Vertical Waves, Parallel Waves, Plane Polarization
Definition
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.
Shaking a string up and down achieves vertical polarization, and the complex wave function is as follows.
$$ \tilde{\mathbf{f}}_\perp (z, t) = \tilde{A} e^{i(kz-\omega t)}\mathbf{\hat{x}} $$
Shaking the string from side to side results in horizontal polarization, and the wave function is as follows.
$$ \tilde{\mathbf{f}}_\parallel (z, t) = \tilde{A} e^{i(kz-\omega t)}\mathbf{\hat{y}} $$
When shaking in any direction $xy-$in the plane $\mathbf{\hat{n}}$, the wave function is
$$ \tilde{\mathbf{f}} = \tilde{A} e^{i(kz-\omega t)}\mathbf{\hat{n}} $$
Here, $\mathbf{\hat{n}}$ is called the polarization vector. It defines the plane in which the wave vibrates. The angle $\theta$ formed by $\mathbf{\hat{n}}$ and $\mathbf{\hat{x}}$ is called the polarization angle. Then, the following holds true.
$$ \mathbf{\hat{n}}=\cos \theta\mathbf{\hat{x}}+\sin \theta \mathbf{\hat{y}} $$
Therefore, a wave vibrating in the direction of $\mathbf{\hat{n}}$ can be represented as the sum of a horizontal wave and a vertical wave.
$$ \mathbf{\tilde{f}}(z,t)=(\tilde{A}\cos\theta)e^{i(kz-\omega t)}\mathbf{\hat{x}}+(\tilde{A}\sin\theta)e^{i(kz-\omega t)}\mathbf{\hat{y}} $$
Such polarization is called linear polarization.