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
So, doubling and dividing by the period, the true power of this model
shallow-water tidal wave is
power = 1⁄2(ρgh2λ) × ω/T = 1⁄2ρgh2v × ω,
where ω is the width of the wavefront. Substituting v = √gd,
power = ρgh2√gd × ω/2 = ρg3/2√dh2 × ω/2.
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 = 1⁄2ρAU3 = 1⁄2ρωd (vh/d)3 = ρ(g3/2/√d) h3 × ω/2.
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
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
If the depth d decreases gradually and the width remains constant such
that there is minimal reflection or absorption of the incoming power, then