有關波浪分散關係式的數值應用討論

分散關係式是探討各種水波問題的重要關係式，因為它可以說明水深h、波浪週期T與波長L間的關係。在線性波理論中，常以h ≥ L/2及h ≤ L/20（或h ≤ L/25），作為深水波及淺水波與一般波浪的分際水深，此一論述方式常出現於海岸工程相關的教科書中。分散關係式中的tanh（）函數具有漸近線的性質，當水深波長比很大時，波長可以用gT^2 /2π計算，其中g是重力加速度；而當水深波長比很小時，波長為(方程式略)。然而，基於電腦軟硬體設備計算品質的提升，一般有關波動的數值解析中，常會以反覆迭代法解析分散關係式，此時將牽涉到有關收斂誤差設定的問題。本研究進行一連串的數值實驗，發現選取不同的容許誤差時，會得到不同的分際水深。也就是說，在波動問題的數值模擬中，由於選定的容許迭代誤差的不同，分際水深不再是h ≥ L/2 及 h ≤ L/20。在設定迭代誤差小於0.001的條件下，深水波的分際水深大約是一倍波長，而淺水波則幾乎不存在。連帶著也發現各相關的波動參數，如波壓、群波速度、水粒子運動速度等均有某一程度的差異。本文的目的是要提醒，由於計算設備的改善，數值計算的精確性大幅提高之際，相關的波浪理論有必要做進一步的確認與回顧，以免發生類似波浪已經受到海床影響時，卻仍被以深水波的特性進行解讀，而造成偏差。

關鍵字

線性波理論；分散關係式；迭代誤差

並列摘要

Linear wave dispersion relation has great importance for many kinds of water wave problems, for it can offer the relation between wave length, L and wave period, T at specific water depth, h from numerical approach. Convensionally, in linear wave theory, h ≥ L/2 and h ≥ 2 L/20 or L/25 were taken as the thresholds of the defined deep water waves and shallow water waves, respectively, from the intermediate water waves. Such theory appars in most coastal engineering textbooks Due to the dispersion relation contains hyperbolic function, tanh(), which has asymtotic nature, the wave length can be calculated directly from gT^2/2π for deep water waves and (The equation is abbreviated) for shallow water waves, where g is gravitational accleration. From the numerical simulations, however, the wave length calculated from the iterations of dispersion relation under different convergence tolerance at specific water depth and wave period is found different from the above mentioned depth limits. That is, the creteria of the deep water waves and the shallow water waves are different from the convensional definitions. This paper shows some results of the numerical of water experiments and points out the interpretation about the numerical results wave problems should be carefully, and suggests that the classifications of water waves in accordance to the ratio of water depth and wave length should be revised.

並列關鍵字

Linear Wave Theory ； Dispersion Relation ； Numerical Iteration ； Numerical Error

國際替代計量

有關波浪分散關係式的數值應用討論

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