Calculating period and cross-section of a beat pattern surface.

[ekarademir]

Homework Statement

Beat pattern is superposition of two sine functions with a little frequency difference. (http://en.wikipedia.org/wiki/Beat_(acoustics)#Mathematics_and_physics_of_beat_tones)

We use this pattern to generate a surface, by adding another beat pattern with 60 degrees difference.

Now, normal beat pattern has a well known super-period. But I need a similar formula for determining super-period of any cross-section from the origin. I need such a formula to use it in simulations.

Homework Equations

A MATLAB code to generate such a surface is

l1 = 0.295; % Period 1
l2 = 0.305; % Period 2

% This is the formula for determining period for beat curve
period = l1*l2/(abs(l1-l2));

x = -period:0.01:period;
y = -period:0.01:period;

[X,Y] = meshgrid(x,y);

S = sin(2*pi*X/l1) + sin(2*pi*X/l2) + sin(2*pi*(X/2+sqrt(3)*Y/2)/l1) + sin(2*pi*(X/2+sqrt(3)*Y/2)/l2);

Resulting surface is

The Attempt at a Solution

Taking the Fourier transform, we can determine the four k values that generate this surface. But one problem is to determine the resulting cross-section pattern.

This is not for a homework, but for a simulation problem I struggle with.

 

[lanedance]

which cross section? can you just set y = constant?

 

[ekarademir]

Hi,

I'm looking for any cross-section. I've managed to draw the cross-section by using the surface formula for y=f(x) function, however, now the problem is determining the super period of that cross-section.

For instance, i want to take a cross-section by placing a line passing through origin with a slope of tan(pi/5), and i want to determine one period of this cross-section (since this cross-section also consists of four sine waves, i assume it has a periodicity, may be I'm wrong).

 

[lanedance]

so if you have z(x,y), and can find the equation of the line you want to examine y=f(x), then look at z(x,f(x))

what do you mean by super period?

 

[ekarademir]

What I mean by super period is the period of the envelope function. If you examine the beat curve

the envelope function has a period of P1*P2/(P1-P2), P1 and P2 are periods of two sine waves that are superposed to produce this beat pattern.

I need a similar formula or a way to calculate the super-period of each cross-section.

Addition of two sines can be rewritten as multiplication of one cosine and one sine functions, then the cosine function forms the envelope. However, I couldn't come up with a similar formula for addition of four sine functions, which produces the resulting cross-section.

 

[lanedance]

ok, yeah i know the envelope, had never heard it called super before...

how about this as a simple example consider the field sin(2*pi*X/L), you could describe this as a wave vector something along the lines of [itex] k = (\frac{2 \pi}{L},0)^T [\itex], ie a wave vector parallel to the x axis.

now consider cross sections, qualitatively the spatial frequency will be largest parallel to the x axis, and become zero along the y.

say we want the cross section along y=x, a unit vector in that direction is [itex] u = (1,0)^T [\itex] then the project ion of the wave vector on the x-axis is
$$u \bullet k = (1,0)^T \bullet (\frac{2 \pi}{L},0) = \frac{2 \pi}{L}$$
which lines up with what we would expect

say we want the cross section along y=x, a unit vector in that direction is [itex] u = \frac{1}{\sqrt{2}}(1,1)^T [\itex]

similarly taking the inner product along y=x, as expected the frequency is reduced by a factor of [itex] \frac{1}{\sqrt{2}} [\itex]

 

[ekarademir]

Thank you, this helped a lot. Last night I realized that I've been approaching this problem too mathematically :) I considered to look it as a solid state problem.

When I widened the plot boundaries, this pattern looked much like a hexagonal lattice. Then all periodicity (or translation symmetry) can be written with two vectors, which transforms into another hexagonal lattice with two wave vectors in the reciprocal space. Then I could calculate the periodicity in any direction by writing the direction vector in terms of these two primitive cell vectors; just as you pointed out. I'll try to work on this direction, if any result comes out, I'll post it in here.

 

[lanedance]

i was more implying you know the wave vector for your 2 inputs waves

you could then project them onto the direction you are interested in and then you have the 2 frequencies interaction on your cross section

It should be an easy matter to add them to get the beat (envelope) frequency

 

[ekarademir]

Hmm, but the problem is there are no two input waves. A beat pattern can be generated by two input waves but this pattern is generated by four input waves (two beat patterns) in any cross-section.

 

[ekarademir]

I guess solid state approach paid off. I've constructed a hexagonal lattice, with two primitive cell vectors, these vectors can be used to build the smallest vector along the direction we want the cross-section to be. If we use integer amount of these vectors, then we get the vector that shows the translation symmetry. Hence, the magnitude of the vector gives the period.

 

[ekarademir]

I have come across a superb paper for just this kind of a problem.

Olof Bryndahl
Journal of Optical Society of America, Vol. 65, N. 6 pp-685-694 (June 1975)
http://www.opticsinfobase.org/abstract.cfm?URI=josa-65-6-685