I came across this notation where the equations were related to waves across a liquid surface as a consequence of moving bodies within that liquid. Of course I'm familiar with dot products, but this notation used:
$$ \mathbf{r}.\mathbf{k} $$
where I've always seen dot products represented as:
$$ r \cdot k $$
Also, given the context that I came across these equations, it's entirely possible that there was some sort of parsing error that resulted in the latex 'dot' being interpreted as a period, but I'm still leaning towards this meaning either a regular dot product or some function that I'm unaware of.
Any help would be phenomenal. Thanks!
This is very likely the standard dot product on $\mathbb{R}^3$. It is common to see $\vec{k} \cdot \vec{r}$ in physics where $\vec{k}$ is the angular wave vector that encodes the wavelength ($\lambda = \frac{2 \pi}{\lvert \vec{k} \rvert}$) and wavefront normal ($\hat{k}$) of a sinusoidal traveling wave. In fact, $\vec{k} \cdot \vec{r}$ should be thought of as the spatial contribution to the phase observed at a position $\vec{r}$ (in general, the phase of a sinusoidal traveling wave is of the form $\vec{k} \cdot \vec{r} - \omega t + \phi$ where $\omega$ is angular frequency, $t$ is time, and $\phi$ is a phase offset). The meaning of the dot product $\vec{k} \cdot \vec{r}$ is readily explained when viewed as $\lvert \vec{k} \rvert \hat{k} \cdot \vec{r}$, where $\hat{k} \cdot \vec{r}$ is the component of the position vector in the direction of phase propagation and $\lvert \vec{k} \rvert$ is the proportionality constant between this distance and the spatial contribution to phase.