Reflection, refraction, and Snell’s law

Prerequisites

Learning objectives

State what happens to energy when a seismic wave meets a layer boundary
Apply Snell’s law to compute the angle of a refracted ray
Explain the critical angle and when total internal reflection occurs
Connect these ideas to what a reflection seismogram records

The whole business of reflection seismology depends on one physical fact: when a wave reaches a boundary between two materials with different acoustic properties, part of its energy bounces back (the reflection) and part continues forward, slightly bent (the transmission, which we also call refraction). What we record at the surface is a long sequence of those echoes — one echo for each boundary the wave met on its way down.

The three things that happen at every boundary

Some energy reflects back at the same angle as the incidence angle (measured from the normal to the boundary). This is what the reflection seismogram records.
Some energy transmits through at a new angle, computed from the velocity contrast. This is the wave that continues downward and may reflect off a deeper boundary.
How the energy splits depends on the contrast in acoustic impedance (density × velocity) across the boundary. Big contrast = bright reflection. Tiny contrast = invisible reflection. We will make this quantitative in §1.2.

For now, focus on the second point — the angle of the transmitted ray. This is governed by Snell’s law, one of the oldest and most reliable statements in physics:

\dfrac{\sin \theta_1}{V_1} = \dfrac{\sin \theta_2}{V_2}

where $\theta_1$ is the incidence angle (measured from the vertical normal to the boundary), $\theta_2$ is the transmitted angle, and $V_1$ , $V_2$ are the wave velocities in the upper and lower media. Rearranging gives the useful working form: $\sin \theta_2 = \sin \theta_1 \cdot \dfrac{V_2}{V_1}$ .

Which way does the ray bend?

If $V_2 > V_1$ (faster lower layer), the transmitted ray bends away from the normal — $\theta_2 > \theta_1$ . The ray spreads out into the faster rock.
If $V_2 < V_1$ (slower lower layer), the transmitted ray bends toward the normal — $\theta_2 < \theta_1$ . The ray steepens.
If $V_2 = V_1$ (no contrast), $\theta_2 = \theta_1$ . No bending, and also no reflection — the boundary is invisible to seismic.

Try the visualizer above. Set $V_1 = 2000$ and $V_2 = 4000$ — a strong soft-over-hard contrast. Slide the incidence angle up from 0°. Notice how the transmitted ray (blue) swings outward much faster than the incident ray (orange). Now keep pushing θ₁ higher. At some angle, the transmitted ray lies along the boundary and beyond that angle, it disappears entirely. That point is called the critical angle.

The critical angle

The critical angle $\theta_c$ is the incidence angle at which the transmitted ray would travel parallel to the boundary ( $\theta_2 = 90^\circ$ ). Setting $\sin \theta_2 = 1$ in Snell’s law:

\sin \theta_c = V_1 / V_2

Beyond the critical angle, all the energy reflects back — this is called total internal reflection. The condition requires $V_2 > V_1$ (the lower layer must be faster), which is why we do not always have a critical angle.

Why does any of this matter for interpretation? Two reasons. First, when we record reflection seismic we usually care about the near-vertical echoes — small incidence angles where rays barely bend. That is why stacked seismic data tries to approximate "normal-incidence" reflectivity. Second, as we push to wider angles (the far offsets in our acquisition), the bending matters more and reflection strength changes with angle. That is AVO (amplitude versus offset), a central topic in Part 5 and the reason angles matter even in post-stack work.

One last piece: when a P-wave hits an inclined boundary, it does not produce just one reflected and one transmitted ray. It produces four — reflected P and S, transmitted P and S. This is mode conversion. We will not compute it in Part 0, but it is a real phenomenon that shows up as weird energy on your seismic when boundaries are steep and the contrast is strong.

References

Sheriff, R. E., & Geldart, L. P. (1995). Exploration Seismology (2nd ed.). Cambridge University Press.
Aki, K., & Richards, P. G. (2002). Quantitative Seismology (2nd ed.). University Science Books.
Yilmaz, Ö. (2001). Seismic Data Analysis (2 vols.). Society of Exploration Geophysicists.
Sheriff, R. E. (2002). Encyclopedic Dictionary of Applied Geophysics. Society of Exploration Geophysicists.