gravity wave, which is usually the principal dynamics in the shallow water depth, it is identified that the infra-gravity waves would be locally forced by nonlinear wave groups, especially at the outer breakpoint. In order to understand the physical nature of infra-gravity wave, the characteristics of energy transport on each chosen wave is examined by means of wavelet and auto-bispectral analysis. Note that the hydrodynamics of detailed analysis on the shoaling process is also discussed in this paper.
The growth rate of the incident shoaling bound long wave varies with water depth with an exponent between 0.25 and 2.5, depending on a normalized bed slope that distinguishes between mild and steep slope regimes. The normalized bed slope is verified to be a parameter which controls the reflection coefficient of infra-gravity waves at the shoreline as a function of remaining variables, breaking water depth and incident long wave height around the shoreline. In the case of a relatively mild slope regime, the dissipation due to long wave breaking in the inner surf zone is more evident than that in a steep slope regime. The spatial evolution of short-wave groups and the relationship between low-frequency wave and short-wave envelope over the sloping bottom is analyzed to understand the time-lags between each component by auto-correlation method at specific locations. While the mean breakpoint meets a node for a free standing long wave, the maximum response of the surf-beat wave is to be expected. However, the obscure nodal structure and the suppression of shoreline motion cause the breakpoint node to shift offshore.
The investigations on the swash motion due to sloping effect, varied incident wave energy and frequency difference illustrate a relationship between individual bores and low-frequency oscillation. The cross-shore structures of long wave components are examined to understand the maximum and minimum responses of outgoing free long waves on the amplification of swash motion. It should be noticed that the uprush and backwash periods appear to be sensitive to the sloping gradient, offshore wave steepness and frequency difference. These experimental data provide a systematic study on the generation and dissipation of low-frequency waves due to complex swash-zone motion in different wave conditions.

Baldock, T. E., P. Holmes and D. P. Horn, 1997. Low frequency swash motion induced by wave grouping. Coastal Engineering, 32, 197-222.
Baldock, T. E. and P. Holmes, 1999. Simulation and prediction of swash oscillation on a steep beach. Coastal Eng., 36, 219-242.
Baldock, T. E., D. A. Huntley, P. A. D. Bird, T. O’Hare and G. N. Bullock, 2000. Breakpoint generated surf beat induced by bi-chromatic wave groups. Coastal Eng., 39, 213-242.
Baldock, T. E. and D. A. Huntley, 2002. Long wave forcing by the breaking of random gravity waves on a beach. Proc. R. Soc. Lond., A458, 2177–2201.
Baldock, T. E. and T. J. O’Hare, 2004. Energy transfer and dissipation during surf beat conditions. Proc. 24th ICCE, ASCE, 1212-1224.
Baldock, T. E., 2006. Long wave generation by the shoaling and breaking of transient wave groups on a beach. Proc. R. Soc. Lond., A, 462, 1853-1876.
Bendat, J. S., and A. G. Piersol, 1986. Random Data, Analysis and Measurement, John Wiley, New York.
Benjamin, T. B., Feir, C. M., 1967. The disintegration of wave trains on deep water: Part 1, Theory. J. Fluid Mech., 27, 417-430.
Battjes, J. A., 1974. Surf similarity. Proc. 14th ICCE, ASCE, 466-480.
Battjes, J. A. and J. P. F. M. Janssen, 1978. Energy loss and set-up due to breaking of random waves. Proc. 16th ICCE, ASCE, 569-587.
Battjes, J. A., H. J. Bakkenes, T. T. Janssen and A. van Dongeren, 2004. Shoaling of subharmonic gravity waves, J. Geophys. Res., 109, C020009, doi:10.1029/ 2003JC001863
Butt, T. and P. Russell, 1999. Sediment transport mechanisms in high energy swash. Mar. Geol., 161, 361-375, 1999.
Carrier, G. F. and H. P. Greenspan, 1958. Water waves of finite amplitude on a sloping beach. J. Fluid Mech., 4, 97-109.
Cox, D. T., W. A. Hobensack and A. Sukumaran, 2001. Bottom stress in the inner surf and swash zone. Proceedings of the 27th International Conference on Coastal Engineering held in Sydney, Australia, July 16-21, 2000, edited by B. L. Edge, Am. Soc. Of Civ. Eng., Reston, Va.
