| The generation of debris flow has some causes. This researche is on intermittent debris flow
surges and due to mathematical approach of wave equation by numerical analysis. The
following wave equation was obtained based on the momentum equation of shallow
water.
∂η′ ′∂η′ ∂2η′ ∂3η′
∂τ′ + a1η ∂ξ′ − a2∂ξ′2 + a3∂ξ′3 = 0
(1)
where, a1 = (3∕2)c0′2, a2 = (1∕2)( )
1∕c0′2 − 1∕2tanθ (c0′∕u0′),
a3 = (1∕2){ 4 2 }
(2 + c0′ )∕(2c0′)− 3∕2 , u0′ = u0∕c0, c0′ = c0∕vp0, c0 = √ --------
gh0cosθ,
η′ = η∕h0, t′ = tvp0∕h0, ξ = ε1∕2(x− vp0t), τ = ε3∕2t, ξ′ = ξ∕h0 = ε1∕2(x′ − t′),
τ′ = ε3∕2t′,
u0, h0 : velocity, depth of steady uniform flow, x : axis of flow direction, t : time, η : variance
of flow surface from depth h0, θ : slope angle of the channel, g : acceleration due to gravity,
ξ, τ : the Gardner-Morikawa transformation of x axis and time, ε : parameter of perturbative
expansion, vp0 : phase velocity, c0 : long wave velocity, ′(with prime) : non-dimensional
variable.
η′ of equation (1) changes depending on the values of a1, a2, a3 on same section of ξ′ and
τ′, and a1, a2 and a3 are function of c0′. c0′ is ratio of long wave velocity and
phase velocity, and c0′ = 1 when phase velocity is equal to long wave velocity.
For c0′ = 1, then a3 = 0, the equation (1) becomes Burgers Equation, the waves
deform to a wave of wave number one with increased phase velocity on progress at
time. Therefor, the wave parts from Burgers equation and becomes the one that
depend on equation (1) , KdV-Burgers equation. When the new phase velocity is
grater than 1.04 times c0′ (long wave velocity), waveform behaves as a solitary
wave. This research shows these processes by some numerical solutions of equation
(1). |