The anti-localization of non-stationary linear waves

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The anti-localization of non-stationary linear waves and its

relation to the localization. The simplest illustrative

problem

Ekaterina V. Shishkinaa, Serge N. Gavrilova,∗, Yulia A. Mochalovaa

aInstitute for Problems in Mechanical Engineering RAS, V.O., Bolshoy pr. 61, St. Petersburg, 199178,

Russia

Abstract

We introduce a new wave phenomenon, which can be observed in continuum and dis-

crete systems, where a trapped mode exists under certain conditions, namely, the anti-

localization of non-stationary linear waves. This is zeroing of the non-localized propa-

gating component of the wave-ﬁeld in a neighbourhood of an inclusion. In other words,

it is a tendency for non-stationary waves to propagate avoiding a neighbourhood of an

inclusion. The anti-localization is caused by a destructive interference of the harmonics

involved into the representation of the solution in the form of a Fourier integral. The

anti-localization is associated with the waves from the pass-band, whereas the localiza-

tion related with a trapped mode is due to poles inside the stop-band. In the framework

of a simple illustrative problem considered in the paper, we have demonstrated that

the anti-localization exists for all cases excepting the boundary of the domain in the

parameter space where the wave localization occurs. Thus, the anti-localization can be

observed in the absence of the localization as well as together with the localization. We

also investigate the inﬂuence of the anti-localization on the wave-ﬁeld in whole.

Keywords: anti-localization, localization, trapped mode, non-stationary waves, linear

waves, vibration, inclusion, defect

∗Corresponding author

Email addresses: shishkina_k@mail.ru (Ekaterina V. Shishkina), serge@pdmi.ras.ru

(Serge N. Gavrilov), yumochalova@yandex.ru (Yulia A. Mochalova)

Preprint submitted to Journal of Sound and Vibration March 7, 2023

arXiv:2210.06736v5 [physics.class-ph] 5 Mar 2023

1. Introduction

It is well known that in an inﬁnite linear (continuum or discrete) almost homoge-

neous system involving a ﬁnite number of inclusions or defects, provided that there is

a stop-band in the dispersion characteristics for the corresponding pure homogeneous

system, one can observe the linear wave localization (see, e.g., studies [1–5] and refer-

ences there). This type of the wave localization is related to the formation of a discrete

part of spectrum of natural frequencies inside a stop-band under certain conditions,

which are fulﬁlled in some domain of the parameter space (we call this the localization

domain). In continuum mechanics the corresponding localized modes are known as the

trapped (or trapping) modes, which have been ﬁrst time discovered by Ursell [6] in the

theory of surface water waves.

In discrete mechanical systems the analogous phenomenon, to the best of our

knowledge, was ﬁrst time described by Montroll and Potts [7], though it was previously

known in physics for non-mechanical systems [8–10]. According to Luongo [11,12],

in physics this type of localization is known as the strong localization. The presence

of frequencies inside the stop-band leads to the possibility to localize non-stationary

waves, i.e., to trap some portions of the wave energy forever near inhomogeneities (in

the absence of dissipation). One can observe undamped localized vibration of an inﬁ-

nite system subjected to an impulse loading. For a discrete mechanical system this was

shown ﬁrst time by Teramoto & Tokeno [13]. For a continuum system Ursell declared

in 1987 [14] that this fact had not been demonstrated, though, in reality, Kaplunov

showed [15] it in 1986 not knowing that the considered system possesses a trapped

mode. The latter fact was discovered later [16,17]. Nowadays, the localization of

non-stationary waves in continuum systems is described in many studies [2,5,18–25].

Trapped modes characterized by natural frequencies inside a stop-band should be

distinguished from so-called embedded trapped modes, see, e.g., review [26]. The

latter ones are characterized by the discrete spectrum of natural frequencies embedded

into the continuous spectrum (i.e., into the pass-band). The localization associated with

the embedded trapped modes is not considered in this paper.

The present paper demonstrates that in the discussed above class of systems (con-

tinuum or discrete), where we can expect the wave localization, a new wave phe-

nomenon can be generally observed. We suggest calling it the anti-localization of non-

stationary linear waves. This is zeroing of the non-localized propagating component

of the wave-ﬁeld in a neighbourhood of an inclusion. In other words, it is a tendency

for non-stationary waves to propagate avoiding a neighbourhood of an inclusion. The

anti-localization is caused by a destructive interference of the propagating harmonics

involved into the representation of the solution in the form of a Fourier integral. The

anti-localization is associated with the waves from the pass-band (including the cut-

off frequency separating the pass-band and the stop-band), whereas the corresponding

localization is due to poles inside the stop-band. In Sect. 2we introduce the simplest

illustrative problem to demonstrate what the anti-localization of non-stationary waves

is, the inﬂuence of the anti-localization to the wave-ﬁeld in whole, and the inﬂuence

of the wave localization to the anti-localization. We show that for the problem under

consideration the anti-localization of non-stationary waves exists in all cases excepting

the boundary of the localization domain. Thus, the anti-localization can be observed in

the absence of the localization as well as together with the localization.

The mechanical system we deal with (an inﬁnite taut string on the Winkler founda-

tion equipped with a discrete mass-spring oscillator) can be considered as an extension

of the system studied in [15]. Note that the identical mechanical system was previ-

ously considered in studies [27,28], where the anti-localization was not discovered.

