| Abstract: | A study is made of the formation of a shock wave (bore), produced by the movement of an initially weak discontinuity in the spatial derivatives of velocity and liquid depth in an area of stationary current in a channel of constant inclination. The formation of shock waves from compression waves was first studied by Riman [1]. Frictional resistance was considered in the Chezy form. The equations obtained therein for determination of the moment in time and spatial coordinates of the point at which the shock wave is formed, as well as the laws for propagation of shock waves are applicable to the problem of one-dimensional transient motion in a gas, the pressure of which is dependent on density. Instantaneous collapse of waves, as well as formation and movement of bores in rivers for an idealized flow model in a channel with horizontal bottom, neglecting friction, were described by Khristianovich, Mikhlin, and Devison [2], and Stoker [3]. Recently in the work of Sachdev and Bhatnagar [4], using numerical integration of the equation for bore intensity, the problem of shock wave propagation in a channel of constant inclination with consideration of fluid resistance in the Chezy form was studied. Gradual wave collapse and the bore formation mechanism were studied by Stoker [3] on the basis of the shallow-water theory. Neglecting friction on the horizontal channel bottom, he calculated the moment of time and coordinates of the point at which the shock wave is formed in the case where the initial disturbance is sinusoidal. The dependence of these values on wave amplitude for a channel of constant inclination was obtained by Jeffrey [5], who also neglected friction on the channel bottom and considered the initial disturbance to be sinusoidal. Lighthill and Whitham [6] discovered that for Froude numbers greater than two, the linear theory led to unlimited growth in the intensity of the flood wave. We note that the studies of flood-wave motion in the region of the first characteristic, performed in [3, 6], differ only in the forms of the resistance laws and dependences of the unknown functions on the variables. Physical peculiarities of various liquid wave motions were also examined by Lighthill in [7].Saratov. Translated from Izvestiya Akademii Nauk SSSR. Mekhanika Zhidkosti i Gaza, No. 2, pp. 62–66, March–April, 1972. |
