# A few questions about wave interactions (conceptual, not technical)

• elegysix

#### elegysix

Suppose there are two spheres freely floating on the surface of an infinitely large and deep body of water.
Suppose these two spheres bob up and down independently and create waves (which may or may not have the same frequency and amplitude) in the water.

Do these spheres always remain bobbing in a cycle around the same position, or do these wave sources always attract/repel each other?

Are there specific conditions to produce attraction/repulsion?
(such as a certain phase difference and distance, or matched frequency, etc.)

Thanks for any insight.

## Answers and Replies

Freely floating spheres as described will respond to each others waves.

Factors governing the resulting motion would depend on the resistance the sphere has to the water, the wavelength of the waves, the amplitude, ... stuff like that.

In general, a traveling wave will transfer momentum to the objects floating in it - pushing them in the direction the wave is travelling. i.e. away from the source. i.e. repulsion. For your setup - the two spheres plus the water make for coupled oscillators (damped) so the motion can get very complicated.

First let's just talk about 1 sphere bobbing and causing waves. What is its effect on a second sphere. I think there is no net water movement away from the first sphere. The second sphere will experience repetitive vertical and horizontal displacement in a repetitive pattern with no net long term displacement. Those patterns are well illustrated here http://www.acs.psu.edu/drussell/demos/waves/wavemotion.html

There doesn't need to be a net water movement away from the sphere for the second sphere to get pushed away. The wave is moving. Floating objects can get pushed along by waves. You can try it out in a bathtub if you like.

The page you have found is showing you the motion of the medium rather than an object like a sphere floating on it in order to illustrate the different forms of wave motion.

I don't understand what transverse mechanism applies a net lateral force to the floating sphere. I'm talking a linear wave caused by a vertical displacement or vibration, not a breaking wave or net current.

It's a travelling wave - traveling waves carry energy and momentum in the direction of the wave.
When the wave encounters the sphere, some is reflected. Conservation of momentum dictates that some momentum got transferred to the sphere.

I'll agree that to the extent that there is reflection there has to be energy transfer. But since the sphere is free to move up and down there will be little reflection. A weightless sphere would not cause a reflection and so would have no net motion. It seems that only to the extent that the sphere appears rigid (has inertia) will there be a reflection.

Wave phenomenon don't carry matter but energy perpendicular to wavefront. In case of water waves Energy is in the form of Kinetic energy of vertical displacement of water molecules ,so does any object placed on it.Hence there would be no attaction/repulsion will be observed.

When you fish with hook, line, sinker, and float ... waves from passing boats make the float bob up and down. Thus it is often called a bobber. The waves don't move the bobber along the path of the wave front.

Best Regards, Ye Olde Fisherman

OK - this is going to get quite long - I'll split it in two.

Part 1. Responding qualitatively.

A weightless sphere would not cause a reflection and so would have no net motion.
It would also have to be frictionless and the fluid has to have zero viscosity. It would displace zero water, and respond instantly to the slightest impulse with infinite acceleration.

A weightless sphere also won't go back down after the passing wave has pushed it up (unless it sticks to the water) - in fact it should keep going.

The simple pictures of floating objects you get in class lead to some very-hard-to-shift misconceptions about buoyancy.

It seems that only to the extent that the sphere appears rigid (has inertia) will there be a reflection.
Sort of. Though "rigid" and "has inertia" are not the same thing.

Objects don't respond instantly to the forces on them - so, in this case, some water in the wave tries to flow around the object (remember the mass transfer is upwards in this example) as well as lifting it. The effect is that the water rises around the object before the object starts lifting - to do this, some must be moving laterally. Remember how "floating" works?

Real water waves are somewhat more complicated (see later).

physycho and UltrafastPED (and meBigGuy's intuition) are correct, though, in that you would mostly expect a very buoyant sphere to bob up and down - maybe a little lateral movement back-and-forth - given no other information that is the way to bet. However, notice: small-amplitude ripples may not make a heavy object bob and being pushed along is not impossible.

Waves do carry energy.

Water waves, and mechanical waves in general, can also carry momentum in the direction of propagation. This means they can push things along by the usual conservation laws. The exact behavior depends on the situation.

BTW: if the idea was to use water waves to model elementary forces as waves, it won't work.

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Part 2. Classical Wave Mechanics Primer.

In longitudinal waves, energy and momentum propagate without bulk transfer of mass.
http://physics.tutorvista.com/waves/transverse-waves.html
... that's the basic HS-level treatment.

Seems that it is not all that obvious that this should be so for transverse waves, however, as evidenced by other threads on the same subject in these forums (and elsewhere). i.e.

However - here is a formal treatment:
http://silverdialogues.fas.nyu.edu/docs/IO/24452/peskin.pdf
... section 3 deals with water waves.
Notice what they have to take into account? And that's a simple model! (i.e. floating objects not considered) Just working out the momentum in the direction of propagation averaged over a single period ... judging from other discussions, people seem to get the idea that, since the mass-movement is entirely transverse, and oscillatory anyway, the period-average momentum would be zero.

The motion of ﬂuid particles in water waves is circular, and one might think that the net momentum in the direction of propagation would be zero. This reasoning is incorrect, however, because of the correlation between the height of the water and the direction of horizontal motion.

... You could think about it this way - a traveling water wave has transverse and longitudinal components. You see it in a dirty ripple tank.

OK - now let's go back to the original questions:
Do these spheres always remain bobbing in a cycle around the same position, or do these wave sources always attract/repel each other?

Are there specific conditions to produce attraction/repulsion?
(such as a certain phase difference and distance, or matched frequency, etc.)

Off the information in the last post - the spheres form a system of coupled oscillators.
The two spheres can be expected to mostly bob in place - moving apart on average.

In the next question, where we drive the water waves with one sphere and leave the other one to float ... again, expect the sphere to bob in place with any net lateral motion to be directed away from the source.

In both cases, the precise scientific answer is "it depends".

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I'll stick with "they bob up and down"

For a simple explanation see http://www.bitesizephysics.com/energywaveslesso.html

And here: http://hyperphysics.phy-astr.gsu.edu/hbase/sound/wavplt.html

You get the same for an electron in a laser beam ... the electron oscillates, and follows a closed circuit back to where it started - not just up and down, but also along the field gradient, pushed first out, then back. But a closed loop.

You need enough energy to drive nonlinear relationships in order for a wave to move a float (or light to move an electron). In this case they call it "wave surfing":
http://www.utexas.edu/news/2013/06/...letop-opens-new-chapter-for-science-research/

This was first done a few years ago, almost simultaneously between several large laser facilities.

Sort of. Though "rigid" and "has inertia" are not the same thing.
I meant to imply rigid or has inerta. It needs to present a discontinuity or boundary. I think your reflections are the key to approaching this. If there are no reflections from the sphere, it will not travel.

I didn't read through all your links so maybe I missed something. I have found no approach to develop those forces other than stokes drift which seems to require nonlinear systems.

Of course the original question is not so simple, especially if you try to visualize how to have two bobbing spheres putting energy into the system without reflections. I'd agree that in any real system they would move away from each other.

"Objects don't respond instantly to the forces on them - so, in this case, some water in the wave tries to flow around the object (remember the mass transfer is upwards in this example) as well as lifting it. The effect is that the water rises around the object before the object starts lifting - to do this, some must be moving laterally."

^ This would appear to be the only net force, small albeit.