Low-frequency Expansion Research Articles

Expansions for semiclassical conformal blocks

We propose a relation the expansions of regular and irregular semiclassical conformal blocks at different branch points making use of the connection between the accessory parameters of the BPZ decoupling equations to the logarithm derivative of isomonodromic tau functions. We give support for these relations by considering two eigenvalue problems for the confluent Heun equations obtained from the linearized perturbation theory of black holes. We first derive the large frequency expansion of the spheroidal equations, and then compare numerically the excited quasi-normal mode spectrum for the Schwarzschild case obtained from the large frequency expansion to the one obtained from the low frequency expansion and with the literature, indicating that the relations hold generically in the complex modulus plane.

Probing de Sitter from the horizon

In a QFT on de Sitter background, one can study correlators between fields pushed to the future and past horizons of a comoving observer. This is a neat probe of the physics in the observer’s causal diamond (known as the static patch). We use this observable to give a generalization of the quasinormal spectrum in interacting theories, and to connect it to the spectral density that appears in the Källén-Lehmann expansion of dS correlators. We also introduce a finite-temperature effective field theory consisting of free bulk fields coupled to a boundary. In matching it to the low frequency expansion of correlators, we find positivity constraints on the EFT parameters following from unitarity.

On a Novel Algorithmic Determination of Acoustic Low Frequency Coefficients for Arbitrary Impenetrable Scatterers

The calculation of low frequency expansions for acoustic wave scattering has been under thorough investigation for many decades due to their utility in technological applications. In the present work, we revisit the acoustic Low Frequency Scattering theory, and we provide the theoretical framework of a new algorithmic procedure for deriving the scattering coefficients of the total pressure field, produced by a plane wave excitation of an arbitrary, convex impenetrable scatterer. The proposed semi-analytical procedure reduces the demands for computation time and errors significantly since it includes mainly algebraic and linear integral operators. Based on the Atkinson–Wilcox theorem, any order low frequency scattering coefficient can be calculated, in finite steps, through algebraic operators at all steps, except for the last one, where a regular Fredholm integral equation with a continuous and separable integral kernel is needed to be solved. Explicit, ready to use formulae are provided for the first three low frequency scattering coefficients, demonstrating the applicability of the algorithm. The validation of the obtained formulae is demonstrated through recovering of the well-known analytical results for the case of a radially symmetric scatterer.

Spin-dependent gravitational tail memory in D=4

We derive the leading spin-dependent gravitational tail memory, which appears at the second post-Minkowskian (2 PM) order and behaves as $u^{-2}$ for large retarded time $u$. This result follows from classical soft graviton theorem at order $\omega\ln\omega$ as a low-frequency expansion of gravitational waveform with frequency $\omega$. First, we conjecture the result from the classical limit of quantum soft graviton theorem up to sub-subleading order in soft expansion and then we derive it for a classical scattering process without any reference to the soft graviton theorem. The final result of the gravitational waveform in the direct derivation completely agrees with the conjectured result.

Tidal response from scattering and the role of analytic continuation

The tidal response of a compact object is a key gravitational-wave observable encoding information about its interior. This link is subtle due to the nonlinearities of general relativity. We show that considering a scattering process bypasses challenges with potential ambiguities, as the tidal response is determined by the asymptotic in- and outgoing waves at null infinity. As an application of the general method, we analyze scalar waves scattering off a nonspinning black hole and demonstrate that the low-frequency expansion of the tidal response reproduces known results for the Love number and absorption. In addition, we discuss the definition of the response based on gauge-invariant observables obtained from an effective action description, and clarify the role of analytic continuation for robustly (i) extracting the response and the physical information it contains, and (ii) distinguishing high-order post-Newtonian corrections from finite-size effects in a binary system. Our work is important for interpreting upcoming gravitational-wave data for subatomic physics of ultradense matter in neutron stars, probing black holes and gravity, and looking for beyond-standard-model fields.

Low-frequency scattering defined by the Helmholtz equation in one dimension

The Helmholtz equation in one dimension, which describes the propagation of electromagnetic waves in effectively one-dimensional systems, is equivalent to the time-independent Schrödinger equation. The fact that the potential term entering the latter is energy-dependent obstructs the application of the results on low-energy quantum scattering in the study of the low-frequency waves satisfying the Helmholtz equation. We use a recently developed dynamical formulation of stationary scattering to offer a comprehensive treatment of the low-frequency scattering of these waves for a general finite-range scatterer. In particular, we give explicit formulas for the coefficients of the low-frequency series expansion of the transfer matrix of the system which in turn allow for determining the low-frequency expansions of its reflection, transmission, and absorption coefficients. Our general results reveal a number of interesting physical aspects of low-frequency scattering particularly in relation to permittivity profiles having balanced gain and loss.

Low-Frequency Expansion Approach for Seismic Data Based on Compressed Sensing in Low SNR

Low-frequency information can reflect the basic trend of a formation, enhance the accuracy of velocity analysis and improve the imaging accuracy of deep structures in seismic exploration. However, the low-frequency information obtained by the conventional seismic acquisition method is seriously polluted by noise, which will be further lost in processing. Compressed sensing (CS) theory is used to exploit the sparsity of the reflection coefficient in the frequency domain to expand the low-frequency components reasonably, thus improving the data quality. However, the conventional CS method is greatly affected by noise, and the effective expansion of low-frequency information can only be realized in the case of a high signal-to-noise ratio (SNR). In this paper, well information is introduced into the objective function to constrain the inversion process of the estimated reflection coefficient, and then, the low-frequency component of the original data is expanded by extracting the low-frequency information of the reflection coefficient. It has been proved by model tests and actual data processing results that the objective function of estimating the reflection coefficient constrained by well logging data based on CS theory can improve the anti-noise interference ability of the inversion process and expand the low-frequency information well in the case of a low SNR.

