Therefore, as found before, (derived with constant volume, but still applicable with a time

varying volume),

1-*n*c

⎛ *n*c -1 ⎞

*nc*

*xA*2 (τ 2 ) = κ cf ⎜

1

⎟ τ2

(7.31)

*nc*

*A*2 ⎠

⎝

where now *A*2 is a function of time. Therefore, define a constant *A*II equal to the initial

mobile water volume of the *U*2 wave, and a new variable ∆2 equal to the total water

volume the *U*2 wave has absorbed, by the equation

*A*2 = *A*II + ∆2

(7.32)

where we know that because the *U*1 wave is undisturbed beneath the *U*1 *U*2 junction, the

total water volume below the junction must be the same as it would be were the *U*2 wave

not present, giving rise to the conclusion that the total water volume above the *U*1*U*2

junction must be equal to the initial volume of the *U*2 wave plus the volume of the *U*1

wave, which would not otherwise have passed the *U*1*U*2 junction point. This means that

*nc*

⎡*x *⎤

*nc*-1

∆2 = ⎢ A2 ⎥ ( nc - 1)τ11-*nc *.

1

(7.33)

⎢ κ cf ⎥

⎣

⎦

By substituting into Equation (7.32) we get

1-*n*c

⎛

⎞

*nc*

⎜

⎟

⎜

⎟

*nc *- 1

*xA*2 (τ 2 ) = κ cf ⎜

1

τ2 ,

⎟

(7.34)

*nc*

*nc*

⎜

⎟

⎡*x *⎤

*nc*-1

⎜ *A*II + ⎢ A2 ⎥ ( nc - 1) (τ1 )1-*nc*

1

⎟

⎜

⎟

⎢ κ cf ⎥

⎣

⎦

⎝

⎠

which can be solved for *x*A2

1-*nc*

*nc*-1

⎡ ⎛τ ⎞ ⎤

1

*nc*

⎛* A * ⎞

*nc*-1

*nc*

1

⎢1 - ⎜ 2 ⎟ ⎥ .

= κ cf τ 2 ⎜ II ⎟

(7.35)

*xA*2

*nc*

⎢ ⎝ τ1 ⎠ ⎥

⎝ *n*c - 1 ⎠

⎣

⎦

This is the form used in the model to find the depth of any wave that has a preceding

wave. Finding the point at which one wave of volume flux completely overtakes its

antecedent proves to be straightforward if the first wave has no predecessor, with the

result being

-1

⎡⎛ *A*

⎤

*nc *-1

⎞

τ 2 = δ ⎢⎜

+ 1⎟ - 1⎥

(7.36)

*II*

*AI*

⎢⎝

⎥

⎠

⎣

⎦

where δ is the time between the formation of wave *U*1 and wave *U*2. This gives us the

ability to calculate the flux as a function of depth and time resulting from a single water

volume input, to calculate the depth and net water volume of a following input wave of

volume flux, and to calculate the depth and time at which one wave of volume flux will

overrun a preceding wave of volume flux. In short, it is now possible to completely

describe the saturation and flux resulting from two point inputs in several compact

equations, without any finite difference or finite element matrix solutions to differential

equations. In order to extend the model to the general case in which many waves may be

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