Disorder of size (polydispersity) and mass of discrete elements or particles in randomly structured media (e.g., granular matter such as soil) has numerous effects on the materials' sound propagation characteristics. The influence of disorder on energy and momentum transport, the sound wave speed and its low-pass frequency-filtering characteristics is the subject of this study. The goal is understanding the connection between the particle-microscale disorder and dynamics and the system-macroscale wave propagation, which can be applied to nondestructive testing, seismic exploration of buried objects (oil, mineral, etc.) or to study the internal structure of the Earth. To isolate the longitudinal P-wave mode from shear and rotational modes, a one-dimensional system of equally sized elements or particles is used to study the effect of mass disorder alone via (direct and/or ensemble averaged) real time signals, signals in Fourier space, energy and dispersion curves. Increase in mass disorder (where disorder has been defined such that it is independent of the shape of the probability distribution of masses) decreases the sound wave speed along a granular chain. Energies associated with the eigenmodes can be used to obtain better quality dispersion relations for disordered chains; these dispersion relations confirm the decrease in pass frequency and wave speed with increasing disorder acting opposite to the wave acceleration close to the source.

Sound wave propagation through matter has been an extensive area of research
(for a textbook example, see

Disordered, heterogeneous and random media cause multiple scattering of
seismic waves that are dispersed, attenuated and localized in space
(

In recent years, wave propagation through granular materials has attracted a
lot of attention. Granular material is heterogeneous with
many discretized units and can be used for other modeling geometrically
heterogeneous media (

A striking characteristic of consolidated granular matter is that grain–grain
forces are arranged and correlated in a linear manner known as force chains
(

The force chains and granular chains which carry the large forces of the
system supposedly support faster sound transmission across granular matter
(

Although it is very simplistic, a polydisperse granular chain can have various kinds of disorder: size, mass or stiffness disorder (

In Sect.

A granular or force chain from a network (schematic).

A one-dimensional chain of

Prestressed chain of granular elements during dynamic wave propagation.

A length scale

The equation of motion for any particle

The repulsive interaction force can be expressed as a power series and can be
expanded about the initial overlap

Using an ansatz for real space and another ansatz for spatial Fourier space
in Eq. (

The initial conditions required to solve various special cases are the
initial displacements (

In wavenumber space (spatial Fourier transform), the initial condition is
specified by

Scaled moments of ensemble-averaged distributions (500 ensembles) used for the one-dimensional chain (256 elements long).

The mass distribution of the monodisperse chain has been selected randomly
from normal (

The participation ratio (

Unscaled moments of ensemble-averaged distributions (500 ensembles) used for the one-dimensional chain (256 elements long).

Moments of ensemble-averaged distributions (10 000 ensembles) used for the one-dimensional chain (256 elements long).

The analytical expression for the dispersion relation in an ordered chain of
particles or elements with linear contact forces are given by
(

From Eq. (

The group velocity is given by

The analytical expressions derived in the previous sections are computed for
chains that are

Equation (

The displacement as a function of time is shown for the 100th
particle in a chain of particles with disorder parameter

In order to quantify the limitations of the linear space–time responses
obtained from Eq. (

The nonlinear increase in

Ensemble-averaged displacements (500 times) of the 150th

Only the mass disorder of the particles in the chain with length 256 is taken
into consideration and

Displacements of 150th

Coherent wave velocities determined through velocity picking. The
peak of the coherent wave packet's velocity

Mechanical waves propagating through disordered media or granular media such
as soil (on the receiver end) can be divided into two parts, the coherent
part and the incoherent part (

Scaled coherent wave velocity picking for different particles before and after localization length for a disordered chain with normal distribution (256 elements long, 500 ensembles).

Tables

Wave speed for common distance of separation (seven particles or
elements) with different disorder

Unscaled coherent wave velocity picking (

Coherent wave velocity calculated from the time taken by the pulse to travel a common distance of separation (seven particles or elements) with time calculated in reference to 5, 10, 70 and 90 % of the peak value and the peak value of the coherent wave packet.

To understand the effect of disorder on wave speed without taking into
account this “source effect”, the velocity based on the time taken by the
pulse for propagating a common distance of seven particles has been computed
in Table

In Fig.

A spatial as well as temporal two-dimensional FFT is carried out for a single
realization of a 256-element-long chain with disorder

Panel

The

Dispersion relation,

Participation ratio or localization length with respect to different
frequencies for 500 ensembles with

Figure

The density of states or density of vibrational modes is an important
quantifying factor in studying the vibrational properties of materials such
as jammed granular media (

An impulse driven wave propagating through a precompressed mass-disordered
granular chain has been studied. Motivation comes from the existence of force
chains which form the backbone network for mechanical wave propagation in
granular materials such as soil. The scaled standard deviation of the mass
probability distribution of the elements or particles of the granular chain has
been identified as the relevant disorder parameter (

Interestingly, on first sight, the dependence of wave speed on magnitude of
disorder looks nonmonotonous. This surprising increase of wave speed for weak
disorder, and decrease for stronger disorder, is due to two different effects
overlapping: the increase of wave speed takes place close to the source (see
Fig.

As another main result, Eq. (

The energy analysis presented in this article can be used for understanding
pulse propagation in disordered, weakly or strongly nonlinear granular chains
and its attenuation, widening and acceleration (experimentally and
numerically investigated in

Data have been generated using the aforementioned theoretical model. The readers can reproduce it by using the equations mentioned in their respective sections.

The energy of the system (chain) can be calculated by vector multiplications
at a particular instance of time, the nonunitary dimension of the vector
gives the respective information of the individual particles. The kinetic
energy of the chain at a particular instant of time is

Multiplicative Factor

If a Hertzian repulsive interaction force is taken into consideration between particles
(

The raw

The unscaled normal distribution is given as

The first scaled moment of the normal distribution is

The Gaussian integral (normalizing condition) can be used, differentiated
with respect to

The unscaled binary distribution is given by

The first scaled moment of the distribution is given as

The second scaled moment of the binary distribution is given as follows:

The unscaled uniform distribution for the mass distribution is given by

The first scaled moment of the distribution is

The second moment of the distribution is given as

From Eqs. (

The manuscript was coauthored by RKS and SL.

The authors declare that they have no conflict of interest.

This article is part of the special issue “Waves in media with preexisting or emerging inhomogeneities and dissipation”. It is a result of the EGU General Assembly 2016, Vienna, Austria, 17–22 April 2016.

This work is part of the Industrial Partnership Programme (IPP) “Computational sciences for energy research” of the Foundation for Fundamental Research on Matter (FOM), which is part of the Netherlands Organisation for Scientific Research (NWO). This research programme is cofinanced by Shell Global Solutions International B.V. We acknowledge support from ESA under contract 4000115113 Soft Matter Dynamics. Edited by: Elena Pasternak Reviewed by: two anonymous referees