Our results are in agreement with observations in studies [15,29–33], which deal with

non-stationary oscillation caused by an impulse source at an inclusion in (discrete or

continuum) systems where the localization of non-stationary waves is possible1; see

more details in Sect. 3(Discussion).

The term “anti-localization” in the sense we use it in the paper was introduced by

Shishkina & Gavrilov in recent study [34], though the term “weak anti-localization”

is commonly known in modern quantum physics. According to [35], the weak anti-

localization is a phenomenon observable in disordered systems, which has been pre-

1In the discrete case it is more correct to speak about quasi-waves, since the perturbations propagate at

an inﬁnite speed

dicted in [36]. The term “weak anti-localization”, as far as we know, has been sug-

gested in [37] as a phenomenon opposite to the weak localization predicted in [38,39]

(see also [40]). The latter one is a spatially localized ampliﬁcation of a stationary

wave-ﬁeld composed of the propagating waves from the pass-band [11,12,41]. The

weak localization is observable only in disordered systems [11,35]. It emerges due

to a constructive wave interference at some inhomogeneities, whereas the weak anti-

localization is caused by a destructive interference. In optics [42] and acoustics [43–45]

the weak localization is known as the coherent back-scattering. At the same time, we

have not found any study on the weak anti-localization in acoustics or wave mechan-

ics, although there are may be some. Note that the strong localization, which is related

with waves from the stop-band, can also be observed in disordered systems, see the

Anderson localization [46–48,41].

Thus, the anti-localization of non-stationary waves discussed in this paper and the

weak anti-localization, which is known in quantum physics, have different nature. In-

deed, the latter one is observable only in disordered systems, whereas we consider

ordered deterministic systems. At the same time, the meaning of the term “anti-

localization” is the same as in quantum physics. In both cases one can see zeroing (or,

strictly speaking, asymptotic weakening) of the wave-ﬁeld near some inhomogeneities.

One more physical phenomenon, which should be distinguished from the anti-

localization of non-stationary waves and, therefore, should be referenced here, is the

blocking of running waves [49–55]. The blocking is observable in diffraction prob-

lems at resonant (or “almost” resonant) frequencies in systems where an embedded

trapped mode can exist under certain conditions. Thus, the blocking is a stationary

phenomenon, which is beyond the scope of our paper.

Note that recent paper [56] introduces the term “anti-localization” while consid-

ering an ordered ﬁnite non-linear mechanical system. The term again means zeroing

of the wave-ﬁeld near an inhomogeneity. Since an ordered mechanical system is un-

der consideration, the phenomenon is not a weak anti-localization. Also, it is not an

anti-localization of non-stationary waves, since the stationary deterministic vibration

is under consideration.

2. The illustrative problem and its solution

We consider transverse oscillation of an inﬁnite taut string on the Winkler elastic

foundation. The string is equipped with a discrete mass-spring oscillator, which is

subjected to an impulse loading. The governing equation in the dimensionless form is

u00 −¨u−u=M¨u+Ku −δ(t)δ(x).(1)

Here and in what follows, we denote by prime the derivative with respect to spatial

coordinate xand by overdot the derivative with respect to time t, non-negative con-

stants Mand K(the problem parameters) are the dimensionless mass and the stiffness

characterizing the oscillator. Zero initial conditions are assumed.

The solution can be represented in the form of the following Fourier integral:

u(x, t) = 1

2πZ+∞

−∞

G(x, Ω) exp(−iΩt) dΩ = 1

2πZΩ∗

Gstop(x, Ω) exp(−iΩt) dΩ

2πZ∞

Ω∗

Gpass(x, Ω) exp(−iΩt) dΩ + c.c.=Istop +Ipass + c.c.(2)

Here G(x, Ω) is the corresponding Green function in the frequency domain [23]:

G(x, Ω) = Gstop(x, Ω) def

=exp(−√1−Ω2|x|)

2√1−Ω2+K−MΩ2,Ω∈S;(3)

G(x, Ω) = Gpass(x, Ω) def

=−exp(i√Ω2−1|x|sign(Ω))

2i sign(Ω)√Ω2−1−K+MΩ2,Ω∈P.(4)

Here S= (−Ω∗,Ω∗)and P= (−∞,−Ω∗)∪(Ω∗,∞)are the stop-band and the pass-

band, respectively; Ω∗= 1 is the cut-off frequency, which separates the bands.

The integral Istop describes, in particular, a localized non-vanishing oscillation

that one can observe in the system under certain conditions. Namely, in the case K >

Ma rough estimate Istop =O(t−1) (t→ ∞)can be obtained (the details of the

mathematical technique can be found in Appendix B). In the case

K < M (5)

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Theanti-localizationofnon-stationarylinearwavesanditsrelationtothelocalization.ThesimplestillustrativeproblemEkaterinaV.Shishkinaa,SergeN.Gavrilova,,YuliaA.MochalovaaaInstituteforProblemsinMechanicalEngineeringRAS,V.O.,Bolshoypr.61,St.Petersburg,199178,RussiaAbstractWeintroduceanewwavephenomenon,wh...

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