有限長度水槽中孤立波造波與傳遞過程之修正線性解析

Abstract

本文根據Dean與Dalrymple（1991）描述造波水槽內部之線性邊界條件，將水槽內部之流速勢分為振盪波與前進波以簡化問題，再以Fourier級數展開，可將問題化為二非齊次常微分方程式，在給定造波初始條件的情況下即可解析有限長度造波水槽自靜止開始造波一段時間後停止之線性暫態解。在給定造波板運動方程式的條件下，水槽內部之波浪由開始造波至造波結束後波浪隨時間自由傳遞的過程均可描述，並以步進速度造波（step-velocity）做為此模式之驗證，同時討論了級數的收斂性、收斂項數與水槽長度之關係、以及方程式中各項之物理意義。 本文並以弱非線性的孤立波造波問題做為解析之對象，由於孤立波造波板速度為一超越函數，造成解析上的困難；本文以 hypergeometric 函數推求常微分方程式之全解，並與理論波形解比較後，發現由於未考慮非線性及分散性過強等問題，使得線性暫態解較理論波形拉長與歪斜，可能無法有效描述孤立波造波問題，故針對線性之分散關係做出修正。以長波條件做為基準，將Fourier級數展開之成份波分為弱分散波（weakly-dispersive wave）與非分散波（non-dispersive wave），加上Hedges（1976）之非線性淺水分散關係式，推得修正後之線性解。並以修正線性解分別對孤立波造波過程後波浪傳遞、尾波及波高衰減現象、以及質量守恆、級數收斂問題進行探討，可發現隨著非線性效應之提高，孤立波造波後尾波效應及波高衰減的量會隨之增加。

The paper presents a linear transient solution for the solitary wave generated by a piston-type paddle in a finite flume based on the boundary condition derived by Dean and Dalrymple (1991). The wavemaker problem is simplified by decomposing the wave potential into two parts. One is the evanescent wave and the other is progressive wave. The possible solution for the evanescent wave is expressed in form of the Fourier series. The governing equations for velocity potential can be written as two nonhomogeneous ordinary differential equations. The corresponding analytic solution for this problem subjects to initial rest condition. The validity of the proposed solution for step-movement of the paddle is first examined well through the convergence of the series and by a comparison with the previous solution for an infinite flume. The solitary wave can be generated considering Goring and Raichlen’s movement of a paddle. The proposed original linear solution for the solitary wave generation is expressed in the hypergeometric function. Two disadvantages of the original solution with large trailing wave and skewed wave profile are found by comparing with the theory of solitary wave derived from Boussinesq’s equation. The difference between the original linear solution and the solitary wave theory results from the nonlinearity and dispersion of generated waves in the flume. A modification on linear wave speed for weakly dispersive wave is considered following Hedges’ (1976) expression. The other dispersive waves are assumed to move with a constant speed as a linear shallow wave. The modified solution keeps the basic conservation of mass and has good agreement with experimental results. Trailing wave increases with the large movement of a paddle, but the wave comparatively crest decay due to nonlinearity.