Elgar, S. R. and T. Guza, 1985. Observations of bispectra of shoaling surface gravity waves. J. Fluid Mech., 161, 425-448.
Elgar, S. R., T. H. C. Herbers and R. T. Guza, 1994. Reflection of ocean surface gravity waves from a natural beach. J. Phys. Oceanogr., 24, 1503-1511.
Elgar, S., B. Raubenheimer, and R. T. Guza, 2001. Current meter performance in the surf zone. J. Atmos. Oceanic Techol., 18, 1735-1746.
Foda, M. A. and C. C. Mei, 1988. Subharmonic generation of edge waves by non-uniform incident waves. Wave Motion, 10(2), 149-160.
Frigaard, P. and M. Brorsen, 1995. A time domain method for separating incident and reflected irregular waves. Coastal Eng., 24, 205–215.
Garrett, C. J. R. and B. Toulany, 1981. Variability of the flow through the Strait of Belle Isle. J. Mar. Res., 39, 163-189.
Guza, R. T. and E. B. Thornton, 1982. Swash oscillation on a natural beach. J. Geophys. Res., 87, 483-491.
Guza, R. T., E. B. Thornton and R. A. Holman, 1984. Swash on steep and shallow beaches. Proc. 19th ICCE, ASCE, 708-723.
Guza, R. T. and E. B. Thorton, 1985. Observations of surf beat. J. Geophys. Res., 90, 3161-3172.
Hasselman, K., W. Munk and G. MacDonald, 1963. In Time Series Analysis. ed. M. Rosenblat, 125-139. Wiley.
Haubrich, R. A., 1965. Earth noises, 5 to 500 milli-cycles per second, 1. J. Geophys. Res., 70, 1415-1427.
Henderson, S. M. and A. J. Bowen, 2002. Observations of surf beat forcing and dissipation. J. Geophys. Res., 107(C11), 3193, doi:10.1029/2000JC000498.
Herbers, T. H., C., S. Elgar, and R. T. Guza, 1994. Infra-gravity frequency (0.005– 0.05 Hz) motions on the shelf, part I: Forced waves. J. Mar. Res., 24, 917– 927.
Holland, K. T., B. Raubenheimer, R. T. Guza and R. A. Holman, 1995. Run-up kinematics on a natural beach. J. Geophys., Res. 100 (3), 4985-4993.
Holland, K. T. and J. A. Puleo, 2001. Variable swash motions associated with foreshore profile change. J. Geophys. Res., 106, 4613-4623.
Huang, M. C., 2004. Wave Parameters and Functions in Wavelet Analysis. Ocean Engineering, 31, 111-125.
Hughes, M. G., 1992. Application of a nonlinear shallow water theory to swash following bore collapse on a sandy beach. J. Coastal Res., 8, 562-578.
Hughes, M. G., G. Masselink and R. W. Brander, 1997. Flow velocity and sediment transport in the swash zone of a steep beach. Mar. Geol., 138, 91-103.
Huntley, D. A., R. T. Guza and A. J. Bowen, 1977. A universal form for shoreline run-up spectra? J. Geophys. Res., 82, 2577-2581.
Hwung, H. H., W. S. Chiang, P. L.-F. Liu and P. Lynett, 2004. Sideband Evolution of Initially Uniform Deep Water Wave. Proc. 29th ICCE, ASCE, 157-168.
Hwung, H. H., W. S. Chiang, 2005. The measurements on wave modulation and breaking. Measurement of Science and Technology, 1921-1928.
Hwung, H. H., Y. H. Lin, and S. C. Hsiao, 2006. The investigation on the generation of infra-gravity wave. Ocean Eng., doi: 10.1016/j.oceaneng.2006.08.012. (in press)
Hwung, H. H., W. S. Chiang and S. C. Hsiao, 2007. Observations on the evolution of wave modulation. Proc. R. Soc. Lond., 463, 85-112.
Hwung, H. H., Y. H. Lin, 2007. The Investigation on the Generation of Infra-gravity Wave. Wave Motion. doi:10.1016/j.wavemoti.2007.03.007 (in press)
Iribarren, C. R., and C. Nogales, 1949. Protection des Ports, paper presented at XVIIth International Navigation Congress, Permanent Int. Assoc. of Navig. Congr., Lisbon, Portugal.
