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Name: James
Status: other
Grade: other
Location: CA
Country: N/A
Date: October 2006

Question:
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).



Replies:
"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 understandings.

"Linear" means a || b = a + b : superimposed wave-fields add without distortion. 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 dispersive?

For example: Microwave Wave guides are not non-linear, but they are dispersive. E-M waves in a microwave wave guide change speed dramatically depending on frequency, from zero (a standing wave) at their low-frequency cutoff, to nearly light speed for much higher frequencies aimed straight down the bore. 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 breakdown levels. Plasma breakdown then changes things a lot. Plasmas are generally pretty non-linear.

The linear, non-dispersive medium is the type most amenable to our mathematics, 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 symbolic math.

The speed-invariant property was emphasized in class to familiarize you with the idea and behavior of this common and basic type of medium.

All real materials such as glass and water, and even gasses, are slightly dispersive 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 extremes we have been able to measure. However, in principle even this medium has limits. 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 inaccurate. At ridiculously high intensities Vacuum Breakdown can occur, generating particles such as electron/positron pairs and consuming some of the photons. So there is no _completely_ linear medium anywhere in nature; they all have limits. 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 change.)

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 forwards direction, 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.

Jim Swenson


James,

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 water density.

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
Meteorologist
Climate Research Section
Environmental Science Division
Argonne National Laboratory



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