Martin S.
 16:23 Jan262009 Plotting dispersion curves in Matlab with contour() Hi everybody,
has anybody ever plotted dispersion curves of plate waves using a contour line method? If yes, you might be familar with the issues I'm facing (see below) trying this. I don't succeed in creating a clean plot of the curves. I only get pieces of them and a lot of points that belong there. Let me explain how I do it and what problems might be the reason for the failure.
Dispersion curves of guided modes in plates are made of couples of frequency and propagation velocity (or wavenumber). Those couples are solutions of the characteristic equation(s) (=dispersion equations) of the plate under consideration. Thus, to plot the dispersion curves, one has to obtain the roots of the characteristic equation(s).
One way to do this in Matlab, is the following:
Let z(f,v)=0 be the characteristic equation. Now we would compute the left side of the equation in the whole frequencyvelocity domain of interest. For example, we could move through that domain by incrementing the frequency in steps of 1kHz and the velocity in steps of 1m/s. In every step, we would store the value z(f,v) in a matrix Z, whose rows and columns would represent constant values of velocity and frequency, respectively. We would have a twodimensional function which could be graphically represented by a surface above the fvplane. In Matlab we would create such a surface plot by simply applying the function surf() on the matrix Z. Similarly, we would apply the function contour(). This would yield contour lines of the function z(f,v). As we want to know where z(f,v) becomes zero, we would specify the niveau of the contour line as zero. This could be done by the function call:
contour(Z, [0 0]).
Now for the issues I'm seeing (and which might be the reason for the unsatisfying results):
1.) The values in Z are complex, but a contour plot can only be generated for real values. This could be easily solved by working with the magnitudes of Z. In Matlab: abs(Z). However, it should be noted that the zeros would then also be minima of z(f,v). Possibly, the contour method wouldn't work well with that.
2.) By using certain values for the increments of f and v, we generally would not find those couples (f,v) where Z becomes EXACTLY zero. Mostly, we would at best come close to it. So we would need to have an idea of what the right order of magnitude for the increments would have to be, to be close enough to the zero level contour lines. Also, we would have to generate the contour plot not only for exactly zero, but also for values more or less close to zero. But this again would result in possibly including points in the plot which do not belong to dispersion curves. And still there might be points which do belong to a curve, but are not yet plotted.
If you have already tried this and succeeded in it or if you have an idea, please tell me about it. I need help. 
Slawomir Mackiewicz
 15:36 Jan272009 Re: Plotting dispersion curves in Matlab with contour() In Reply to Martin S. at 16:23 Jan262009 (Opening).
Hi Martin
The clear reason of your failure in solving the plate waves characteristic equation by using Matlab contour routine is taking the absolute value of function Z(f,V).
This type of substitution would be right for exact, analytical methods of solution but not for your approach based on extrapolation of numerical data. If you take Abs(Z) practically all points of Abs(Z(f,v)) matrix have positive values and Matlab contour routine almost never finds zero crossing points (all extrapolated values between positive values are also positive). You seem to have noticed that problem in your issue 2).
If you dont want to solve the characteristic equation in the full complex domain you can still improve your solution by:
1. Generating the contour plots for value epsilon slightly greater than zero.
2. Tightening the frequency and velocity steps of your Z(f,v) matrix.
The optimal values for these parameters you can establish by the trial and error technique.
Best Regards
Slawomir 
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