What are phasors and how do they relate to determining the wavelength of a wave?

[Syrus]

Homework Statement

My physics text states: "We can represent a string wave (or any other type of wave) vectorially with a phasor."

Now, for phasors the amplitude and angular frequency are given. If this is the case though, how is k, the angular wave number (and hence the wavelength), determined? The text also later states "[t]his variation corresponds to the sinusoidal variation in the displacement of any point along the string as the wave passes through it."

So does a phasor then represent the wave itself (and there is some means of solving for the wavelength of the wave it describes) or does is only represent the simple harmonic motion of a point of a medium through which a wave passes?

Homework Equations

The Attempt at a Solution

 

[ehild]

A wave traveling along a straight line can be written as A cos (ωt-ks+θ). (s is the position along the straight line.) You know that cosine and sine are the projections of the unit vector to the x-axis and y axis, respectively. A cos (wt-ks+θ) can be represented as a vector of length A making the angle (ωt-ks+θ) with the x axis. As the time changes or the position is different you get a vector with different angle. So the wave can be thought as a rotating vector, a phasor. The phasor represents a wave or vibration as a vector of magnitude equal to the amplitude, and angle equal to the phase. If you have more waves of the same ω and k, but with different amplitudes A and phase constants θ, you can represent them as vectors of different magnitudes inclined with different angles to the x axis, and rotating together. The rotation can happen in time with angular frequency ω or with length, with period λ (k=2π/λ) You can use the phasors to add up vibrations or waves.

ehild