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

  1. Jun 23, 2011 #1
    1. The problem statement, all variables and given/known data
    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.

    2. Relevant equations

    A MATLAB code to generate such a surface is

    Code (Text):

    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
    2jdjh1i.jpg


    3. 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.
     
  2. jcsd
  3. Jun 23, 2011 #2

    lanedance

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    which cross section? can you just set y = constant?
     
  4. Jun 23, 2011 #3
    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).
     
  5. Jun 23, 2011 #4

    lanedance

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    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?
     
  6. Jun 23, 2011 #5
    What I mean by super period is the period of the envelope function. If you examine the beat curve
    Beat.png

    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.
     
  7. Jun 23, 2011 #6

    lanedance

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    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
    [tex] u \bullet k = (1,0)^T \bullet (\frac{2 \pi}{L},0) = \frac{2 \pi}{L} [/tex]
    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]
     
    Last edited: Jun 23, 2011
  8. Jun 24, 2011 #7
    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.
     
  9. Jun 24, 2011 #8

    lanedance

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    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
     
  10. Jun 24, 2011 #9
    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.
     
  11. Jun 24, 2011 #10
    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.
     
  12. Jul 5, 2011 #11
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