# Amplitude of sine wave

1. Apr 20, 2012

### jeffy

Hi,

I want to calculate the amplitude of the following:

V(t) = Acos(2wt)+Bcos(3wt)+Ccos(4wt)+Dcos(5wt)

Not really sure what approach I can take to do this. I was thinking of using some combination of the double angle formula to simplify each part, but I don't think that will necessarily help me find the amplitude of V(t)

suggestions?

tia

edit: wanted to add that each of the cosines actually has an associated phase, but for demonstration purposes, i did not include them here.

2. Apr 22, 2012

### a4b3c2d1e0f

Conceptually, for two terms you would have:

f(t) = A*cos(awt + @) + B*cos(bwt + ß)
Take the derivative to find the extreme and equate to 0:
d/dt f(t) =
-A*sin(awt + @)*aw -B*sin(bwt + ß)*bw = 0
Divide both sides by -w and rearrange
-(A*a)/(B*b)*sin(awt + @) = sin(bwt + ß)
Now you have a nonlinear equation which you have to solve for 't' .
I guess one possible way would be to use series expansion for sin, although i'd rather do 2nd order expansion for cos, by modifying these sines to cos, and doing the expansion around several points, something of the form:
http://www.wolframalpha.com/input/?i=series+expansion+A*cos(x+a)+around+x=p+order+2
where you can see that you'd then be dealing with 2nd order polynomials.
I guess its a pain doing all that analytically, so I'd rather suggest making some code that'd simply find the amplitude, by checking several points of this summed up function in 1 period.

3. Apr 22, 2012

### chiro

If your codomain for t is the whole positive real line, then it would be a lot easier to basically treat it like a number theory problem.

What I mean by this is that your maximum for the cos will happen when the argument = 2npi where n is an integer. Now what you can do is find when all of them meet this requirement at the same point.

This is the same as factorizing the arguments. Lets multiply firstly the coeffecients of the t terms which are 2w,3w,4w,5w which gives us 2wx3wx4wx5w = 90w^4. Now we will need 90w^4t to be 2npi for some integer n. This needs to be solved taking into account the nature of w considering if it is rational, irrational, transcendental and so on.

Lets consider that w = 2 for the moment. Then 90x2^4t = 2npi. Let t = pi, then we get 90x8 = n = 720. If we plug these in to your equation we get a value of A+B+C+D at that point which is one maximum value. You could do the same sort of thing to find when you get -(A+B+C+D) at a different set of points.

So in other words, what I am saying is that if you can factor the arguments and get a solution then you can show that the amplitude is always the sum of the amplitudes and this should always happen if you are considering sums of cosines or sines over a real domain.

If you want to find the argument, this is a different matter though.