My question is based on the 1981 paper “Computation of wave fields in inhomogenous media” by Cerveny et.al. (I will add any appropriately needed information to this posted question, so there is no need to obtain the paper).

They search for solutions to the wave equation

$ $ \frac{\partial^2u}{\partial{x^2}} + \frac{\partial^2u}{\partial{z^2}} = \frac{1}{V(x,z)^2}\frac{\partial^2u}{\partial{t^2}}$ $

where $ V(x,z)$ is known. The $ \mathbb{R}^2$ Cartesian coordinates $ (x,z)$ are used. An arbitrary ray $ \Omega \subset \mathbb{R}^2$ is taken, and parameterized by arclength $ s$ increasing from an initial point $ s = 0$ at $ t = 0$ . The real plane is parameterized based on $ \Omega$ by the arclength $ s$ and the unit normal $ n$ off the ray.

For non-straight rays $ \Omega$ there are irregular regions (where different pairs $ (s,n)$ that describe the same point in $ \mathbb{R}^2$ ), and the complement in $ \mathbb{R}^2$ is called the regularity region.

They then obtain the first fundamental form in terms of a small segment $ dr$ of arclength as

$ $ dr^2 = h^2 ds^2 + dn^2$ $

where

$ $ h = 1 + \frac{n}{v}v_n$ $

for $ v(s) = V(s,0)$ and

$ $ v_n = \left[\frac{\partial V(s,n)}{\partial n}\right]_{n=0}.$ $

My question: How did they obtain the first fundamental form? I know that

$ $ dr^2 = F_{ss} ds^2 + 2F_sF_n ds dn + F_{nn} dn^2$ $

where $ F(s,n) = (x,z)$ is the coordinate transformation and $ F_s$ is the first derivative of $ F$ wrt $ s,$ $ F_{ss}$ the second, $ F_{sn}$ the mixed derivative. However, they obtain the first fundamental form with arbitrary array $ \Omega$ .

Let me know if you need more information.

Reference: Červený, Vlastislav, Mikhail M. Popov, and Ivan Pšenčík. “Computation of wave fields in inhomogeneous media—Gaussian beam approach.” Geophysical Journal International 70.1 (1982): 109-128.