SME
Oceanography
Waves and tides
Use this for “A 10-second swell crosses water 500 m deep. How long are its waves, how fast do they travel, and is the water deep for them?”
Diagram
d
L
=
500
156
=
3.2:
deep
water
(d
>
L
2
):
the
orbits
are
circles
that
shrink
with
depth
and
never
reach
the
floor.
Wave speed c = 15.6 m/s.
Wave heights are not to scale.
This is an example. Type your own values to start a new problem.
L (m)
=
g
×
(
T (s)
)
2
÷
(2π)
c (m/s)
=
L (m)
÷
T (s)
d
Water depth
example
q
Depth ÷ wavelength
worked out (not typed)
Formulas
L
=
9.81
×
T
2
2 × π
156.131
=
9.81 ×
10
2
2 × π
c
=
L
T
15.6131
=
156.131
10
q
=
d
L
3.2024
=
500
156.131
Assumptions
•
Deep water means d > L ÷ 2: the floor lies below the wave’s orbits and does not slow it.
•
In deep water L = gT² ÷ 2π with g = 9.81 m/s², so storm swell sorts itself by period.
•
Where d < L ÷ 2 the floor drags on the wave and the shallow-water rules apply instead.
Step-by-step
WE KNOW
T = 10 s, d = 500 m
FIND
wavelength (L), wave speed (c), depth ÷ wavelength (q)
Step 1 · Find wavelength (L)
Use L = 9.81 × T² ÷ (2 × π)
In deep water the wavelength grows with the square of the period.
L
=
9.81 ×
10
2
2 × π
L
=
9.81
×
100
2π
L
=
981
2π
L
=
156.131
m
Step 2 · Find wave speed (c)
Use c = L ÷ T
A wave moves one wavelength each period.
c
=
156.131
10
c
=
15.6131
m/s
Step 3 · Find depth ÷ wavelength (q)
Use q = d ÷ L
Past
1
2
the
water
is
deep
for
this
wave;
under
1
20
it
is
shallow;
between,
intermediate.
q
=
500
156.131
q
=
3.2024
Check
Put the numbers back in. Both sides should match.
156.131
=
9.81 ×
10
2
2 × π
✓
15.6131
=
156.131
10
✓
3.2024
=
500
156.131
✓
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Tsunami speed and travel time
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