1.  INTRODUCTION

Wave attenuation in prismatic channels is often attributed to bottom friction. While friction undoubtedly plays a leading role, it is by no means the only mechanism responsible for wave attenuation. This is exemplified by the well-known theory of kinematic waves, which states that while kinematic waves are governed by bottom friction, they do not attenuate.

The objective of this paper is to throw additional light onto the physical mechanisms responsible for wave attenuation in open channel flow. Following the approach of Ponce and Simons (4), this paper focuses on the identification of the term or group of terms which will cause a free-surface flow wave to dissipate. The conclusions may prove of interest to researchers and engineers practicing in the area of unsteady open channel now.

2.   GOVERNING EQUATIONS

The governing equations for one dimensional unsteady flow in prismatic channels of rectangular cross section, expressed in terms of unit width, are (2):

Equation of continuity:

∂d            ∂u           ∂d _____  + d ____  + u ____  = 0   ∂t             ∂x           ∂x

(1)

and equation of motion

1       ∂u            u       ∂u           ∂d            ____  _____  +   ____  _____  +   _____  +   Sƒ  - So  =  0   g       ∂t            g       ∂x            ∂x

(2)

in which u = mean velocity; d = flow depth; g = gravitational acceleration; S_ƒ = friction slope, S_o = bottom slope; x = space; and t = time. In uniform flow, S_ƒ = S_ƒ = S_o, and S_ƒ is related to the bottom shear stress τ_o by the following (1):

τo            Sƒ  =  ______             γ do

(3)

in which γ = density of water; and d_o = equilibrium flow depth.

The perturbation equations corresponding to Eqs. 1 and 2, respectively, are (3):

∂d'             ∂u'              ∂d' _____  + do _____  + uo _____  = 0   ∂t               ∂x               ∂x

(4)

1       ∂u'           uo     ∂u'           ∂d'                τ'o       d' ____  _____  +   ____  _____  +   _____  + So ( ____  - ____ )  =  0   g       ∂t            g       ∂x            ∂x                 τo       do

(5)

in which, generally, variable ƒ has been expressed as ƒ = ƒ_o + ƒ'; in which ƒ_o = the equilibrium value; and ƒ' = the small perturbation to ƒ. In order to make Eqs. (4) y (5) mathematically tractable, the bottom shear stress is related to the mean velocity. In general:

1            τ  =  ___ ƒρu 2          8

(6)

in which ƒ = Darcy-Weisbach factor friction and ρ = mass density of water. Along with Eq. 6, Eq 5 converts to:

1       ∂u'           uo     ∂u'           ∂d'                  u'       d' ____  _____  +   ____  _____  +   _____  + So ( 2 ___  - ___ )  =  0   g       ∂t            g       ∂x            ∂x                   uo      do

(7)

In order to keep track of all the terms in Eqs. 4 and 7, they are recast as follows:

∂d'                ∂u'                 ∂d' r _____  + v do _____  + w uo _____  = 0      ∂t                  ∂x                  ∂x

(8)

e       ∂u'         a uo    ∂u'              ∂d'                     u'       d' ____  _____ +   _____  _____  +   p _____  + k So ( 2 ___  - ___ )  =  0   g       ∂t             g       ∂x              ∂x                      uo      do

(9)

in which the coefficients r, v, w, e, a, p and k can take values of either 1 or 0, depending upon whether its associated term is considered or neglected in the analysis. The coefficient r affects the rate-of-rise term in the continuity equation, while v and w affect the prism storage and wedge storage terms, respectively. The coefficient e affects the local acceleration term, a the convective acceleration term, p the pressure gradient term, and k the kinematic term (friction and bed slope).

The transformation of the system of Eqs. 8 and 9 to the frequency domain is accomplished by seeking a solution in sinusoidal form such that (4):

d'            ____ = d* exp [i (σ* x* - β* t*)]  do

(10)

u'            ____ = u* exp [i (σ* x* - β* t*)]  uo

(11)

in which d_* and u_* = dimensionless depth and velocity amplitude functions, respectively; σ_* = dimensionless wave number, defined as σ_* = (2π/L)(d_o/S_o); and β_* = a complex propagation factor, with L = wavelength and i = (-1)^1/2.

