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Mathematics Algebra Wolfram Language

Help with harmonic addition theorem

ok ok

Posted 11 months ago

Hello, It is well known that the linear combination, or harmonic addition, of sine and cosine waves is equivalent to a single sine (or cosine) wave with a phase shift and scaled amplitude. I did some online search and found: http://mathworld.wolfram.com/notebooks/Trigonometry/HarmonicAdditionTheorem.nb Basically the two identities I want to implement as functions are: a cos(q)+b sin(q)==Sqrt[a^2+b^2] sgn(a) cos(tan^-1(-(b/a))+q) a cos(q)+b sin(q)==-Sqrt[a^2+b^2] sgn(a) sin(tan^-1(-(b/a))+q-[Pi]/2) I used mathematica to verify them (see attached notebook). I need two functions (one for each identity) that would convert the sum of sine and cos wave to either a single sine or cos wave with a phase shift and scaled amplitude. I am a Mathematica newbie and any help with the syntax would be much appreciated. Thanks, Attachments: HarmonicAddition...nb

POSTED BY: ok ok

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Gianluca Gorni

Gianluca Gorni, University of Udine

Posted 11 months ago

This is more complicated, but it should work in more cases: convertToCos[a_.Cos[q_] + b_.Sin[q_]] := Sign[a] Sqrt[a^2 + b^2] Cos[q + ArcTan[-b/a]]; convertToCos[b_.Sin[q_]] := -b Cos[q + Pi/2]; convertToCos[a_.Cos[q_]] := a Cos[q]; For example convertToCos[2 Cos[theta] + Sin[theta]] gives Sqrt[5] Cos[theta - ArcTan[1/2]]

This is more complicated, but it should work in more cases:

convertToCos[a_.*Cos[q_] + b_.*Sin[q_]] := 
  Sign[a] Sqrt[a^2 + b^2] Cos[q + ArcTan[-b/a]];
convertToCos[b_.*Sin[q_]] := -b Cos[q + Pi/2];
convertToCos[a_.*Cos[q_]] := a Cos[q];

For example

convertToCos[2 Cos[theta] + Sin[theta]]

gives

Sqrt[5] Cos[theta - ArcTan[1/2]]

POSTED BY: Gianluca Gorni

Bill Nelson

Posted 11 months ago

Does this example help? HarmonicAddition[a_Cos[q_]+b_Sin[q_]]:=Sqrt[a^2+b^2]Sign[a]Cos[ArcTan[-b/a]+q] and then HarmonicAddition[2 Cos[theta]+3 Sin[theta]] returns Sqrt[13]*Cos[theta - ArcTan[3/2]] Please test that in all the ways that you might use that and see if it does what you need.

POSTED BY: Bill Nelson

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