SME
Oceanography · Problem type
Wave energy and power
Use this for “Waves 2 m high arrive every 10 s. How much energy does each square meter of sea hold, and how much power reaches each meter of coast?”
Diagram
Crest to crest is one wavelength, 156.1 m.
The crest rises 1 m above the middle.
This is an example. Type your own values to start a new problem.
E (J/m²)
=
ρ
×
g
×
(
H (m)
)
2
÷
8
P (kW/m)
=
E (J/m²)
×
cg (m/s)
H is the wave height, crest to trough, twice the amplitude the picture marks; P is the power each meter of crest carries.
T
Period
example · s
Formulas
E
=
1,025
×
9.81
×
H
2
8
5,027.625
=
1,025
×
9.81
×
2
2
8
cg
=
9.81
×
T
4 × π
7.8065
=
9.81
×
10
4 × π
P
=
E
×
cg
1,000
39.2484
=
5,027.625
×
7.8065
1,000
Assumptions
•
Deep-water waves of one height and period; seawater is 1,025 kg/m³ and g = 9.81 m/s².
•
The energy, half in the moving water and half in the raised surface, is E = ρgH² ÷ 8 per m² of sea.
•
It travels at the group speed cg = gT ÷ 4π, half the wave speed.
Step-by-step
WE KNOW
H = 2 m, T = 10 s
FIND
wave energy (E), group speed (cg), wave power along the crest (P)
Step 1 · Find wave energy (E)
Use E = 1,025 × 9.81 × H² ÷ 8
The energy grows with the square of the height: twice as high holds four times as much.
E
=
1,025
×
9.81
×
2
2
8
E
=
1,025
×
9.81
×
4
8
E
=
10,055.25
×
4
8
E
=
40,221
8
E
=
5,027.625
J/
m
2
Step 2 · Find group speed (cg)
Use cg = 9.81 × T ÷ (4 × π)
In
deep
water
the
energy
moves
at
half
the
wave
speed
gT
2π
.
cg
=
9.81
×
10
4 × π
cg
=
98.1
4π
cg = 7.80655 m/s
Step 3 · Find wave power along the crest (P)
Use P = E × cg ÷ 1,000
The
energy
per
m
2
carried
along
at
the
group
speed,
in
watts,
then
watts
to
kilowatts.
P
=
5,027.625
×
7.80655
1,000
P
=
39,250
1,000
P
=
39.2484
kW/m
Check
Put the numbers back in. Both sides should match.
5,027.625
=
1,025
×
9.81
×
2
2
8
✓
7.80655
=
9.81
×
10
4 × π
✓
39.2484
=
5,027.625
×
7.80655
1,000
✓
← Previous lesson
Tsunami speed and travel time
Next lesson →
Spring and neap tides from the Sun and Moon
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
Related lessons
Waves and tides
Main lesson
Tsunami speed and travel time
Use this for “A tsunami crosses 3,000 km of ocean 4,000 m deep. How fast does it travel, and how many hours does it take?”
Spring and neap tides from the Sun and Moon
Use this for “The Sun is 1.989 × 10³⁰ kg at 1.496 × 10¹¹ m and the Moon 7.35 × 10²² kg at 3.844 × 10⁸ m. How big is the Sun’s tide next to the Moon’s?”