Tại một vị trí nhất định trong lòng chất lỏng chuyển động, ở một thời điểm nhất định, vận tốc của một phần tử chất lỏng đo được gọi là vận tốc tức thời, kí hiệu u. Riêng đối với dòng chảy rối, vận tốc điểm tức thời này luôn thay đổi về hướng và trị số nên ta thường thay bằng giá trị trung bình trong một thời gian T nhất định gọi là vận tốc trung bình thời gian. . | Gravity Wave Steepening Bores and Breakers 145 Reimann invariant and propagation velocity V called the characteristic velocity 7 u 2pgh V u pgh . This can be verified by substituting into and by using to evaluate the time derivatives of u and h. The characteristic equation has a general solution 7 r e for the composite coordinate e x t called the characteristic coordinate defined implicitly by e V C t x . The demonstration that this is a solution comes from taking the t and x derivatives of @te tdịVỡte V 0 @xe tdịVỡxe 1 with dịV dV de solving for dte and dx substituting them into expressions for the derivatives of 7 dt7 dị w - @-r . V 7x dị Tdxe @v . 7 ị 1 dịVt 7 ị xe 1 dịVt and finally inserting the latter into @eF dt7 V@x7 2 V V 0 . 1 dị V t The function r is determined by an initial condition e x 0 x r e 7 x 0 . Going forward in time 7 preserves its initial value r e but this value moves to a new location X e t e V e t by propagating at a speed V e . The speed V is ạ gH after neglecting the velocity and height departures u h H from the resting state in these approximate values for V are the familiar linear gravity wave speeds for equal and oppositely directed propagation Sec. . When the fluctuation amplitudes are not negligible then the propagation speeds differ from the linear speeds and are spatially inhomogeneous. In general two initial conditions must be specified for the second-order partial differential equation system . This is accomplished by specifying conditions for 7 x and 7- x that then have independent solutions 7 e . A particular solution for propagation in the x direction is I 2pgH r 2 pH dr 146 Rotating Shallow-Water and Wave Dynamics W 4 pH- pH h X t H e u X t 2 pH- P H X t V t V pgH- 2 pH where H x H g x 0 is the initial layer thickness shape. Here V 0 whenever H 4 9 H and both V and u increase with increasing H. When h is larger X t progresses faster and vice versa. For an .
