---
title: "Seismic Wave Reflection and P-to-S Conversion at an Interface Interactive Lab | Sandbox Physics"
description: "What happens to a P wave when it hits a rock boundary at an angle?"
canonical_url: "https://sandboxphysics.com/labs/seismic-interface-conversion/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/seismic-interface-conversion/"
last_modified: "2026-10-31"
---

FR36-A · Elastic waves at a rock boundary

# Seismic Wave Reflection and P-to-S Conversion at an Interface

**What happens to a P wave when it hits a rock boundary at an angle?** It splits into up to four waves: a reflected P wave, a reflected S wave, a transmitted P wave and a transmitted S wave. Shared horizontal slowness fixes their angles, and the boundary conditions fix how the energy divides. Past a critical angle a transmitted wave stops carrying energy into the lower rock.

A P wave from a buried shot meets a flat boundary between two rocks. Watch it split into reflected and transmitted P and S waves, and drag the source to change the angle.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

Model assumptions

Two elastic half-spaces in welded contact, in two dimensions (P-SV motion only). The wavefield comes from a fourth-order staggered-grid velocity-stress finite-difference solver with an absorbing sponge; the rays, angles and energy fractions come from the exact plane-wave boundary conditions (Zoeppritz equations), solved as a complex linear system of four equations. Energy fractions are vertical energy fluxes normalized by the incident flux; past a critical angle that wave carries no energy away. Rock speeds and densities are assumed, illustrative values. The scene shows P motion in orange and S motion in cyan; curvature of the wavefronts, the head wave and the free surface above the window are not part of the plane-wave coefficients.

Simulated shot record with ray-theory arrival times

- direct P (ray theory)
- reflected P
- converted P to S
- head wave

Energy partition against incidence angle (plane-wave reference)

- reflected P
- reflected S
- transmitted P
- transmitted S
- sum of the four

More readouts and the model record

- **P critical angle**
- **S critical angle**
- **Ray parameter**
- **Transmitted P angle**
- **Reflected S angle**
- **Transmitted S angle**
- **Reflected P amplitude ratio**
- **Reflected P phase**
- **Normal-incidence reflection, impedance formula**
- **Sum of the four energy fractions**
- **Grid points per shortest wavelength**

The record saves the medium pair, the angle, the exact coefficients and the simulated traces.

[Continue the course: seismic imaging from waves to inversion](https://sandboxphysics.com/stories/seismic-imaging/)

Physics tutorial

## Seismic Wave Reflection and P-to-S Conversion at an Interface

**Background** A solid carries two kinds of body wave. P waves squeeze and stretch the rock along their direction of travel; S waves shear it sideways and travel more slowly. A fluid has no shear stiffness, so it carries only P waves.

**Why it matters** Reflection seismology reads rock layers from the echoes of waves that hit boundaries. The amplitude of each echo depends on the angle and on both rocks, so the split into $\mathrm P$ and $\mathrm S$ waves is the first thing to understand.

Start with the essentials

- **Focus question**: What happens to a P wave when it hits a rock boundary at an angle?
- **One-sentence intuition**: All four outgoing waves share the same horizontal slowness, so their angles follow from Snell’s law. How much energy each carries comes from requiring displacement and traction to be continuous at the boundary, and the four energy fractions always add up to one.

### Core mathematical model

#### Snell’s law for P and S waves

$$
p=\frac{\sin i_1}{\alpha_1}=\frac{\sin j_1}{\beta_1}=\frac{\sin i_2}{\alpha_2}=\frac{\sin j_2}{\beta_2}
$$

Here $\alpha$ and $\beta$ are the P and S speeds above (1) and below (2); $i$ are P-ray angles and $j$ S-ray angles, both measured from the normal. A transmitted wave has a critical angle when its speed exceeds the incident P speed.

#### Normal incidence

$$
R=\frac{\rho_2\alpha_2-\rho_1\alpha_1}{\rho_2\alpha_2+\rho_1\alpha_1},\qquad E_{\rm R}+E_{\rm T}=R^2+\frac{\rho_2\alpha_2}{\rho_1\alpha_1}T^2=1
$$

At zero angle no S wave is created, and the reflection follows from the two acoustic impedances alone. The Lab checks its finite-difference wavefield against this value.

#### Energy flux across the boundary

$$
E_k=\frac{\rho_k\,c_k\cos\theta_k}{\rho_1\,\alpha_1\cos i_1}\,\lvert A_k\rvert^2,\qquad \sum_k E_k=1
$$

Outgoing wave $k$ with amplitude ratio $A_k$, speed $c_k$ and angle $\theta_k$ carries this share of the incident energy flux across the boundary. An evanescent wave past its critical angle carries none.

### Common difficulties

#### Amplitude is not energy

**Typical misconception** A transmitted amplitude larger than one means energy was created.

**Better mental model** Energy flux also depends on density, speed and angle. The flux fractions add up to one even when an amplitude ratio exceeds one.

### Run the experiment

1. 01

   #### Cross the critical angle

   Choose Past the critical angle, then drag the incidence angle from 10 degrees up to 40 degrees.

   **What to observe:** Near 30 degrees the transmitted P energy drops to zero and the head wave appears in the shot record. The reflected waves take the energy instead.

2. 02

   #### Make S from P

   Choose P turns into S and switch the wavefield to S only.

   **What to observe:** The incident wave is pure P, yet cyan S fronts leave the hit point both upwards and downwards: they are created at the boundary.

3. 03

   #### Remove the shear from one side

   Choose Water over rock and compare with Softer rock below.

   **What to observe:** With water above, no S wave is reflected, but the rock below still carries a converted S wave. When the lower rock is slower, there is no critical angle.

Physical conclusion

One shared horizontal slowness sets the angles; the boundary conditions set the energy split.

## Related experiments and articles

Seismic imaging · Reading path

### Seismic waves and imaging, from rock boundaries to tomography

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **Why a solid rock carries two kinds of seismic wave**](https://sandboxphysics.com/stories/seismic-imaging/p-and-s-waves/)
2. [Article **Snell’s law and the critical angle for seismic waves**](https://sandboxphysics.com/stories/seismic-imaging/snell-law-and-critical-angles/)
3. [Article **How a P wave shares its energy at a rock boundary**](https://sandboxphysics.com/stories/seismic-imaging/mode-conversion-and-energy-flux/)

Motion & dynamics · Starter path

## From a falling ball to an orbit

Start by measuring gravity and splitting motion into components. Then add contact forces and collisions, and follow the same laws all the way to an orbit.

[Start here → **Free Fall & Measuring Gravity** Capture a few drops and fit the camera positions to estimate gravitational acceleration. Then add air damping and watch the fit drift from the truth. Observe · 14 min](https://sandboxphysics.com/labs/free-fall/)

1. [1 Free Fall & Measuring Gravity](https://sandboxphysics.com/labs/free-fall/)
2. [2 Projectile Flight Designer](https://sandboxphysics.com/labs/projectile-flight/)
3. [3 Inclined Plane & Friction](https://sandboxphysics.com/labs/inclined-plane/)
4. [4 One-Dimensional Collision Bench](https://sandboxphysics.com/labs/collision-bench/)
5. [5 Mass–Spring Oscillator](https://sandboxphysics.com/labs/mass-spring/)
6. [6 How Orbits Take Shape](https://sandboxphysics.com/labs/orbit/)

[View the full path →](https://sandboxphysics.com/topics/mechanics/#starter-path)