Vector Addition of Sine Waves

How the heck can you use vector addition if they are not vectors? It is really very simple. For example, consider two waves intersecting (combining with each other; they are additive), and the contribution of one is y₁ = 3.0sin(w t) and the other is y₂ = 2.0sin(w t + p /6) where the units of amplitude are arbitrary. They are out of phase by p /6 radians (30^o); if y₂ is at its peak, y₁ will peak at time p /(6w ) later.

Here is the idea that turns a difficult problem into an easy one: Consider two vectors, magnitudes 2.0 and 3.0, angle p /6 radians between them, rotating with the same angular speed w . Their y components (see below) would be y₁ = 3.0sin(w t) and y₂ = 2.0sin(w t + p /6), which is exactly what we want. So we find the vector sum. If the magnitude of the resultant is R and the angle from the 3 vector is f , then our sum is y = Rsin(w t + f ), the y component of the resultant. In this case we get R = 4.8 and f = 12^o = 0.21 rad. I bet there is a way to do this with trig identities, but don't ask me how.

These vectors are called phasors. You can handle any number of waves by this technique, but they must be the same frequency if you want to avoid a big mess. They are useful in the study of AC circuits, also.

(To make it simpler, I put the 3 in the x direction.) R_x = 3 + 2cos30^o = 4.732, R_y = 2sin30^o = 1, R = (R_x² + R_y²)^1/2, and f = ATN(R_y/R_x).

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