Explicit dynamics: Rayleigh surface wave speed in a half-space (Lamb problem)
Explicit dynamics (waves in a continuum)PASS · 1.31%A vertical Ricker force acts on the free surface of an elastic half-space (a plane-strain slab, one HEX8 element thick, absorbing right and bottom faces). The Rayleigh pulse is tracked by three surface geophones; the speed from the cross-correlation lag is compared with the root of the Rayleigh equation (about 0.919 c_s for nu = 0.25) and with the engineering formula c_s(0.862 + 1.14 nu)/(1 + nu).A vertical Ricker force acts on the free surface of an elastic half-space (a plane-strain slab, one HEX8 element thick, absorbing right and bottom faces). The Rayleigh pulse is tracked by three surface geophones; the speed from the cross-correlation lag is com… show full text
- Tolerance:
- 5%
Problem statement
- Geometry
- Slab 1500 × 750 × 5 m, 300 × 150 × 1 HEX8 elements (h = 5 m, 18 elements per Rayleigh wavelength at 10 Hz); symmetry on the x = 0 face and both z faces (plane strain).
- Material
- Soil: E = 5 GPa, nu = 0.25, rho = 2000 kg/m³.
- Boundary conditions
- Two vertical point forces (Ricker 10 Hz) on the top surface at x = 0 (z = 0 and z = h); absorbing right (x = 1500 m) and bottom faces.
- Measured quantity
- Vertical velocity at surface geophones x = 600, 900, 1200 m; cross-correlation lag a→c and a→b in the Rayleigh window.
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=1500000 dy=750000 dz=5000
mesh hex nx=300 ny=150 nz=1
explicit endTime=1.7704565 dtScale=0.8 saveStride=1000000 bulkQ1=0 bulkQ2=0
bc explicit_symmetry surface number 1
bc explicit_symmetry surface number 5
bc explicit_symmetry surface number 6
bc explicit_absorbing surface number 2
bc explicit_absorbing surface number 3
bc explicit_force at=0,750000,0 vector=0,-5e5,0 ricker=10
bc explicit_force at=0,750000,5000 vector=0,-5e5,0 ricker=10
explicit geophone at=600000,750000,0 name=a
explicit geophone at=900000,750000,0 name=b
explicit geophone at=1200000,750000,0 name=c
solve explicit
probe explicit_wave_speed geophone=a geophone2=c component=vy method=xcorr from=0.639449 to=1.52046 id="cR"
probe explicit_wave_speed geophone=a geophone2=b component=vy method=xcorr from=0.639449 to=1.19416 id="cR_ab"
probe explicit_steps id="steps"Results comparison
| Reference value | c_R = 919.402 m/s (root of the Rayleigh equation)approximate formula 917.6 m/s |
| Our result | c_R = 909.447 m/s |
| Deviation | 1.31% |
| Tolerance | 5% |
| Status | PASS · 1.31% |
| Element family | Solid element |
| Mesh details | 300×150×1 HEX8 (plane strain slab, h = 5 m, 990 steps);: webcae_solve (native) |
Updated: 2026-10-22 · version not recorded