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.

Reflected P energy $14.3\,\%$

Reflected S energy $42\,\%$

Transmitted P energy $0\,\%$

Transmitted S energy $43.7\,\%$

Split one P wave into four

Incidence angle (degrees)

i_1

40

Lower rock P speed (m/s)

\alpha_2

4000

Lower rock S speed (m/s)

\beta_2

2200

Wavefield shown

Drag the source or the gold hit point on the interface to change the angle.

Model, units and sources

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

The stage window is 2.5 km wide and 1.5 km deep; rays are drawn to scale, and their widths follow the energy fractions.

Read the course chapter behind this Lab

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

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

BackgroundA 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 mattersReflection 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 misconceptionA transmitted amplitude larger than one means energy was created.

Better mental modelEnergy 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

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.
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.
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.

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

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

Core mathematical model

Snell’s law for P and S waves

Normal incidence

Energy flux across the boundary

Common difficulties

Amplitude is not energy

Run the experiment

Cross the critical angle

Make S from P

Remove the shear from one side

Related experiments and articles

Seismic waves and imaging, from rock boundaries to tomography