Physlib.ClassicalMechanics.WaveEquation.HarmonicWave

For a non-zero $d$-dimensional wave vector $\mathbf{k}$, this definition computes its direction as a unit vector. Given $\mathbf{k} \in \text{WaveVector } d$ and the condition $h: \mathbf{k} \neq 0$, the direction is obtained by normalizing the vector: \[ \mathbf{\hat{k}} = \frac{\mathbf{k}}{\|\mathbf{k}\|} \] where $\|\mathbf{k}\|$ is the Euclidean norm in $\text{Space } d$.

The function `harmonicWave` defines a time-harmonic wave in a $d$-dimensional space. Given an amplitude function $a(\omega, r)$, a phase function $g(\omega, r)$, a dispersion relation $\omega(k)$ that maps a wave vector $k$ to an angular frequency, and a specific wave vector $k$, the value of the wave at time $t$ and position $r$ is given by: $$a(\omega(k), r) \cos(\omega(k) t - g(\omega(k), r))$$ where $a, g: \mathbb{R} \to \mathbb{R}^d \to \mathbb{R}$ represent the frequency-dependent amplitude and phase offset respectively, and $\omega: \text{WaveVector } d \to \mathbb{R}$ determines the angular frequency from the wave vector.

The function `transverseHarmonicPlaneWave` defines a monochromatic, time-harmonic plane wave propagating in the $z$-direction within a 3-dimensional space. Given a wave vector $\mathbf{k} = (0, 0, \frac{\omega}{c})$, amplitudes $f_{0x}$ and $f_{0y}$, angular frequency $\omega$, and phase offsets $\delta_x$ and $\delta_y$, the wave at time $t$ and position $\mathbf{r}$ is represented as a vector in $\mathbb{R}^3$: $$\mathbf{\psi}(t, \mathbf{r}) = f_{0x} \cos(\omega t - \mathbf{k} \cdot \mathbf{r} + \delta_x) \mathbf{e}_x + f_{0y} \cos(\omega t - \mathbf{k} \cdot \mathbf{r} + \delta_y) \mathbf{e}_y$$ where $\mathbf{e}_x$ and $\mathbf{e}_y$ are the unit vectors along the $x$ and $y$ axes respectively, and $\mathbf{k} \cdot \mathbf{r}$ is the standard inner product in $\mathbb{R}^3$. The wave is transverse because its oscillations occur in the $xy$-plane, perpendicular to the direction of propagation $\mathbf{k}$.

Let $c > 0$ be the wave speed and $\omega > 0$ be the angular frequency. Let the wave vector be $\mathbf{k} = (0, 0, \frac{\omega}{c}) \in \mathbb{R}^3$, which points in the $z$-direction. For any amplitudes $f_{0x}, f_{0y}$ and phase offsets $\delta_x, \delta_y$, the transverse harmonic plane wave $\mathbf{\psi}(t, \mathbf{r})$, defined as: $$\mathbf{\psi}(t, \mathbf{r}) = f_{0x} \cos(\omega t - \mathbf{k} \cdot \mathbf{r} + \delta_x) \mathbf{e}_x + f_{0y} \cos(\omega t - \mathbf{k} \cdot \mathbf{r} + \delta_y) \mathbf{e}_y$$ is equal to a general plane wave with speed $c$ and direction $\mathbf{\hat{k}} = (0, 0, 1)$, whose profile function $\mathbf{f} : \mathbb{R} \to \mathbb{R}^3$ is given by: $$\mathbf{f}(p) = f_{0x} \cos\left(-\frac{\omega}{c}p + \delta_x\right) \mathbf{e}_x + f_{0y} \cos\left(-\frac{\omega}{c}p + \delta_y\right) \mathbf{e}_y$$ where $\mathbf{e}_x$ and $\mathbf{e}_y$ are the unit vectors along the $x$ and $y$ axes respectively. This shows that the transverse harmonic representation is equivalent to the general plane wave expression under the dispersion relation $\|\mathbf{k}\| = \omega/c$.