水文学基礎 Fundamentals of Hydrology (流出 Runoff) Answer
The kinematic wave equations for overlandflow on a rectangular slope are derived as
∂h ∂q
+
=r
∂t ∂x
(1)
q = αhm
(2)
Suppose that the flow depth h is 0 anywhere on a rectangular slope at time t = 0 and rainfall
intensity r takes a constant value r0 for t ≥ 0. The slope length is L. In this case, Calcurate the time
that the characteristic curve starting at the top of slope at t = 0 reaches the slope end.
矩形平面上のキネマティックウェーブモデルを考える。時刻 t = 0 で水深 h は斜面のどこでも 0 であ
り、降雨強度 r は、時刻 t ≥ 0 において一定値 r0 をとるとする。斜面長は L とする。このとき、t = 0
に斜面上端を発した特性曲線法が斜面下端に到達する時間を求めよ。
(Answer)
The characteristic curve starting from the top of the slope at t = 0 is given by setting x0 = 0,
h0 = 0 and t0 = 0 as
x(t) = αr0m−1 tm
(3)
Setting x(t) = L and solving the equation for t, we obtain

1/m
 L 
t = Tc =  m−1 
αr
(4)
0
The time Tc is called the time of concentration. The time of concentration is an indicator that
explains the peak occurrence time of flow rate, which is a function of rainfall intensity and the
catchment characteristics.

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