Modeling the Dispersion of Waves in a Multilayered Inhomogeneous Membrane with Fractional-order Infusion

Ali Rab^*
*Correspondence: Ali Rab, Department of Mathematics and Statistics, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia, Email:

Introduction

The study of wave dispersion in materials is a crucial aspect of material
science, particularly for designing structures that need to absorb, transmit,
or dissipate mechanical waves. In the context of membranes, such as those
used in membranes for medical devices, musical instruments, or engineered
acoustic materials, the dispersion of waves plays a significant role in
understanding how waves propagate through different layers with varying
properties. A multilayered membrane with inhomogeneous material properties
adds another layer of complexity to the problem, which is particularly
challenging to model using traditional methods. A relatively new and exciting
avenue of research in wave propagation is the inclusion of fractional calculus
to describe materials that exhibit non-local and memory effects. This is
especially relevant when dealing with materials that show complex, nonlinear,
or viscoelastic behavior. In this article, we explore the modeling of wave
dispersion in multilayered inhomogeneous membranes with fractional-order
infusion, providing a comprehensive overview of the theoretical framework,
mathematical modeling, and potential applications of this approach [1-3].

Description

Fractional calculus is a powerful mathematical tool that generalizes
traditional calculus by incorporating non-integer derivatives and integrals. It
is particularly useful for modeling systems that exhibit memory or hereditary
effects, such as viscoelastic materials, anomalous diffusion, and systems
with non-local interactions. In the case of wave propagation, fractional-order
differential equations can be employed to model damping, dispersion, and
other complex behaviors that are difficult to describe with traditional integerorder
models. For instance, in viscoelastic materials, the relationship between
stress and strain is often better described by a fractional order derivative
rather than a simple linear model. This fractional approach introduces a
non-local element into the material behavior, where the response at a given
point depends not only on the current state but also on past states, which is
reflective of memory effects. In the context of wave propagation, fractionalorder
infusion refers to modifying the standard wave equation by incorporating
fractional derivatives in time or space. For example, in viscoelastic layers of a
membrane, a fractional derivative with respect to time may be used to account
for the memory effect in the material response, and a fractional derivative with
respect to space could capture the inhomogeneities or complex structures
of the medium. The dispersion relation, which characterizes how wave
frequency and wavenumber kkk are related in the system, can be derived from
the modified wave equation. In a fractional medium, the dispersion relation
will depend on both the material properties and the fractional orders, and it will
show different characteristics from traditional wave propagation in integerorder
media [4,5].

Conclusion

Modeling wave dispersion in multilayered inhomogeneous membranes
with fractional-order infusion provides a powerful framework for analyzing
complex wave propagation in materials with non-local or memory effects.
By incorporating fractional calculus into the wave equations, it becomes
possible to describe the intricate behavior of waves in systems that exhibit
viscoelasticity, anomalous dispersion, and complex material inhomogeneities.
This approach holds great promise for a range of applications, from acoustic
metamaterials to medical devices, seismic studies, and non-destructive
testing, offering more accurate models for real-world materials and structures
that traditional models cannot fully capture. The use of fractional-order models
is an exciting frontier in material science, and its potential for revolutionizing
wave propagation theory and its applications is immense.

Acknowledgement

None.

Conflict of Interest

None.

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