2D Harmonic Oscillator example

[fog37]

Hello Forum,

The 1D harmonic oscillator is an important model of a system that oscillates periodically and sinusoidally about its equilibrium position. The restoring force is linear. There is only one mode with one single frequency omega_0 (which is the resonant frequency).

What about the 2D oscillator? are a vibrating drum membrane or a rectangular membrane an examples of 2D harmonic oscillator? In that case there are many modes, each having a frequencies that is an integer multiple of the fundamental frequency omega_0, correct? Also each mode has a sinusoidal temporal behavior at every point in space even if the full spatial dependence is not sinusoidal (Bessel functions for circular membrane)...

So, in going from 1D to 2D, we go from having one mode to having countably infinite modes because we go from one degree of freedom to infinite (why countable if the membrane is made of infinite points? The DOF should be infinite) degrees of freedom.

thanks,
fog37

 

[Simon Bridge]

As you have pointed out, 1D HO is characterized by the as equation of form: $m \ddot x \hat\imath = -k x\hat\imath$
This basically says that the restoring force is directly proportional to the the distance x from some reference point.
i.e. it is a central force problem in 1D.

Extended to more than 1 dimension you get: $m \ddot{\vec r} = -k\vec r$ where $\vec r$ has the desired number of dimensions. You recover the 1D problem by putting $\vec r = x\hat\imath$

Something like a drum or the surface of a pond is a system of coupled oscillators in 3D.
Coupled oscillators in a line - a string say - has countably infinite modes. Can you see why - even though there are infinite points each it's own oscillator?

Aside: a pendulum is a 2D system - it can be represented as a 1D system if we fix the radius, then it is a "rigid rotator". The kinds of motion are "stationary", "oscillating", and "spinning". The "single-mode" of the simple pendulum comes from the small-angles approximation.

 

[Orodruin]

In addition to the HO Simon wrote down with
$$
m \ddot {\bf r} = -k {\bf r},
$$ which has the same frequency for all modes, k can in general be replaced by a symmetric positive definite matrix. The eigenvalues of this matrix will be related to the different frequencies of oscillation.

 

[Simon Bridge]

... because you can have a different value of k for each different dimension used ;)

 

[vjacheslav]

Sorry for me interrupting, but is it possible to make 1D HO from 2,3,4D?

 

[fog37]

Thanks everyone.

So a system has as many modes as degrees of freedom (DOF).
The oscillating drum as infinite DOFs and it is composed of an infinite number of coupled oscillators.

However, a system composed of an infinite number of oscillators does not necessarily have infinite DOF: the rigid pendulum is an example. Being connected, coupled does not imply that the motion of each oscillator is constrained. The only way to reduced the number of DOF is to have contraints that reduce the number of variables need to describe the system.

Correct? It is strange to think that two (or more) connected, coupled oscillator are not automatically constraining, affecting, each other's behavior.

I am confused. Doesn't each mode have a different k? For example, the different modes of a vibrating drum have different energy and different temporal frequency omega. The wavevector k is a vector (k_x, k_y, kz).

Thanks,
fog37

 

[Orodruin]

I am confused. Doesn't each mode have a different k? For example, the different modes of a vibrating drum have different energy and different temporal frequency omega. The wavevector k is a vector (k_x, k_y, kz).

Not necessarily. Some modes may be degenerate and have the same frequency. This would occur, for example, for a square drum or for a 2D harmonic oscillator with rotational symmetry.

 

[olivermsun]

I am confused. Doesn't each mode have a different k? For example, the different modes of a vibrating drum have different energy and different temporal frequency omega. The wavevector k is a vector (k_x, k_y, kz).

If you think of the motion in the oscillator as a superposition of traveling plane waves, then all the modes contain some combination of different k's. For example, the 1-d modes on a string are associated with opposed pairs of waves with wave vectors ±k such that the boundary conditions (e.g., y=0 at both ends) are satisfied.

In the case of the drum, plane waves might not be the easiest way to think about the problem; cylindrical coordinates would make more sense because of the circular symmetry.

 

[fog37]

yes, olivermsun,

I agree: it is hard to decompose the arbitrary vibration of a circular membrane into plane waves that satisfy the boundary conditions (zero motion at the drum edge). In fact, a better family of basis functions should be the bessel functions, correct? Maybe...

That said, we should be able to construct any arbitrary vibration using whatever family of basis functions we want. The choice of the basis functions should be just a matter of mathematical convenience...Is that true?

Also, does the concept of k vector only apply to plane waves?

 

[olivermsun]

Well, I guess $\vec{k} = (k_x, k_y, k_z)$ in Cartesian coordinates would normally be for plane waves, but I'm sure there are other ways of expressing waves using other basis functions.

 

[Simon Bridge]

To see the harmonics on a drum.
http://homepages.ius.edu/kforinas/S/Percussion.html

You want to use cylindrical-polar coordinates $z(r,\theta)\sin(\omega t+\delta)$ ... or some such for stationary modes.

Boundary conditions for a drum radius R, $z(R,\theta)=0,\; z(r,\theta) = z(r,\theta+2\pi),\; \frac{\partial}{\partial \theta}z(r,\theta) = \frac{\partial}{\partial \theta} z(r,\theta+2\pi)$ ... that sort of thing. In general, a drum may have varying mass density along it's membrane. It's usually best to look at the simpler cases while you are working out the maths though.

Note: this is not a 2D HO, but an infinite array of coupled 1D HOs.
The modes are usually represented by a 2D vector, and they are not harmonic.

Aside: from an nD HO, you can often just take a projection to get the 1D case.
i.e. take the 2D rigid rotator and project one end onto a line.

 

[jtbell]

For more mathematical detail of the circular membrane (with uniform surface mass density), see

http://en.wikipedia.org/wiki/Vibrations_of_a_circular_membrane

This is the situation in which I first learned about Bessel functions (the solution of the radial part of the wave equation in this case) as an undergraduate.

 

[fog37]

In quantum mechanics, is the quantized simple harmonic oscillator equation solved to find the electronic energy of a system, the vibrational energy, or the rotational energy?

For what specific situation do we use the quantized SHO?

 

[Simon Bridge]

The SHO potential is studied for the insights it provides into more difficult problems, and also as an approximation for more complicated potentials (i.e. band-edges in solid state for low temperatures). So yeah - it is for electronic energy levels in some situations.