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

^{1}⁄_{4}*ρgh*^{2}*λ*.

(G.5)

So, doubling and dividing by the period, the true power of this model

shallow-water tidal wave is

power = ^{1}⁄_{2}(*ρgh*^{2}*λ*) × *ω*/*T* = ^{1}⁄_{2}*ρgh*^{2}*v* × *ω*,

(G.6)

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

power = *ρgh*^{2}√*gd* × *ω*/2 = *ρg*^{3/2}√*d**h*^{2} × *ω*/2.

(G.7)

Let’s compare this power with the kinetic-energy flux *K*_{BV}. 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

*K*_{BV} = ^{1}⁄_{2}*ρAU*^{3} = ^{1}⁄_{2}*ρωd* (*vh*/*d*)^{3} = *ρ*(*g*^{3/2}/√*d*) *h*^{3} × *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

K_{BV} |
= | ρω(g^{3/2}/√d) h^{3} |
= | h |

power | ρg^{3/2}h^{2}√dω |
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.

If the depth *d* decreases gradually and the width remains constant such

that there is minimal reflection or absorption of the incoming power, then