Janssen, T. T., J. A. Battjes and A. R. van Dongeren, 2003. Long waves induced by short waves groups over a sloping bottom. J. Geophys. Res., 108(C8), 3252, doi:10.1029/2002JC001515.
Katoh, K. and S. Nakamura, 1992. Generation of infra-gravity waves in breaking process of wave groups. Proc. 23rd ICCE, ASCE, 990-1003.
Karunarathna, H., A. Chadwick and J. Lawrence, 2005. Numerical experiments of swash oscillations on steep and gentle beaches. Coastal Eng., 52, 497-511.
Kim, Y. C. and E. J. Powers, 1979. Digital bi-spectral analysis and its application to nonlinear wave interactions. IEEE Trans. Plasma Science, 1, 120-131.
Kostense, J. K., 1984. Measurement of surf beat and set-down beneath wave groups. Proc. 19th ICCE, ASCE, 724-740.
Lake, B. M., H. C. Yuen, H. Rungaldier and W. E. Ferguson, 1977. Non-linear deep-water waves: theory and experiment. Part 2. Evolution of a continuous train. J. Fluid Mech., 83, 1, 49-74.
Lancaster, P. and K. S. Alkauskas, 1996. Transform Methods in Applied Mathematics. John Wiley, New York.
Lin, Y. H., S. C. Hsiao and H. H. Hwung, 2007. The generation and dissipation of low-frequency waves. J. Geophys. Res. (under review)
Lin, Y. H., S. C. Hsiao and H. H. Hwung, 2007. Swash Motion Driven by Bi-chromatic Wave Groups over Sloping Bottoms. Ocean Eng. (paper under review)
List, J. H., 1991. Wave groupiness variation in the nearshore. Coastal Eng., 15, 475-496.
List, J. H., 1992. A model for the generation of two dimensional surf beat. J. Geophys. Res., 97, 5623-5635.
Longuet-Higgins, M. S. and R. W. Stewart, 1962. Radiation stress and mass transport in gravity wave, with application to “surf beats”. J. Fluid Mech., 13, 481-504.
Longuet-Higgins, M. S., 1986. Eulerian and Lagrangian aspects in the nearshore. J. Fluid Mech., 173, 683-707.
Madsen. P. A., O. R. Sorensen and H. A. Schäffer, 1997. Surf zone dynamics simulated by a Boussinesq type model: Part 2. Surf beat and swash oscillations for wave groups and irregular waves. Coastal Eng., 32, 289-319.
Mase, H., 1995. Frequency down-shift of swash oscillations compared to incident waves. J. Hydraulic Res., 33, 397-411.
Munk, W. H., 1949. Surf beats. Trans. Am. Geophys. Union, 30, 849-854.
Masselink, G. 1995. Group bound long waves as a source of infra-gravity energy in the surf zone. Continental Shelf Research, 15, 1525-1547.
Masselink, G. and M. Hughes., 1998. Field investigation of sediment transport in the swash zone, Continental Shelf Research, 18, 1179-1199.
Mellville, W. K., 1982. The instability and breaking of deep-water waves. J. Fluid Mech., 114, 1-30.
O’Hare, T. J. and D. A. Huntley, 1994. Bar formation due to wave groups and associated long waves. Mar. Geol., 116, 313-325.
Okihiro, M., R. T. Guza, and R. J. Seymour, 1992. Bound infra-gravity waves. J. Geophys. Res., 97(C7), 11453-11469.
Okihiro, M. and R. T. Guza, 1995. Infragravity energy modulation by tides. J. Geophys. Res., 100, 16,143-16,148.
Oltman, J. M. and R. T. Guza, 1987. Infra-gravity edge wave observations on two California beaches. J. Phys. Oceanogr., 17(5), 644-663.
Phillips, O. M., 1977. The dynamics of the upper ocean. Cambridge University Press, 336.
Raubenheimer, B., R. T. Guza, S. Elgar and N. Kobayash, 1995. Swash on a gently sloping beach. J. Geophys. Res., 100(5), 8751-8760.
