The Lighthill and Whitham (1955) subcritical flow (F0 < 1) solution of the linear dynamic
wave equation (eq 7) for steady-uniform initial flow with a unit-step increase in φ at the
upstream boundary is
x
t-
I1(βz1/2 ) -α(t -t′)
x
c+
φ(x, t) = x ⋅ A(x)∫
φ(0, t′)dt′ + e - Hx
φ 0, t -
e
(14)
c+
1/2
z
0
where
x
x
z = t - t′ -
t - t′ -
c+
c-
(1 + F / 2)
2
0
α=
2η
(
) (
)
1/ 2
1/ 2
1 - F02 / 4
1 - F02
β=
2η
β e F0 x / 4ηc0
A(x) =
(1 - F )c
2
0
0
1 1 - F0 / 2
H=
2ηc0 1 + F0
.
I1 is a first-order modified Bessel function of the first kind, and φ(0,t) is the upstream
boundary condition that is equal to 0 for
x
t-
<0
c+
and equal to 1 otherwise.
Given x and t we can obtain φ with eq 14, but different dependent variable designations
would be helpful for studying the solution. For example, insights could be obtained from
wave profiles at specified times and from constant φ trajectory traces on the x-t plane. Also,
it would be useful to calculate and relate the celerity of a point on the dynamic wave profile
to the dynamic wave and kinematic wave celerities. To obtain these results we write eq 14
as an implicit function:
x
t-
I1(βz1/ 2 ) -α(t -t′)
x
c+
F(x, t, φ) = x ⋅ A(x) ∫
φ(0, t′) dt′ + e - Hxφ 0, t - - φ(x, t) = 0.
e
(15)
c+
z1/ 2
0
The dynamic wave profile celerity of a point with constant φ can be obtained by differenti-
ating F(x,t,φ) and setting the result to zero as
dx - Ft
cdyn =
=
(16)
dt
Fx
where Fx and Ft are the partial derivatives of F(x,t,φ) with respect to x and t, respectively.
Dividing eq 16 by the kinematic wave celerity yields dimensionless profile celerity
cdyn
cdyn =
(17)
~
.
ck
5
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