(v σ*) u* + (w σ* - r β*) d* = 0

(12)

[(2 k + i Fo2 (a σ* - e β*)] u* + (i p σ* - k) d* = 0

(13)

in which F_o = equilibrium flow Froude number, defined as F_o = u_o / (gd_o)^1/2. Equations 12 and 13 constitute a homogeneous system of linear equations. The nontrivial condition for the determinant of the coefficient matrix yields the following characteristic equation:

(er Fo2) β*2 + [-σ* (ar + ew) Fo2 + 2ikr ] β* - [σ*2 (pv - aw Fo2) + iσ*k (v + 2w)] = 0

(14)

A close look at Equation 14 reveals that if k = 0, all imaginary terms drop out of it, i.e., gravity waves are not subject to attenuation. On the other hand, if e = a = p = 0, the equation can also be expressed in real terms only. Kinematic waves, therefore, are not subject to attenuation either. Attenuation is produced by the existence of both real and imaginary terms in Eq. 14, i.e., k = 1, and either e, a, or p are equal to 1.

Equation 14 is a second order algebraic equation with imaginary terms. Here in its solution is carried out in two stages: first, by neglecting local acceleration, e = 0; and secondly, by considering the complete solution.

3.   NEGLECT OF LOCAL ACCELERATION

With e = 0, Eq. 14 reduces to:

r (-aσ*Fo2 + 2ik) β* - [σ*2 (pv - awFo2) + iσ*k (v + 2w)] = 0

(15)

Solving for β_*:

v                   2w                       p           w                                                v           σ* ( ___ ) { ( 1 + ____ ) 2k2 - [ ( ___ ) - ( ___ ) Fo2 ] a2σ*2Fo2 } - iσ*2 ( ___ ) k ( 2p + aFo2 )                     r                      v                       a            v                                                r β* = __________________________________________________________________________________________________                                                        4k2 + a2σ*2Fo2

(16)

The propagation characteristics are the dimensionless celerity, c_*, an the logarithmic decrement, δ, defined as following (4):

c           β*R c* = ______ = ______            uo          σ*

(17)

and

2π β*l δ  =  ________           | β*R |

(18)

in which c = the wave celerity; β_*R = the real part of β_*; and β_*l = the imaginary part of β_*. Therefore:

v                    2w                      p           w          ( ___ ) { ( 1 + _____ ) 2k2 - [ ( ___ ) - ( ___ ) Fo2 ] a2σ*2Fo2 }              r                      v                       a            v c* = ______________________________________________________________                                         4k2 + a2σ*2Fo2

(19)

and

2πσ*k (2p + aFo2) δ =  -  _____________________________________________________                        2w                       p            w             ( 1 + _____ ) 2k2 - ([ ( ___ ) - ( ___ ) Fo2 ] a2σ*2Fo2                         v                         a            v

(20)

Wave attenuation is caused by the nonzero value of δ. Therefore. from Eq. 20, it is clear that k = 1 is a necessary condition for wave attenuation. However, if p = a = 0, there will be no wave attenuation, regardless of k. Therefore, wave attenuation will occur when k = p = 1, or k = a = 1. It is concluded that, in the absence of local acceleration, a wave attenuates due to the interaction of the kinematic term with the pressure gradient or convective acceleration term, or both.

4.   COMPLETE SOLUTION

The solution of Eq. 14 leads to:

1       a      w               k                           p          v       1          k β*  =  σ* [ ___ ( ___ + ___ ) - iζ ( ___ ) ] + σ* { [ ( ___ ) ( ___ ) ____ - ( ___ )2 ζ 2                   2       e       r                e                          e          r       Fo2       e      1      a      w                       k          v       w           k          a + ___ ( ___ - ___ )2 ] + iζ [ ( ___ ) ( ___ + ___ ) - ( ___ ) ( ___ ) ) ] }1/2      4      e       r                       e          r        r            e          e

(21)

in which

1 ζ  =  ________          σ* Fo2

(22)

is related to the kinematic flow number of Woolhiser and Liggett (5).