Raubenheimer, B., 2002. Observations and predictions of fluid velocities in the surf and swash zones. J. Geophys. Res., 107, 3190, doi:10.1029/2001JC001264.
Raubenheimer, B., S. Elgar and R. T. Guza, 2004. Observations of swash zone velocoties: A note on friction coefficients. J. Geophys. Res., 109, C01027, doi:10.1029/2003JC001877.
Ruessink, B. G. 1998 Bound and free infragravity waves in the nearshore zone under breaking and non-breaking conditions. J. Geophys. Res. 103, 12 795-12 805.
Schäffer, H.A. and I. A. Svendsen, 1988. Surf beat generation on a mild slope beach. Proc. of the 21th International Conference on Coastal Engineering, 1058-1072.
Schäffer, H. A., 1993. Infra-gravity waves induced by short wave groups. J. Fluid Mech., 247, 551-588.
Shen, M. C. and R. E. Meyer, 1963. Climb of a bore on a beach, 3, Run-up. J. Fluid Mech., 16, 113-125.
Sheremet, A., R. T. Guza, S. Elgar and T. H. C. Herbers, 2002.Observations of nearshore infragravity waves: 1. Seaward and shoreward propagating components. J. Geophys. Res., 107(C8), 3095, doi:10.1029/2001JC000970.
Sobey, R. J. and H. B. Liang, 1986. Complex envelope identification of wave groups. Proc. 20th ICCE, 752-766. New York: American Society of Civil Engineers.
Symonds, G., D. A. Huntley and A. J. Bowen, 1982. Two-dimensional surf beat: long wave generation by a time-varying breakpoint. J. Geophys. Res., 87, 492-498.
Thomson, J. S., Elgar, B. Raubenheimer, T. H. C. Herbers and R. T. Guza, 2006. Tidal modulation of infragravity waves via nonlinear energy losses in the surf zone. Geophys. Res. Lett., 33, L05601, doi:10.1029/2005GL025514.
Torrence C. and G. P. Compo, 1998. A practical guide to wavelet analysis. Bulletin of the American Meteorological Society, 79, 61–78.
Tucker, M. J., 1950. Surf beats: sea wave of 1 to 5 min. period. Proc. R. Soc. London, Ser. A 202, 565-573.
Tulin, M. P., T. Waseda, 1999. Laboratory observations of wave group evolution, including breaking effects. J. Fluid Mech., 378, 197-232.
van Dongeren, A. R., 1997. Quasi 3-D modeling of nearshore hydrodynamics, Rep. CACR-97-04, 243, Cent. For Appl. Coastal Res., Univ. of Del., Newark.
van Dongeren, A. R., H. J. Bakkenes, and T. T. Janssen, 2002. Generation of long waves by short wave groups, Proc. of 28th ICCE, edited by J. Smith McKee, World Sci., River Edge, N. J.
van Dongeren, A. R., J. van Noorloos, K. Steenhauer, J. Battjes, T. Janssen and A. Reniers, 2004. Shoaling and shoreline dissipation of subharmonic gravity waves. Proc. of 29th ICCE, ASCE, 1225-1237.
van Dongeren, A. R., J. Battjes, T. Janssen, J. van Noorloos, K. Steenhauer, G. Steenberen and A. Reniers, 2007. Shoaling and shoreline dissipation of low-frequency waves. J. Geophys. Res., 112, C02011, doi:10.1029/2006JC003701.
Voulgaris, G. and J. H. Trowbridge, 1998. Evaluation of the acoustic Doppler velocimeter (ADV) for turbulence measurements. J. Atmos. Oceanic Technol., 15, 272-289.
Watson, G. and D. H. Peregrine, 1992. Low frequency waves in the surf zone. Proc. 23rd ICCE, ASCE, 818-831.
Watson, G., T. C. D. Barnes and D. H. Peregrine, 1994. The generation of low frequency waves by a single wave group incident on a beach. Proc. 24th ICCE, ASCE, 776-790.
Wright, L. D., A. D. Short, 1984. Morpho-dynamic variability of surf zones and beaches: a synthesis. Mar. Geol., 56, 93-118.
Wright, L. D., P. Nielsen, N. C. Shi and J. H. List, 1986. Morphodynamics of a bar-trough surf zone. Mar. Geol., 70, 251-285.