〰️ Viscoelastic Wave Propagation

Waves in a Maxwell viscoelastic medium. Below the crossover frequency ωc = G/η they diffuse; above they propagate. Watch dispersion curves and attenuation change with relaxation time.

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Upper: displacement field · Lower: dispersion diagram · P pause · R reset

How it Works

A sinusoidal displacement is driven at the left boundary. The simulation propagates a 1D shear wave through a Maxwell viscoelastic medium using the finite-difference time-domain method. 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 exponentially (diffusive regime). At high frequencies (ω >> ωc) the real part of k dominates — coherent waves travel at speed c = sqrt(G/ρ) (elastic regime). The lower panel shows the live dispersion curve.

τ = η / G [relaxation time]
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. In viscoelastic media it arises from viscous dissipation. The spatial attenuation coefficient α = Im(k) grows with viscosity.

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.

About this simulation

A finite-difference solver steps stress and velocity across a 300-point 1D grid using the Maxwell constitutive law dσ/dt = G(dv/dx) − σ/τ, driven by a sinusoidal displacement at the left edge. Because relaxation time τ = η/G sets a crossover frequency ωc = G/η, the same medium can look like a diffusing liquid or a ringing elastic solid depending purely on how fast you shake it.

🔬 What it shows

A driven displacement wave in the top panel and a live Re(k)/Im(k) dispersion diagram below, with a marked crossover frequency ωc separating the diffusive regime from the propagating-wave regime.

🎮 How to use

Adjust Shear Modulus G, Viscosity η, Drive Frequency ω and Amplitude A with the sliders; watch the Regime readout flip between Elastic and Viscous, or press P/R to pause/reset.

💡 Did you know?

Silly putty is the classic party trick for this physics: pull it slowly and it flows like a liquid (below ωc), but snap it fast and it shatters like a brittle elastic solid (above ωc).

Frequently asked questions

Why does the Regime readout switch between Elastic and Viscous?

The code compares the drive frequency omega to the crossover wc = G/eta computed each frame; when omega exceeds wc it labels the medium Elastic (wave), otherwise Viscous (diffuse), matching Maxwell-model theory.

What does the orange line overlaid on the displacement plot show?

It plots the internal stress field sigma[i], which the Maxwell constitutive equation evolves separately from displacement, revealing how stress and strain can be out of phase in a viscoelastic medium.

How is the dispersion curve in the lower panel computed?

For each frequency on the x-axis the code evaluates the complex wavenumber from k^2 = rho*omega^2*(1+i/(omega*tau))/G, plotting Re(k) in blue (propagation) and Im(k) in red (attenuation).

Why does increasing viscosity change the crossover frequency?

Relaxation time tau = eta/G grows with viscosity, and since wc = 1/tau, a higher eta value pushes the crossover frequency lower, making the medium behave viscously over a wider frequency range.

What happens at the right edge of the displacement grid?

The code copies the second-to-last grid value into the last one every step, implementing a simple absorbing boundary so waves exit without reflecting back into the domain.