# Mechanical waves encountering a change in Density

• I
Suppose a ocean wave encountered a section of ocean which had a higher level of aeration from gas such as methane escaping from the seafloor.

Due to the aerated sections apparent lower density would the wave travel slower through the aerated section than its propagation speed thru pure seawater ?

• Stavros Kiri

## Answers and Replies

That's a good question! I think the answer is no. Although generally in water waves the water density comes in (in the speed formula), however for deep waters (like in ocean) it doesn't really matter. See e.g.
http://practicalphysics.org/speed-water-waves.html

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• sophiecentaur
sophiecentaur
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The problem is that the approximate formula for speed that's used for ocean waves is
v= √(gλ/2π)
but the wavelength can change for different densities. We really need a formula that uses the frequency because that doesn't change.
The same sort of thing as in the OP could happen at the interface between fresh and sea water, which is a very common situation.

• Stavros Kiri
The problem is that the approximate formula for speed that's used for ocean waves is
v= √(gλ/2π)
but the wavelength can change for different densities. We really need a formula that uses the frequency because that doesn't change.
We can use v = λf (f the frequency) and eliminate λ.(*) Still density doesn't appear, so the answer is still no.
Note: in the reference I gave above there are some typos (mixing up and confusing greek and latin symbols g, γ and r, ρ). [But it's pretty obvious what they mean in the equations. Anyone interested for a specific formula question, I could correct it here.]

(*) we get v = g/2πf

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sophiecentaur
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We can use v = λf (f the frequency) and eliminate λ.(*)
You can't solve this equation unless you know two of the variables. Another equation is needed - derived from the equation of motion. Too hard for me guv'nor.

You can't solve this equation unless you know two of the variables.
I meant using both equations ... to eliminate λ (and find v in terms of f only). And in fact ...
(*) we get v = g/2πf
Note: velocity depending on λ shows dispersive medium ...
For ocean waves, as you said, the dispersion relation is:
v= √(gλ/2π)
(That's the 2nd equation I used earlier above to switch to f ...)

sophiecentaur
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Oh yes. I see that now. It's still upsetting my reality check, though, that there is no change at the interface between two media. But I suppose that there would be a change of level between density changes (even in calm water) and a flow of the less dense over the more dense.

• Stavros Kiri
Just from a conservation of energy viewpoint, I would think _something_ happens.
Change in amplitude perhaps? (in the interface between the two media - with λ, f and velocity v staying the same | that's my best bet, based on the equations above ...) [Note: f (frequency) generally depends on the source.]
However, that's just my idea (about the amplitude [etc.]), and I do not claim it's true yet. I tried looking at the paper (1st round) but it seems I need a lot of work with it to even see if it helps.
Also (to everyone) take notice and cf. the other related thread of the OP:
https://www.physicsforums.com/threads/non-newtonian-newtonian-fluid-interface.942080/

Thanks for all your replies guys , i always like when a thread gets a few good minds involved. I too have been unsettled by the notion that nothing happens at the interface.

I have searched for alternative ways to express Wave velocity and came across this for sound waves. Not being a surface wave i'm not sure if we can apply ??
http://www.phys.uconn.edu/~gibson/Notes/Section5_1/Sec5_1.htm

Wave velocity
We have already discussed a fair amount the velocity of a wave. As mentioned above, this is a curious property of a wave – it is always in motion and always travels with the same speed, the wave velocity. The velocity of a wave depends on properties of the medium is a rather simple way: We discussed the restoring force above – that is the force tending to return the medium to is equilibrium. The density of the medium is just what it sounds like. Waves in a thick liquid, like molasses, will travel more slowly than waves in a thin liquid, like water (assuming, of course, that the restoring force is the same). The units on restoring forces and densities are a bit complicated, but the combination of the square root of the ratio of the two does work out to be length/time, as it should.

As well as the discussed density change, I could definitely see Aeration ~ really being being a buoyant force opposing gravity as the restoring force- having an effect.

(in the speed formula), however for deep waters (like in ocean) it doesn't really matter. See e.g.
http://practicalphysics.org/speed-water-waves.html

Stavros's link had the following blurb the density, r, does not appear because if it increases, the force acting and the mass to be moved both increase by the same factor, with no net effect on the response time of water ahead of a wave front.

This makes sense but only if the density change is consistent from origin ie comparing two different liquids otherwise we do have higher mass pushing a smaller mass in front (which would suggest acceleration )

Is there a way Celerity could be expressed in terms of bulk modulus as that's whats really changing due to the aeration ?? I was just trying to keep the question as simple as possible .

Edit : I found an explanation in terms of Bulk Modulus of why a sound wave travels faster in Water (a denser medium) from
https://physics.stackexchange.com/q...udinal-waves-how-velocity-varies-with-density

#### Attachments

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• Stavros Kiri
I too have been unsettled by the notion that nothing happens at the interface.
But why speed only? (For ocean waves we've shown it does not depend on the density, but just on the frequencey, which characterizes the source [e.g. wind].)
we get v = g/2πf
Also take a look at
It's still upsetting my reality check, though, that there is no change at the interface between two media. But I suppose that there would be a change of level between density changes (even in calm water) and a flow of the less dense over the more dense.
Just from a conservation of energy viewpoint, I would think _something_ happens.
Change in amplitude perhaps? (in the interface between the two media - with λ, f and velocity v staying the same | that's my best bet, based on the equations above ...) [Note: f (frequency) generally depends on the source.]
However, that's just my idea (about the amplitude [etc.]), and I do not claim it's true yet.
After doing the basic physics (for ocean waves, smoothly transiting to different densities), I think the only correct answer is that velocity v, wavelength λ and f (frequency) all stay the same in the transition, but there will be a change of amplitude in the wave (as different densities will allow the wave to strech more [or less etc.]).

