Ocean Wave Speed and Frequency
Date: October 2006
At school I was taught:
The speed of waves propagating through a medium is determined not by
the frequency or amplitude of the wave but by the properties of the medium.
Why is it then that waves in the ocean can travel at different
speeds depending on their wave length? Apparently you can
approximate the wave speed by v=1.25*sqrt(lambda).
"The speed of waves propagating through a medium is determined not
by the frequency or amplitude of the wave but by the properties of the medium."
Not true for all, but for many.
It is true for ideal media which are "linear" and "non-dispersive".
Those words might have flown by you in class, or likely they were omitted.
Either way you caught it with our question, and can proceed to further
"Linear" means a || b = a + b : superimposed wave-fields add without
In this case the presence of one wave does not bend or destroy or capture or
frequency-shift the other.
Each wave pretends the other does not exist.
Any two different waves can sail through each other at an angle, without
interacting, in a linear medium.
Linearity has the consequence that amplitude does not affect the speed.
After all, one large wave of amplitude 2 is the same as
two waves of amplitude 1, overlaid in the same place and time.
Two waves of strength 1 do not affect each other,
so a wave of amplitude 2 will do the same thing as
(travel the same speed as) a wave of amplitude 1.
The ideal medium is "linear", and funkier media are "non-linear".
You can imagine that waves of falling dominoes would be very non-linear,
usually of amplitude "1" or "0", almost no place in between, >1, or < 0.
"Dispersive" means the wave-speed changes depending on wave-frequency.
So "non-dispersive" means frequency does not affect the speed.
Dispersion is why a glass prism can spread a narrow beam of white light
into a rainbow of colors.
The ideal medium is "non-dispersive".
Notice that "Linear" and Dispersive" are two different qualities.
For each kind of wave ask yourself both questions: Is it non-linear? Is it
For example: Microwave Wave guides are not non-linear, but they are
E-M waves in a microwave wave guide change speed dramatically depending on
from zero (a standing wave) at their low-frequency cutoff,
to nearly light speed for much higher frequencies aimed straight down the
However, amplitude does not affect the speed in wave guides at all, for at
least several orders of magnitude,
from infinitesimal thermal noise levels, almost to screaming plasma
Plasma breakdown then changes things a lot. Plasmas are generally pretty
The linear, non-dispersive medium is the type most amenable to our
enabling us to derive and understand many things about it.
So it is considered the basic concept of a medium, the first you will be
taught about in school,
Other kinds of media are considered perturbations from it,
and those are coped with mainly by numeric computing rather than by
The speed-invariant property was emphasized in class
to familiarize you with the idea and behavior of this common and basic type
All real materials such as glass and water, and even gasses, are slightly
and very slightly non-linear for electromagnetic waves.
I would suggest that the purest known linear and non-dispersive wave-medium
is electromagnetic waves in vacuum.
I think they are linear and non-dispersive to the farthest and finest
we have been able to measure. However, in principle even this medium has
At low intensities, I do not know if there is any theoretical limit,
other than the quantization hindrance: it takes a long time to measure
a wave amplitude having a very low rate of photons, and is always somewhat
At ridiculously high intensities Vacuum Breakdown can occur, generating
such as electron/positron pairs and consuming some of the photons.
So there is no _completely_ linear medium anywhere in nature; they all have
But it is a good thought, and it really works for many things.
Ocean waves are pretty linear, at least until the wave gets tall (slopes
greater than, say, 30 degrees).
(You can see that small ripples in a pool cross over each other without
But they are dispersive, following the formula you mentioned.
I think it has something to do with the wave-motion reaching transversely
down into the water by an amount proportional to the wave length.
That has the effect of giving long waves stronger coupling into the
which helps them get there a little faster.
Most waves in a volume (3-D waves) do not have such a transverse dimension
to spread into.
But many kinds of surface-defined waves (2-D) are dispersive this way.
Wave speed is proportional to wavelength times wave frequency,
for a medium with no boundary restrictions.
In the open ocean there are no boundary restrictions such as
there are near a beach or in shallow water, thus the relationship above
should hold. Amplitude is not involved, but unlike what you said,
wave speed is dependent on frequency, as well as wavelength.
If you hold the frequency constant, wavelength will
determine the wave speed.
Wave speed is also dependent on the density of the
medium. The denser the medium (for the same phase of a medium,
in this case a liquid), the slower the speed of a wave
of a particular frequency. Since sea water can vary in density
(density is determined by temperature, depth, and salinity),
waves in the ocean can vary in wave speed depending on the sea
For a wave with a large amplitude (that would reach deep into the
ocean), the top of the wave (near the ocean surface)
could move faster than the bottom of the wave (which would be in
denser water lower down), resulting in the bottom of the wave
lagging the top of the wave. This can result in a tilted wave
after a while and possibly a reduced amplitude and longer wavelength.
David R. Cook
Climate Research Section
Environmental Science Division
Argonne National Laboratory
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Update: June 2012