Horizontal slowness in seismology

Question

I am working on a seismology assignment and it starts with the fundamentals of seismology such as the wave equation, the slowness... such as the question below:

write an equation describing the horizontal slowness (reciprocal of the velocity). This becomes a vector quantity for propagation in multiple directions.

I haven't fully grasped these concepts yet. And I looked around for lecture notes and this is what I found: MIT-introduction to seismology and

I am not sure which one is the answer to this question or what is even the difference between the two because they both seem to represent the slowness

Answer 1

The slowness vector $\mathbf{s}=[\xi,\eta]$ (your pdf "$\mathbf n=[a,b]$" ) consists of the horizontal component $\xi$ (or your $a$) and vertical component $\eta$ (or your $b$), then we have: \begin{align} s=|\mathbf{s}|=\sqrt{\xi^2+\eta^2}=\frac{1}{c} \end{align} which satisfies Pythagoras law $s^2=\xi^2+\eta^2$. Explicitly, the horizontal and vertical components are related to the ray direction by \begin{align} \sin(\theta)&=\frac{\mathrm ds}{\mathrm dx}=c\frac{\mathrm dt}{\mathrm dx}=\frac{c}{c_x}=c\xi\\ \cos(\theta)&=\frac{\mathrm ds}{\mathrm dy}=c\frac{\mathrm dt}{\mathrm dy}=\frac{c}{c_y}=c\eta. \end{align} Therefore \begin{align} \xi&=\frac{1}{c_x}=\frac{\sin(\theta)}{c}\\ \eta&=\frac{1}{c_y}=\frac{\cos(\theta)}{c}. \end{align}