When C₁ = v / r ; C₂ = w / r ; C₃ = a / e ; C₄ = p / e ; and C₅ = k / e ; Eq. 21 reduces to:

1                                                             1                     1 β*  =  σ* [ ___ ( C3 + C2 ) - i ζ C5 ] + σ* [ (C1 C4) ____ - (C5 ζ)2 + ___ ( C3 - C2 )2                   2                                                            Fo2                   4 + i ζ C5 (C1 + C2 - C3) ]1/2

(23)

And when C₆ = ( C₂ + C₃)/2; C₇ = C₁ C₄; C₈ = (C₅)²; C₉ = ( C₃ - C₂)² /4; and C₁₀ = C₅ ( C₁ + C₂ - C₃); Eq. 23 reduces to:

C7 β* = σ* ( C6 - i ζ C5 ) + σ* ( ___ - C8 ζ 2 + C9 + i ζ C10 )1/2                                              Fo2

(24)

Finally, when A = ( C₇/F_o² ) - C₈ ζ ² + C₉; B = ζ C₁₀; C = ( A + B )^1/2; D = [( C + A )/2 ] ^1/2; and E = [( C - A )/2 ]^1/2; the following roots of β_* are:

β*1 = σ* ( C6 + D) - i σ* ( ζ C5 - E)

(25)

β*2 = σ* ( C6 - D) - i σ* ( ζ C5 + E)

(26)

and the dimensionaless celerity and logarithmic decrements of the two components of dynamic wave are:

c*1 = C6 + D

(27)

2π ( ζ C5 - E ) δ1 = - ________________                 | C6 + D |

(28)

c*2 = C6 - D

(29)

2π ( ζ C5 + E ) δ2 = - ________________                 | C6 - D |

(30)

The attenuation of the primary dynamic wave is characterized by δ₁. There are two conditions under which δ₁ = 0. These are: (1) C₅ = E = 0; and (2) ζC₅ = E. For C₅ = 0, it is necessary that k = 0. For E = 0, it is necessary that C = A, i.e., B = 0, which leads to C₁₀ = 0, or C₅ = 0. Therefore, for k = 0; and C₅ = E = δ₁ = 0, the primary dynamic wave is not subject to attenuation.

The case of ζC₅ = E can be shown to translate into the condition that F_o = 2. Neutral stability of primary dynamic waves will then occur in this condition. Therefore, it follows that for ζC₅ = E, δ₁ = 0.

Similar conditions can be formulated for secondary dynamic waves. For C₅ = E = 0; k and δ₂ = 0, leading to the conclusion that the secondary dynamic waves do not attenuate in the absence of friction and bottom slope. Secondary waves, in general, though, attenuate for all Froude numbers, as indicated by Eq. 30.

4.   SUMMARY

The nature of wave attenuation in prismatic channels is clarified by using the tools of the linear stability theory. Wave attenuation is shown to be caused by the interaction of the kinematic term (friction and bottom slope) with the local acceleration term. When the latter is absent or negligible, wave attenuation is caused by the interaction of the kinematic term with the pressure gradient or convective acceleration terms, or both.

REFERENCES

1. Henderson, F. M. 1966. Open Channel Flow, The MacMillan Co., New York, N. Y.

2. Liggett, J. A. 1975. Basic Equations of Unsteady Flow, in Unsteady Flow in Open Channels, K. Mahmood, and V. Yevjevich, eds., Water Resources Publications, Fort Collins, Colo.

3. Lighthill. M. J. and G. B. Whitham. 1955. "On Kinematic Waves I. Flood Movement in Long Rivers", Proceedings of the Royal Society of London. A229, 281-316.

4. Ponce, V. M. and D. B. Simons. 1977. "Shallow Wave Propagation in Open Channel Flow", Journal the Hydraulics Division, ASCE, Vol. 103, No, HY12, Proc. Paper 13392, Dec., 1461-1476.

5. Woolhiser, D. A. and J. A. Liggett. 1967 "Unsteady One-Dimensional Flow over a Plane - The Rising Hydrograph", Water Resources Research, Vol. 3, No. 3, 753-771.