Now, in your other thread, things are different. Not only the density changes, but also there are two significantly different media that actually connect with a thin elastic boundary. That's substantially different, [and I am sure in that case there will also be changes in velocity etc.], and/because there is also going to be refraction and reflection (Snell's law etc.) ...

olivermsun
Science Advisor
It would help if the intent of the original question were made clearer.

Does the OP want to know what would happen if a surface gravity wave propagates through a section of ocean with an aerated/different density near the surface? Is it a "deep water" (short) or a "shallow water" (long) wave, or either?

Or is the question about how density (e.g., aerated pockets) affects a breaking wave when it comes crashing down over the blob with different density

Regarding internal gravity waves and acoustic waves, both of those obviously are affected when they propagate through varying density. If those are being considered as well, are sharp density interfaces being considered? (IMHO they should be for those types of waves.)

"is it a deep water"
Ocean is always deep ...
both of those obviously are affected when they propagate through varying density.
Not the speed of ocean water waves (see earlier posts and formulae).

olivermsun
Science Advisor
The ocean is not always deep, and certainly not relative to the wavelength of gravity waves propagating through it.

The speed of ocean surface water waves is affected mostly by depth changes, not by density. However, since the topic is mechanical waves encountering a change in density, then one has to ask whether acoustic waves and internal gravity waves are being considered. If so, then they certainly are affected.

Hi All

Again thanks for your replys

EDIT : reading more about the phase and group velocities I can probably limit my enquiry to shallow water waves where I understand that these become equal.

Does the OP want to know what would happen if a surface gravity wave propagates through a section of ocean with an aerated/different density near the surface? Is it a "deep water" (short) or a "shallow water" (long) wave, or either?

Yes this is the really the query- If aeration (above 1/2 wavelength) depth changes either the celerity in transition or shallow I didn't refer or intend to refer to breaking waves .

Ocean is always deep ...

Not the speed of ocean water waves (see earlier posts and formulae).

I understand your derivation from the formulas im just worried there might be an assumption that underlies to simply for normal applications...i know Density changes are not something that has reason to be intensely studied but around the edges i have found some papers that suggest there is a relationship.

Below is a summary from a linked paper studying aeration levels effect on Tsunamis that perhaps might be of interest -

Wave-propagation velocity increases with the increase in bulk modulus, and the decrease in mass density of sea water, while tsunami speed increases with the increase in the ocean depth. Comparisons between wave propagation velocity and tsunami speed have been carried out in details. The vertical and horizontal dynamic water-attenuation factors increase with the increase in the frequencies of the wave, the square root of water density and with the decrease in the square root of the bulk modulus of sea water. Initially, dynamic water heights are calculated with different values of air percentages, soils and frequencies, and finally, the tsunami amplitudes at the coast lines are also estimated. Shoaling amplification factor depends on many factors, such as the refraction rays, bulk modulus, density of water, and water depths

http://www.iitk.ac.in/nicee/wcee/article/WCEE2012_0335.pdf

Now, in your other thread, things are different.

Id really like to keep away from the other thread as both @Chestermiller and @Andy Resnick has suggested it is to complex to make simple predictions

This was an attempt to simplify my query

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• Stavros Kiri
olivermsun
Science Advisor
...reading more about the phase and group velocities I can probably limit my enquiry to shallow water waves where I understand that these become equal.
...
If aeration (above 1/2 wavelength) depth changes either the celerity in transition or shallow I didn't refer or intend to refer to waves .
If you are considering shallow water waves without breaking (i.e., approximately linear waves) then the assumption is that the wavelength is much greater than the depth of the water, in which case I'm not sure it makes sense to have an aerated layer that's 1/2 a wavelength deep. In any case I wouldn't expect any physically plausible vertical layering (stratification) to have much effect on these types of waves.

I understand your derivation from the formulas im just worried there might be an assumption that underlies to simply for normal applications...i know Density changes are not something that has reason to be intensely studied but around the edges i have found some papers that suggest there is a relationship.

Below is a summary from a linked paper studying aeration levels effect on Tsunamis that perhaps might be of interest -

Wave-propagation velocity increases with the increase in bulk modulus, and the decrease in mass density of sea water, while tsunami speed increases with the increase in the ocean depth. Comparisons between wave propagation velocity and tsunami speed have been carried out in details. The vertical and horizontal dynamic water-attenuation factors increase with the increase in the frequencies of the wave, the square root of water density and with the decrease in the square root of the bulk modulus of sea water. Initially, dynamic water heights are calculated with different values of air percentages, soils and frequencies, and finally, the tsunami amplitudes at the coast lines are also estimated. Shoaling amplification factor depends on many factors, such as the refraction rays, bulk modulus, density of water, and water depths

http://www.iitk.ac.in/nicee/wcee/article/WCEE2012_0335.pdf
The paper seems to be considering several effects, including the propagation of the associated seismic wave through the bottom sediment layer and also the final run-up of the tsunami on the beach where the amplitude of the tsunami is becoming comparable to the total water depth. Here I do think you have to start considering strong nonlinearity, breaking effects, significant suspension of sediment and aeration, etc. Is this what you are interested in?

• Stavros Kiri