# Out of phase

• Nov 28th 2007, 09:51 PM
Atomic_Sheep
Out of phase
To find the phase angle of a simple sin wave we take two orthagonal components the sin and the cos and then find the arctan of the ratio... thats easy enough. My question is this... lets say we have 2 sin waves that we add together that are out of phase and have different periods... is it still possible to find the phase angle of such a function? Is it still going to be increasing by 1 degree for every moment of the cylce (depending on how long the period is of course)?
• Nov 29th 2007, 03:39 AM
topsquark
Quote:

Originally Posted by Atomic_Sheep
To find the phase angle of a simple sin wave we take two orthagonal components the sin and the cos and then find the arctan of the ratio... thats easy enough. My question is this... lets say we have 2 sin waves that we add together that are out of phase and have different periods... is it still possible to find the phase angle of such a function? Is it still going to be increasing by 1 degree for every moment of the cylce (depending on how long the period is of course)?

$\displaystyle sin(A) + sin(B) = 2 sin \left ( \frac{A + B}{2} \right ) cos \left ( \frac{A - B}{2} \right )$

Note that this is no longer a sine wave. (However the "envelope" of the wave is a sine wave.)

-Dan
• Nov 30th 2007, 01:05 AM
Atomic_Sheep
Hmm that didnt really answer my question... in fact it raised another one... I know if you put sin(A) + sin(B) into a computer it can sum them pretty easily... but I just want to do it by hand and haven't been able to do so using the trig identities... I know the identity you have provided but I just want to get to it... just out of interest really.

I'm still not quite sure though of how the algorithm would look like that will enable the computer to know exactly where the extrema of the function would be i.e. of the function sin(A) + sin(B)... essentially using the 2 sin wave summation function... I need to know where the points of extrema are... I'm using 2 sin waves at the moment to just get my heard around it and then I'm going to increase the complexity of it by adding more and more sin waves but the problem of needing to know where the points of extrema are remains. I'm of course assuming that I know both the amplitude and the period of all the sin waves that I plan to sum.

Essentially I'm not even sure whether knowing the phase angle of such a complex function would enable me to know where the extrema are going to be but from the simple example that I've seen where just one sin wave is used thats obviously the easiest way to know where in the period you are and hence pretty easy to determine where the extrema are... but as the number of sin waves in the function increases I'm not sure whether thats going to be enough.
• Nov 30th 2007, 05:24 AM
topsquark
Quote:

Originally Posted by Atomic_Sheep
Hmm that didnt really answer my question... in fact it raised another one... I know if you put sin(A) + sin(B) into a computer it can sum them pretty easily... but I just want to do it by hand and haven't been able to do so using the trig identities... I know the identity you have provided but I just want to get to it... just out of interest really.

I'm still not quite sure though of how the algorithm would look like that will enable the computer to know exactly where the extrema of the function would be i.e. of the function sin(A) + sin(B)... essentially using the 2 sin wave summation function... I need to know where the points of extrema are... I'm using 2 sin waves at the moment to just get my heard around it and then I'm going to increase the complexity of it by adding more and more sin waves but the problem of needing to know where the points of extrema are remains. I'm of course assuming that I know both the amplitude and the period of all the sin waves that I plan to sum.

Essentially I'm not even sure whether knowing the phase angle of such a complex function would enable me to know where the extrema are going to be but from the simple example that I've seen where just one sin wave is used thats obviously the easiest way to know where in the period you are and hence pretty easy to determine where the extrema are... but as the number of sin waves in the function increases I'm not sure whether thats going to be enough.

If, for example, you have the function
$\displaystyle f(x) = sin(x) + sin(x + 1)$
then you simply take the first derivative, set it equal to 0, and solve for x.

There is a nice neat formula for summing an arbitrary number of sine waves, but what it is momentarily escapes me. I do know that if you are summing a series of sine waves with a spectrum of frequencies that they form "wave-packets" (ie. the envelope is a sine wave form and the function oscillates between the two sides of the envelope), but I'm not certain what happens in regard to the phase. It may well be that such an arbitrary problem can't be done in general. But since you've got the computer anyway, you should be able to program the computer how to do the problem numerically.

-Dan
• Nov 30th 2007, 02:59 PM
Atomic_Sheep
Taking a derivative is a good idea... didn't think of that (Doh)... I know what envelopes your talking about but I dont think they are much help in my case. And I'm pretty sure I know what you're talking about in regards to solving the problem numerically. I'll play around with these things see what works and what doesnt... thanks.