Energy and Human Ambitions on a Finite Planet, 2021a

16 Small Players 284
It’s not an equation to remember, just a way to keep track of the physics
as we build toward a final/useful result. Next, we want to underst<strong>and</strong>
the power delivered as the waves come <strong>and</strong> come. We get a new one
every Δt 6 s. And we also reasoned before, in slightly different form,
that Δt λ/v, where v is the velocity of the wave: 3 m/s in our example.
The power looks like:
P GPE E GPE
Δt
ρλlgA2
16Δt
ρlgA2 v
. (16.5)
16
This is how much power the potential energy part of waves contributes
as the waves pile onto the shore. But waves also have kinetic energy.
It turns out that kinetic energy <strong>and</strong> potential energy are balanced in a
wave—which is perpetually sloshing back <strong>and</strong> forth between potential
<strong>and</strong> kinetic forms, much as happens in the motion of a pendulum. So
the total power is just double P GPE ,or
P tot ρlgA2 v
. (16.6)
8
It is a little awkward to have to specify the length of the wave, but we
needed it to make sense of the mass involved. At this point, let’s switch
to expressing the power per unit length of the wave, or P/l.
P tot
l
ρgA2 v
. (16.7)
8
Notice that this expression does not actually depend on the wavelength,
in the end. The only measures of the wave that enter are the amplitude
<strong>and</strong> velocity. 26
For our example of 1 m amplitude <strong>and</strong> 3 m/s velocity, we compute a
power per unit length of 3,750 W/m. Okay, this is a new unit, <strong>and</strong> it
looks vaguely encouraging. Blow dryers, toaster ovens, space heaters,
or similar power-hungry appliances consume about 1,800 W of power
when running full blast, so 3,750 W/m is roughly equivalent to having
two such appliances plugged in <strong>and</strong> running for every meter of length
along the wave, or coastline. It seems like a bonanza: Our collective hair
will be dry in no time! Take a moment to picture a beach cluttered with
a power-hog appliance plugged in every 0.5 m all down the beach, all
running at full power. That’s what the waves can support, <strong>and</strong> it seems
pretty impressive.
26: The velocity of near-shore waves is set
only by the depth, d, of the water (v √ gd).
We will use this 3,750 W/m figure from
here on for our rough analysis, but it should
be borne in mind that larger wave amplitude
has a quadratic effect on power, <strong>and</strong>
waves are not all 1 m peak-to-trough!
But what we care about, in the end, is how much total power the waves
can deliver: how many terawatts? So we need to multiply the wave P/l
value by a length along the wave, or a shoreline length.
Example 16.4.1 How much wave power arrives on the U.S. Pacific
coast if the whole coastline is experiencing 1mamplitude waves at a
© 2021 T. W. Murphy, Jr.; Creative Commons Attribution-NonCommercial 4.0 International Lic.;
Freely available at: https://escholarship.org/uc/energy_ambitions.