Explicit dynamics: spherical P wave from a smooth point source, absolute amplitude and 1/r decay

Explicit dynamics (waves in a continuum)PASS · 4.59%

A curl-free Gaussian body-force field (a smooth centre of expansion) radiates a pure P wave in an octant model with three symmetry planes and three absorbing faces. Peak radial displacement on the axis at four distances is compared with the closed-form exterior field of the equivalent point source u_r = -Q/(4 pi M) [s'(tau)/(c r) + s(tau)/r^2], Q = A (2 pi)^{3/2} sigma^3, where the Ricker source function is filtered by the Fourier transform of the Gaussian (an equivalent Ricker with a_eff). The near-field term and the 1/r decay are included; nothing is fitted.

Tolerance:: 5%

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Problem statement

Geometry: Octant 600 × 600 × 600 m, 80³ HEX8 elements (h = 7.5 m, 29 cells per wavelength at the Ricker peak 8 Hz); symmetry planes x = y = z = 0; absorbing faces x = y = z = 600 m.
Material: Soil: E = 5 GPa, nu = 0.25, rho = 2000 kg/m³.
Boundary conditions: Radial source A = 1e5 N/m², sigma = 15 m (2 cells), Ricker 8 Hz at the origin (octant of the full-space source).
Measured quantity: Geophones on the x axis at r = 150, 262.5, 375, 450 m; peak |u_x|.

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=600000 dy=600000 dz=600000
mesh hex nx=80 ny=80 nz=80
explicit endTime=0.68462813 dtScale=0.8 saveStride=1000000 bulkQ1=0 bulkQ2=0
bc explicit_symmetry surface number 1
bc explicit_symmetry surface number 3
bc explicit_symmetry surface number 5
bc explicit_absorbing surface number 2
bc explicit_absorbing surface number 4
bc explicit_absorbing surface number 6
bc explicit_source at=0,0,0 sigma=15000 value=100000 ricker=8
explicit geophone at=150000,0,0 name=r0
explicit geophone at=262500,0,0 name=r1
explicit geophone at=375000,0,0 name=r2
explicit geophone at=450000,0,0 name=r3
solve explicit
probe explicit_geo_peak geophone=r0 component=ux id="u0"
probe explicit_geo_peak geophone=r1 component=ux id="u1"
probe explicit_geo_peak geophone=r2 component=ux id="u2"
probe explicit_geo_peak geophone=r3 component=ux id="u3"
probe explicit_steps id="steps"

Results comparison

Reference value	u_x peak at r = 150/262.5/375/450 m: 1.220e-56.681e-64.602e-63.812e-6 m
Our result	1.276e-56.929e-64.755e-63.928e-6 m
Deviation	4.59%
Tolerance	5%
Status	PASS · 4.59%
Element family	Solid element
Mesh details	80³ = 512000 HEX8 (octant 600 m, h = 7.5 m, 3 symmetry planes + 3 absorbing faces), 256 steps;: webcae_solve (native)

Updated: 2026-10-22 · version not recorded