trigonometry
How to express the following equation into a sine wave equation ?
$$S(t) = 4 + 3 \sin 100 \pi t + 5 \sin 200 \pi t$$
I understand that, for the sine wave with same frequency and different amplitude,
I can use the formula
$$\sin (\omega t)+ A_{2}\sin (\omega t)=(A_{1}+A_{2})\sin (\omega t)$$
but how to add sine wave that has different frequency ?
Thanks!
If you want the absolute maximum amplitude and the $\beta$s are such that the peaks can line up, you can just sum the individual waves. They don't line up perfectly very often and less often as more sine waves are included. Another approach is to do a root sum square of the individual amplitudes. This gives you a better feeling for the real amplitudes you will see, but will not be the maximum. If you go to Alpha you can plot it or again. The first has all the peaks lining up at $t=0$, the other has some delays added.
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.
Best Answer
You can't add two sines of different frequencies. This is a fundamental limit which much of math and physics takes advantage of.
Its also the basis of the fourier transform.
To put it simply, sines of different frequencies are orthogonal.