i.e.

(9.7)

Obtain the mathematical solution

A units check is useful for equation (9.7). Referring back to chapter 4, we can see that both sides have dimension L/T. Note also that an initial state is required (as is the case for any differential equation). Equation (9.7) is a first-order equation and a suitable initial condition might be h (0) = 0, i.e. the gutter is dry and it starts to rain. There is a mathematical difficulty with this condition, however, in that equation (9.7) is ‘singular’ at h = 0, i.e. we cannot evaluate h ′(0) when h = 0 is substituted. This situation may be avoided by working with dt /d h instead, or taking h (0) = 1.0 cm, for example. What would be the effect of trying other starting values?

Some data have been given above for this model, but a full list is now convenient: a = 0.075 m; b = 6.0 m; d = 12.0 m; g = 9.81 m s −2 ; A = 0.0025 πm 2 ; α = 30°. Substituting these values into equation (9.7) gives

(9.8)

Solving the differential equation (9.8) will give us h(t), the height of water in the gutter at any time. Our problem is now in the form, given the rainfall rate r(t), find the depth h(t).

Two possible expressions for the rainfall rate r(t) are as follows:

or

The first model is the equivalent of steady persistent rain which sets in over a long period. Either the gutter overflows as it cannot cope, or a steady state will arise where the depth of water in the gutter h(t) settles to a constant value of less

than 0.075 m. From equation (9.8) the steady state can be mathematically predicted by examining what happens if h ′ ( t ) = 0, which applies when there is a steady state:

For example, r = 0.025 cm s −1 a steady-state situation occurs at h 5 cm, as can be seen from Table 9.11 and from the graph (Figure 9.12).

For the second model (illustrated in Figure 9.13) the rain profile represents a short heavy burst which rises to a peak at 20 s and then decreases to zero at 40 s. Thus we would expect the gutter to fill up rapidly before the level falls off. This is the behaviour predicted by the model when equation (9.8) is rewritten as