What is the Resultant Polarization Angle for Two Perpendicular Harmonic Waves?

[eprparadox]

Homework Statement

Two harmonic transverse waves of the same frequency with displacements at right
angles to each other can be represented by the equations:

y = yo*sin(wt-kx)
z = zo*sin(wt-kx + phi)

where yo and zo are nonzero constants

The equations represent a plane polarized wave if phi equals
(a) sqrt(2)
(b) 3pi/2
(c) pi/2
(d) pi/4
(e) 0

Homework Equations

The Attempt at a Solution

I'm kind of lost. If the two displacements are at a 90 degrees to each other then the phi term will only push one sine out of phase with the other one, but the two waves will still have displacements that are 90 degrees apart. So i have no idea how you can ever get a plane polarized wave since the wave would have to be sitting in one plane and these waves are not. The answers are given so I know the correct one, but I want to understand how to solve.

Thanks a lot.

 

[alphysicist]

Hi eprparadox,

Homework Statement

Two harmonic transverse waves of the same frequency with displacements at right
angles to each other can be represented by the equations:

y = yo*sin(wt-kx)
z = zo*sin(wt-kx + phi)

where yo and zo are nonzero constants

The equations represent a plane polarized wave if phi equals
(a) sqrt(2)
(b) 3pi/2
(c) pi/2
(d) pi/4
(e) 0

Homework Equations

The Attempt at a Solution

I'm kind of lost. If the two displacements are at a 90 degrees to each other then the phi term will only push one sine out of phase with the other one, but the two waves will still have displacements that are 90 degrees apart. So i have no idea how you can ever get a plane polarized wave since the wave would have to be sitting in one plane and these waves are not.

But the idea is that for the correct phi value the combined wave will oscillate in one plane.

For example, let's say that at some particular time, you find the vector sum of the y and z waves, and the resultant wave makes an angle of 30 degrees (just as an example) with the z axis.

For the correct phi value, the vector sum will make the same angle with the z axis for all times, and thus the resultant wave will only oscillate in a plane (a plane that is tilted with respect to the coordinate planes that the individual y and z waves oscillate in).

Does that make sense?

 

[eprparadox]

Thanks so much for the response. I think it makes sense. So the answer is that phi is zero and so if that's true, then the two waves are in phase entirely but just with displacements at 90 degrees to each other.

Does this mean that the plane in which the wave is polarized is at a 45 degree angle to the y-axis or the z-axis?

Thanks again.

 

[alphysicist]

Thanks so much for the response. I think it makes sense. So the answer is that phi is zero and so if that's true, then the two waves are in phase entirely but just with displacements at 90 degrees to each other.

And more importantly, when phi=0 the ratio of the magnitudes of the y and z waves is constant.

Does this mean that the plane in which the wave is polarized is at a 45 degree angle to the y-axis or the z-axis?

No, you find the direction of the resultant the way you would for the vector addition of any perpendicular vectors: by using the inverse tangent. (In fact, that's how you know that the answer is phi=0, because when you take the inverse tangent of z/y, for example, you want the trig functions to cancel.)