logo

Vertical Waves, Parallel Waves, Plane Polarization 📂Physics

Vertical Waves, Parallel Waves, Plane Polarization

Definition

5.JPG

A wave whose direction of propagation and direction of vibration are perpendicular to each other is called a transverse wave. Conversely, a wave whose direction of propagation and direction of vibration are parallel to each other is called a longitudinal wave.

Explanation

The phenomenon of a wave vibrating in a specific direction is called polarization. Since there are two directions perpendicular to the direction of propagation for a transverse wave, it can be said to have two states of polarization.

6.JPG

Shaking a string up and down achieves vertical polarization, and the complex wave function is as follows.

f~⊥(z,t)=A~ei(kz−ωt)x^ \tilde{\mathbf{f}}_\perp (z, t) = \tilde{A} e^{i(kz-\omega t)}\mathbf{\hat{x}}

Shaking the string from side to side results in horizontal polarization, and the wave function is as follows.

f~∥(z,t)=A~ei(kz−ωt)y^ \tilde{\mathbf{f}}_\parallel (z, t) = \tilde{A} e^{i(kz-\omega t)}\mathbf{\hat{y}}

7.JPG

When shaking in any direction xy−xy-in the plane n^\mathbf{\hat{n}}, the wave function is

f~=A~ei(kz−ωt)n^ \tilde{\mathbf{f}} = \tilde{A} e^{i(kz-\omega t)}\mathbf{\hat{n}}

Here, n^\mathbf{\hat{n}} is called the polarization vector. It defines the plane in which the wave vibrates. The angle θ\theta formed by n^\mathbf{\hat{n}} and x^\mathbf{\hat{x}} is called the polarization angle. Then, the following holds true.

n^=cos⁡θx^+sin⁡θy^ \mathbf{\hat{n}}=\cos \theta\mathbf{\hat{x}}+\sin \theta \mathbf{\hat{y}}

Therefore, a wave vibrating in the direction of n^\mathbf{\hat{n}} can be represented as the sum of a horizontal wave and a vertical wave.

f~(z,t)=(A~cos⁡θ)ei(kz−ωt)x^+(A~sin⁡θ)ei(kz−ωt)y^ \mathbf{\tilde{f}}(z,t)=(\tilde{A}\cos\theta)e^{i(kz-\omega t)}\mathbf{\hat{x}}+(\tilde{A}\sin\theta)e^{i(kz-\omega t)}\mathbf{\hat{y}}

Such polarization is called linear polarization.