Summing sines of different frequencies

Question

Is there a general formula for solving the following equation: $$A \sin(Bt+C) + D \sin(Et+F) = G \sin(Ht+I)$$

All constants on the left side of the equation are known (t is a variable). Is there a formula for calculating G, H and I? Is this even solvable in general?

I searched the web and found some semi-relevant hits, like here, but it didn't really seem to answer my question.

score 1 · Answer 1 · answered Mar 28 '17 at 23:35

1

I'm not sure about a general formula, but to your answer wether it is solvable in general: the function $sin(3x+4)+sin(6x+7)$ is clearly not harmonic (you can see this by simply plotting it), providing a counterexample to the statement.

answered Mar 28 '17 at 23:35

Tyron

1,212

score 1 · Answer 2 · answered Mar 28 '17 at 23:42

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And, of course, if $B/E$ is irrational, the resulting curve is not even periodic.

answered Mar 28 '17 at 23:42

marty cohen

107,799

score 0 · Accepted Answer · answered Mar 28 '17 at 23:33

0

If you mean $G$, $H$, $I$ to be constant (i.e. independent of $t$), then the answer is No. The sum of sinusoids of different frequencies is never a sinusoid.

answered Mar 28 '17 at 23:33

Robert Israel

448,999

Thanks. Are there solutions if we allow G, H or I to depend on t? – Jens Mar 28 '17 at 23:41
Sure: $G = A \sin(B t + C) + D \sin(Et+F)$, $H = 0$, $I = \pi/2$. – Robert Israel Mar 29 '17 at 00:03

Summing sines of different frequencies

3 Answers3