3.7. Diffusive-Wave Equation Derivation

If advection and inertia terms are neglected (i.e., the 1st and 2nd terms in LHS of (3.5.2)), the simplified momentum equation is

(3.7.1)\[\frac{\partial h}{\partial x} + S_f = S_0\]

or equivalently,

(3.7.2)\[S_f = S_0 - \frac{\partial h}{\partial x}\]

where

• \(h\) is flow height [m],

• \(S_f\) is friction slope [m/m],

• \(S_0\) is bed slope [m/m].

Based on Eq (3.5.3), the conveyance relation is written as

(3.7.3)\[Q = K\sqrt{S_f}\]

A key point is that \(S_f\) is not generally independent of \(A\) in the full diffusive-wave relation, because \(h\) is related to \(A\) in Eq (3.7.2). Therefore, if we take the total differentials of Eq (3.7.3), we can write

(3.7.4)\[dQ = \left(\frac{\partial Q}{\partial A}\right)_{S_f} dA + \left(\frac{\partial Q}{\partial S_f}\right)_A dS_f\]

The first term gives the kinematic-wave celerity:

(3.7.5)\[C = \left(\frac{\partial Q}{\partial A}\right)_{S_f} = \sqrt{S_f}\frac{\partial K}{\partial A} = \frac{Q}{K}\frac{\partial K}{\partial A}\]

The subscript \(S_f\) means that \(C\) is the partial derivative of \(Q\) with respect to \(A\) while holding \(S_f\) fixed.

The second term derivative is

(3.7.6)\[\left(\frac{\partial Q}{\partial S_f}\right)_A = \frac{K}{2\sqrt{S_f}} = \frac{K^2}{2Q}\]

Therefore,

(3.7.7)\[dQ = C\,dA + \frac{K^2}{2Q}dS_f\]

Using the diffusive-wave momentum approximation,

(3.7.8)\[dS_f = -d\left(\frac{\partial h}{\partial x}\right)\]

For a prismatic channel,

(3.7.9)\[dA = w\,dh\]

where \(w\) is the channel top width. Thus,

(3.7.10)\[dh = \frac{dA}{w}\]

and approximately,

(3.7.11)\[dS_f = -\frac{1}{w}\frac{\partial dA}{\partial x}\]

Substituting into the total variation of \(Q\) gives

(3.7.12)\[dQ = C\,dA - \frac{K^2}{2Qw}\frac{\partial dA}{\partial x}\]

Define the hydraulic diffusivity as

(3.7.13)\[D = \frac{K^2}{2Qw}\]

Then

(3.7.14)\[dQ = C\,dA - D\frac{\partial dA}{\partial x}\]

This shows that the dependence of \(S_f\) on the water-surface slope is not ignored. Instead, it becomes the diffusion term in the final equation. Differentiating with respect to time gives

(3.7.15)\[\frac{\partial Q}{\partial t} = C\frac{\partial A}{\partial t} - D\frac{\partial^2 A}{\partial x \partial t}\]

From continuity (Eq (3.5.1)),

(3.7.16)\[\frac{\partial A}{\partial t} = q_l - \frac{\partial Q}{\partial x}\]

Differentiating this expression with respect to \(x\) gives

(3.7.17)\[\frac{\partial^2 A}{\partial x \partial t} = \frac{\partial q_l}{\partial x} - \frac{\partial^2 Q}{\partial x^2}\]

Substituting into the time derivative of \(Q\),

(3.7.18)\[\frac{\partial Q}{\partial t} = C\left(q_l - \frac{\partial Q}{\partial x}\right) - D\left(\frac{\partial q_l}{\partial x} - \frac{\partial^2 Q}{\partial x^2}\right)\]

Rearranging,

(3.7.19)\[\frac{\partial Q}{\partial t} + C\frac{\partial Q}{\partial x} = D\frac{\partial^2 Q}{\partial x^2} + Cq_l - D\frac{\partial q_l}{\partial x}\]

If lateral inflow is spatially uniform within the reach, or if \(\partial q_l / \partial x\) is neglected, this reduces to

(3.7.20)\[\frac{\partial Q}{\partial t} + C\frac{\partial Q}{\partial x} = D\frac{\partial^2 Q}{\partial x^2} + Cq_l\]