How it Works
A sinusoidal displacement is driven at the left boundary. The simulation propagates a 1D shear wave through a Maxwell viscoelastic medium with a staggered velocity–stress finite-difference scheme (Virieux 1986): velocity lives on the grid nodes and stress on the half-nodes, so both spatial derivatives are centred over a single cell and the odd/even checkerboard mode of a collocated stencil cannot appear. The sub-step obeys the CFL condition c·Δt/Δx ≤ 1 with c = sqrt(G/ρ), and the relaxation term σ/τ is integrated exactly, so the scheme stays stable even when τ is much shorter than a time step. The Maxwell model gives a frequency-dependent complex shear modulus G*(ω) = G·iωτ/(1+iωτ), which determines the complex wave number k(ω) = ω·sqrt(ρ/G*(ω)).
At low frequencies (ω << ωc = 1/τ) the imaginary part of k dominates — disturbances decay within a fraction of a wavelength (diffusive regime). At high frequencies (ω >> ωc) the real part of k dominates — coherent waves travel at a phase speed ω/Re(k) that approaches c = sqrt(G/ρ) (elastic regime). The domain is 20 length units long, which at the default settings is about three wavelengths and 4.3 attenuation lengths, so the wave visibly decays as it crosses the medium instead of dying at the source. The dashed red marker on the displacement panel is 1/Im(k), where the envelope has fallen to 1/e. The lower panel shows the live dispersion curve.
ω_c = 1/τ = G/η [crossover frequency]
G*(ω) = G · iωτ / (1 + iωτ)
k = ω √(ρ / G*(ω))
Frequently Asked Questions
What is a viscoelastic material?
A viscoelastic material exhibits both viscous (liquid-like) and elastic (solid-like) behaviour depending on the timescale of deformation. Examples include polymer melts, biological gels, and silly putty.
What is the Maxwell viscoelastic model?
The Maxwell model represents viscoelasticity as a spring (elastic modulus G) and dashpot (viscosity η) in series. Stress relaxes exponentially with relaxation time τ = η/G.
What is relaxation time in viscoelasticity?
Relaxation time τ = η/G is the time for stress to decay to 1/e of its initial value after an imposed step strain. It separates the elastic (fast) regime from the viscous (slow) regime.
Why do waves diffuse at low frequency in a Maxwell fluid?
At frequencies below ωc = G/η = 1/τ, the material has time to relax and behaves as a viscous liquid. Disturbances decay diffusively rather than propagating as coherent waves.
Why do waves propagate at high frequency in a Maxwell fluid?
At frequencies above ωc, the material cannot relax before the wave passes and behaves elastically, supporting propagating shear waves at speed c = sqrt(G/ρ).
What is the dispersion relation for viscoelastic waves?
For a Maxwell fluid: k² = ρω²/G·(1 + i/(ωτ)). The real part gives phase speed, the imaginary part gives spatial attenuation.
What is attenuation in wave propagation?
Attenuation is the decay of wave amplitude with distance: the amplitude falls as exp(−αx) with α = Im(k). In a Maxwell fluid raising the viscosity η lengthens the relaxation time τ = η/G, which makes the medium more elastic, so α falls and the attenuation length 1/Im(k) → 2τ·sqrt(G/ρ) grows. (It is the Kelvin–Voigt model, with its dashpot in parallel, whose attenuation grows with viscosity.) The Atten. length 1/Im(k) readout shows this directly.
Why is the phase speed not simply sqrt(G/ρ)?
sqrt(G/ρ) is the unrelaxed (infinite-frequency) shear-wave speed, and the Maxwell medium only reaches it as ωτ → ∞. At finite ωτ the medium is softer than G, so the true phase speed is ω/Re(k) with k from the Maxwell dispersion relation, and it falls below sqrt(G/ρ) — approaching sqrt(2ωη/ρ) in the diffusive limit. The Phase speed readout reports ω/Re(k) itself, the same quantity plotted in the dispersion panel.
What are practical examples of viscoelastic wave effects?
Viscoelastic wave phenomena are important in seismic attenuation in sedimentary rock, ultrasound in biological tissue, shock absorption in polymer foams, and vibration damping in composite structures.
What is the Deborah number?
The Deborah number De = τ/tobs compares the material relaxation time to the observation timescale. De >> 1 means elastic behaviour; De << 1 means viscous behaviour.
How does the Kelvin-Voigt model differ from the Maxwell model?
The Kelvin-Voigt model places spring and dashpot in parallel, giving a creep response but no stress relaxation. The Maxwell model places them in series, giving stress relaxation but unlimited creep.