Explicit dynamics: reflection and transmission of a P wave at a layer interface

Explicit dynamics (waves in a continuum)PASS · 0.64%

A Ricker pulse crosses the interface between soil and a stiffer rock layer (and in a second run, a softer layer). The reflected and transmitted velocity amplitudes are compared with the impedance contrast: |R_v| = |Z1 - Z2|/(Z1 + Z2), T_v = 2 Z1/(Z1 + Z2), Z = rho c_p; the arrival time of the reflection must equal 2(x_i - x_g)/c_p. The script shows the stiff-layer run; the soft-layer run is executed by the runner.

Tolerance:: 5%

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Problem statement

Geometry: Column 2000 × 2.5 × 2.5 m, 800 HEX8 elements (h = 2.5 m); the interface at x = 900 m is a material boundary between elements.
Material: Layer 1 (x < 900 m): E = 5 GPa, nu = 0.25, rho = 2000. Layer 2: stiff E = 20 GPa, rho = 2500 (Z2/Z1 = 2.24); or soft E = 1.2 GPa, nu = 0.3, rho = 1600 (Z2/Z1 = 0.46).
Boundary conditions: Face x = 0: Ricker 15 Hz pressure 1 MPa plus an absorbing boundary (the source does not re-reflect); face x = 2000 m: absorbing; lateral faces confined.
Measured quantity: Geophones at x = 500 m (incident and reflected pulses, windows around the expected arrivals) and x = 1100 m (transmitted pulse); peak velocities and peak times.

WebCAE Terminal

Paste these commands into the Terminal panel inside WebCAE. They use the same text DSL as the application; run one command per line:

units mm
material E=5000000000 nu=0.25 density=2000
box dx=2000000 dy=2500 dz=2500
mesh hex nx=800 ny=1 nz=1
explicit endTime=1.2005553 dtScale=0.8 saveStride=1000000 bulkQ1=0 bulkQ2=0
explicit layer axis=x at=900000 E=20000000000 nu=0.25 density=2500
bc explicit_pressure surface number 1 value=1e6 ricker=15
bc explicit_absorbing surface number 1
bc explicit_absorbing surface number 2
bc explicit_symmetry surface number 3
bc explicit_symmetry surface number 4
bc explicit_symmetry surface number 5
bc explicit_symmetry surface number 6
explicit geophone at=500000,0,0 name=near
explicit geophone at=1100000,0,0 name=far
solve explicit
probe explicit_geo_peak geophone=near component=vx from=0.288675 to=0.488675 id="inc"
probe explicit_geo_peak_time geophone=near component=vx from=0.288675 to=0.488675 id="tinc"
probe explicit_geo_peak geophone=near component=vx from=0.750555 to=0.950555 id="ref"
probe explicit_geo_peak_time geophone=near component=vx from=0.750555 to=0.950555 id="tref"
probe explicit_geo_peak geophone=far component=vx from=0.584165 to=0.784165 id="tr"

Results comparison

Reference value	R_v = \|Z1−Z2\|/(Z1+Z2), T_v = 2Z1/(Z1+Z2), lag = 2(x_i−x_g)/c_p
Our result	stiff: R=0.3816 (теория 0.382)soft: R=0.3684 (теория 0.366)
Deviation	0.64%
Tolerance	5%
Status	PASS · 0.64%
Element family	Solid element
Mesh details	800×1×1 HEX8 (h = 2.5 m), two material layers (explicit layer), two explicit (stiff and soft layer);: webcae_solve (native)

Updated: 2026-10-22 · version not recorded