on swings.) So to compute the total energy all we need to do is compute
one of the two – the potential energy per wavelength, or the kinetic energy
per wavelength – then double it. The potential energy of a wave (per
wavelength and per unit width of wavefront) is found by integration to be

14ρgh2λ.

(G.5)

So, doubling and dividing by the period, the true power of this model
shallow-water tidal wave is

power = 12(ρgh2λ) × ω/T  = 12ρgh2v × ω,

(G.6)

where ω is the width of the wavefront. Substituting v = √gd,

power = ρgh2gd × ω/2  = ρg3/2dh2 × ω/2.

(G.7)

Let’s compare this power with the kinetic-energy flux KBV. Strikingly, the
two expressions scale differently with the amplitude h. Using the ampli-
tude conversion relation (G.3), the crest velocity (G.2), and A = ωd, we can
re-express the kinetic-energy flux as

KBV = 12ρAU3 =  12ρωd (vh/d)3 =  ρ(g3/2/√dh3 × ω/2.

(G.8)

So the kinetic-energy-flux method suggests that the total power of a shallow-
water wave scales as amplitude cubed (equation (G.8)); but the correct for-
mula shows that the power scales as amplitude squared (equation (G.7)).

The ratio is

KBV  =  ρω(g3/2/√dh3  =  h
power ρg3/2h2dω d
(G.9)

Because h is usually much smaller than d (h is about 1 m or 2 m, while d
is 100 m or 10 m), estimates of tidal power resources that are based on the
kinetic-energy-flux method may be much too small, at least in cases where
this shallow-water cartoon of tidal waves is appropriate.

Moreover, estimates based on the kinetic-energy-flux method incor-
rectly assert that the total available power at springs (the biggest tides)
is eight times greater than at neaps (the smallest tides), assuming an am-
plitude ratio, springs to neaps, of two to one; but the correct answer is
that the total available power of a travelling wave scales as its amplitude
squared, so the springs-to-neaps ratio of total-incoming-power is four to
one.

Effect of shelving of sea bed, and Coriolis force

If the depth d decreases gradually and the width remains constant such
that there is minimal reflection or absorption of the incoming power, then