Is Huygens's principle valid for inertial waves?

Question

Inertial waves are a type of waves possible in rotating fluids (with constant angular speed $\Omega$), and the driving force behind the propogation of these is the Coriolis force; the fact that the coordinates system that rotates together with the fluid is always changing while the fluid parcels tend to keep moving in straight lines (the inertia principle) means that they are affected by the coriolis acceleration (Notice: all discussions about inertial waves refer to the physical picture as seen from the rotating frame of reference).

Now, the introduction of an axis of rotation naturally makes the physical system non-isotropic; it has a preffered direction (the rotation axis). This kind of conical symmetry finds it's most direct consequence in the dispersion relation for inertial waves ($\omega = 2\Omega cos\theta$), which shows that the frequency of an inertial wave depend only on it's direction of travel (the direction of the wave vector, $\hat{k}$) - the angle between this direction and the rotation axis.

Now, i had great difficulties to understand how inertial waves behave when they reflect off from solid boundaries. My basic difficulty is that i didn't succeed in conceptualizing the basic principles of wave theory for the fundamentally non-isotropic behaviour of inertial waves - Huygens's principle and the principle of minimal time (from these principles one can derive the laws of reflection and refraction for isotropic space). When i attempted to derive a "reflection law" for inertial waves (which can be quite intriguing as the reflected ray needs not be in the plane spanned by the normal to the surface and the incident ray), my attempts usually terminated at my basic misunderstanding of how each point on the front of an inertial wave acts as a source on a new, secondary wave.

So my questions are:

• Can someone offer an illuminating picture of how each point on a wave front acts as a source of secondary waves? Obviously the secondary waves cannot be spheres centered at this point. This part of my question deals with the Huygens's principle.

• What is the mathematical form of the law of reflection from solid boundaries for inertial waves? - given the incident ray direction $\hat{i}$, the axis of rotation direction $\hat{\Omega}$ and the normal direction to the reflecting plane $\hat{n}$, what is the direction of the reflected ray $\hat{r}$?

Answer 1

I can explain some general things about inertial oscillation.

I will first discuss inertial oscillation, at the very end I will turn to the case of wave reflection.

(I'm not sure whether 'Inertal wave' and 'inertial oscillation' are used interexchangebly, or whether some authors may use them for related but not identical concepts.)

Mirjam Glessmer has worked for several years at the rotating platform facility in Grenoble. The Grenoble rotating platform has a diameter of 13 meters, and can be filled with water.

Mirjam Glessmer describes an occasion when they accidentally triggered inertial oscillation, and she took some video footage.

The footage is not good quality, but it's the only footage I'm aware of inertial oscillation in a lab setting.

On how to set up a platform for solid body rotation

blue arrow: gravity
red arrow: normal force
green arrow: resultant force

You see a sequence of three images (implemented as a gif animation.) In order to sustain solid body rotation: at each distance to the central axis the appropriate centripetal force must be exerted. The cross section of the floor of the platform must be in the shape of a parabola. The angular velocity of the platform must be in tune with the slope of the parabola.

In the images the slope of the platform is very much exaggerated.

(About the use of animations: yes, the constant motion is distracting. I suggest a copy/paste from the webpage to a plain-text editor, so you can read at ease. (I think there are also browser extensions available that allow you to halt any animation.)

In the animation below an object is placed on the platform, it has zero velocity with respect to the platform; the object is circumnavigating the central axis, co-rotating with the platform. Wherever the object is placed, it experiences precisely the required centripetal force, the platform is designed to have that property.

In the animation the panel on the left shows the motion with respect to the inertial coordinate system. The panel on the right shows the motion with respect to a coordinate system that is co-rotating with the platform.

In the animation four quadrants are drawn on the rim of the platform. The purpose is to help keep track of motion.

In the animation below the object on the platform has been given a velocity with respect to the platform.

The small arrow represents the centripetal force that arises from the slope of the platform. The object is still circumnavigating the central axis, but the shape of the motion is an ellipse now. As a result the motion with respect to the rotating platform is along a small circle. Is the small circle actually a circle? Mathematically, yes.

The motion along the ellipse is mathematically described as two harmonic oscillations, perpendicular to each other. Transform that motion to the co-rotating coordinate system: the result is a small circle.

In your question you mention: 'it tends to keep moving in a straight line'.

Well: when the kinetic energy associated with the inertial oscillation dissipates then the no-longer-any-dissipation-available state is one where the object is going round in a circle.

If you would remove the centripetal force you would also remove any opportunity for inertial oscillation. That is: the centripetal force is a necessary condition for inertial oscillation to occur.

Given the presence of the centripetal force: motion is along an ellipse. The larger the eccentricity of the ellipse, the larger the inertia circle.

With this omnipresent centripetal force the new rule is: 'it tends to move along a circle'.

Wave reflection

Here is my position on this: the rotation effect affects only propagation. Any motion with respect to the co-rotating coordinate system will tend to move along an inertia circle.

Just by itself reflection is not affected by the rotation of the system as a whole.

That said: I suppose that given that the incoming propagation is curvilinear and that the outgoing propagation is curvilinear it's going to be hard to keep track. But still, I don't see the process of reflection itself being affected by the rotation of the system as a whole.