The actual frequency of a wave that is the sum of sine waves with two different frequencies.

Question

I am just trying to understand something.

Suppose you had two sine waves with two different frequencies, same amplitude and no phase difference.

Freq 1: 1 hz and Freq2: 2 hz and you added them together, how would you calculate the actual frequency of the resultant waves?

Is there a quick formula I can stick it into?

Thanks

Answer 1

Saying no phase difference does not make sense if the frequencies are different. If you have the first rising zero crossings together, then next one will not be. The result will not be a sine wave, so there will not be one frequency. The period will be the least common multiple of the to periods, assuming one exists. You can use the function sum identities to see that you have a product of two sine waves, one at the average of the two frequencies and one at half the difference. In your example you would have the product of a $1.5 Hz$ wave and a $0.5 Hz$ wave.

It is easier to see if the frequencies are closer. The envelope is half the difference frequency and the high frequency is the average.

Answer 2

The period of an oscillation is the reciprocal of the frequency.

If the ratio of the two periods is rational, write it in lowest terms as $a/b$. the sum is periodic with period the least common multiple of $a$ and $b$. Whether you want to call the reciprocal of that period the frequency depends on the application. It won't be the frequency of a pure sine wave even if you began with those.

When the ratio is irrational I think you are out of luck.

Answer 3

I was looking for something else and saw this question. Sorry my answer is very late.

The short answer is that the average frequency of two sine waves of same amplitude but different frequency (phase is irrelevant) is just the average of the two frequencies. Interestingly, if the amplitude is different, the average frequency of the resulting sum wave is just the same as the frequency of the higher amplitude wave.

To do this mathematically, you can do something a little funny (this is what I was working on when I came across this question). If you create an equation for the argument (think phase) of the sum, you can get that with the following equation:

$$\arctan\left(\frac{A\sin(ax)+B\sin(bx)}{A\cos(ax)+B\cos(bx)}\right)$$

The trick is to take the derivative of this, since the derivative of phase is frequency. For the case where $$A=B$$, it reduces to the average of A and B. Otherwise you would need to integrate to get the average frequency. This is easy for rational frequencies but for irrational you have to integrate to infinity... but the rule above should hold.