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Advanced Math/Calculating the position of the nodes on a waveform


I have a waveform of y1(x)= 1.2*cos(1.2*x)+1.8*cos(1.8*x), and i need to find the position of the nodes.

I found the nodes for y2(x)= sin(x*1.2)+sin(x*1.8) with either x = nπ/(1.5) or x = (2n+1)*(π/0.6), where n = 0, 1,2 3, and so on.

but i can't find an easy way for the first one.

Tristan, it is indeed harder to find nodes for y1(x) than for y2(x). I didn't get a closed form solution but the maybe the following will help.

y1(x) = 1.2cos(1.2x) + 1.8cos(1.8x). The difference in the amplitudes of the cosines is of course what makes it tricky.

We want to find x such that y1(x) = 0 or, letting a = 1.2 and b = 1.8 for now

acos(ax) = -bcos(bx).

My approach is to introduce an increment d to the angle ax and to introduce a factor of π' = (2n+1)π to make the cosine negative so that

bcos(bx) = bcos(ax+d+π') which gives (it helps to draw 2 unit circles of radius a and b and plot these quantities)

d = (b-a)x-π',

which is fine except now we need another independent equation relating d and x to solve the problem. Again, using the unit circles,

acos(ax) = bcos(ax+d)     <- no π this time.

Using a trig identity, get

tan(ax) = sin(d)/(cos(d)-a/b)

Substituting the expression for d gives a complicated expression involving x, call it f(x). I don't think it can be solved in closed form for x (it transcendental) so the next step would be to get an approximate value for x by cross plotting

tan(ax) = f(x)

and seeing at what value of x they cross. This should give you the node. Sorry I can't give you a cleaner answer. Let me know if you want me to work some more on this. Good luck.  

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randy patton


college mathematics, applied math, advanced calculus, complex analysis, linear and abstract algebra, probability theory, signal processing, undergraduate physics, physical oceanography


26 years as a professional scientist conducting academic quality research on mostly classified projects involving math/physics modeling and simulation, data analysis and signal processing, instrument development; often ocean related

J. Physical Oceanography, 1984 "A Numerical Model for Low-Frequency Equatorial Dynamics", with M. Cane

M.S. MIT Physical Oceanography, B.S. UC Berkeley Applied Math

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