Direct measurement of unsteady microscale Stokes flow using optically driven microspheres

We present the first direct measurement of the orbits of microscale passive tracers in the unsteady flow created by an oscillating microsphere. Using a time-shared optical trap to position tracers in several concentric arcs around a driven central sphere, we find that the tracers exhibit elliptical Lissajous figures whose orientation and aspect ratio are in quantitative agreement with a scaling law that arises from a low-frequency expansion of the underlying unsteady Stokes equations. The implications of these results for the understanding of metachronal waves in ciliated microorganisms are discussed.

Soft radiation from scattering amplitudes revisited

We apply the recently developed formalism by Kosower, Maybee and O’Connell (KMOC) [12] to analyse the soft electromagnetic and soft gravitational radiation emitted by particles without spin in D ≥ 4 dimensions. We use this formalism in conjunction with quantum soft theorems to derive radiative electro-magnetic and gravitational fields in low frequency expansion and upto next to leading order in the coupling. We show that in all dimensions, the classical limit of sub-leading soft (photon and graviton) theorems is consistent with the classical soft theorems proved by Sen et al. in a series of papers. In particular in [11] Saha, Sahoo and Sen proved classical soft theorems for electro-magnetic and gravitational radiation in D = 4 dimensions. For the class of scattering processes that can be analyzed using KMOC formalism, we show that the classical limit of quantum soft theorems is consistent with the D = 4 classical soft theorems, paving the way for their proof from scattering amplitudes.

Periodic forcing of a large turbulent separation bubble

Abstract

Measurements of pressure and velocity fluctuations in a family of turbulent separation bubbles

Abstract

Open quantum systems and Schwinger-Keldysh holograms

We initiate the study of open quantum field theories using holographic methods. Specifically, we consider a quantum field theory (the system) coupled to a holographic field theory at finite temperature (the environment). We investigate the effects of integrating out the holographic environment with an aim of obtaining an effective dynamics for the resulting open quantum field theory. The influence functionals which enter this open effective action are determined by the real-time (Schwinger-Keldysh) correlation functions of the holographic thermal environment. To evaluate the latter, we exploit recent developments, wherein the semiclassical gravitational Schwinger-Keldysh saddle geometries were identified as complexified black hole spacetimes. We compute real-time correlation functions using holographic methods in these geometries, and argue that they lead to a sensible open effective quantum dynamics for the system in question, a question that hitherto had been left unanswered. In addition to shedding light on open quantum systems coupled to strongly correlated thermal environments, our results also provide a principled computation of Schwinger-Keldysh observables in gravity and holography. In particular, these influence functionals we compute capture both the dissipative physics of black hole quasi- normal modes, as well as that of the fluctuations encoded in outgoing Hawking quanta, and interactions between them. We obtain results for these observables at leading order in a low frequency and momentum expansion in general dimensions, in addition to determining explicit results for two dimensional holographic CFT environments.

Investigation on measurement method for hydrodynamic noise in centrifugal pumps using reverberation tanks

Pumps are considered as a primary source of acoustic energy in industrial piping. At present, the measurement of the hydrodynamic noise of the pump is done by installing the sensors in the pipeline attached to the pump. This technique is based on the fact that plane waves are formed inside the pipeline and the pulsating pressure travels in the form of sound waves. However, in actual there are no plane waves in an elastic pipe due to the phenomenon of low frequency cut-off. In fact, this pulsating pressure inside the pipeline will result in further vibration of the pipeline. This paper proposes a method for measuring the hydrodynamic noise of the pump by using the reverberation tank. The vibration of a pipeline attached to a centrifugal pump can be avoided by replacing the elastic pipeline with a flexible hose. The water outlet of the hose is then placed in the reverberation tank fitted with hydrophone arrays. The hydrodynamic noise of the water pump is measured using low frequency expansion technique of the reverberation method. Experiments show that this method can be used effectively to differentiate between the hydrodynamic noise and other associated noises of a pump.Pumps are considered as a primary source of acoustic energy in industrial piping. At present, the measurement of the hydrodynamic noise of the pump is done by installing the sensors in the pipeline attached to the pump. This technique is based on the fact that plane waves are formed inside the pipeline and the pulsating pressure travels in the form of sound waves. However, in actual there are no plane waves in an elastic pipe due to the phenomenon of low frequency cut-off. In fact, this pulsating pressure inside the pipeline will result in further vibration of the pipeline. This paper proposes a method for measuring the hydrodynamic noise of the pump by using the reverberation tank. The vibration of a pipeline attached to a centrifugal pump can be avoided by replacing the elastic pipeline with a flexible hose. The water outlet of the hose is then placed in the reverberation tank fitted with hydrophone arrays. The hydrodynamic noise of the water pump is measured using low frequency expansion technique of t...

Near-field inverse electromagnetic scattering problems for ellipsoids

The scattering problems of time-harmonic electromagnetic plane waves by a penetrable or an impenetrable ellipsoidal scatterer are considered. A low-frequency formulation of the corresponding direct scattering problems using the Rayleigh approximation is described. A method for solving inverse electromagnetic scattering problems for ellipsoids using near-field data is presented. A finite number of measurements of the zeroth low-frequency approximation of the electric scattered field lead to specify the orientation as well as the size of the ellipsoids. This method is applied for the cases of the perfectly conductive, the impedance, the lossless dielectric and the lossy dielectric ellipsoids. Especially for the case of penetrable ellipsoids, using additionally the leading order term of the low-frequency expansion of the magnetic scattered field, the physical parameters of the ellipsoids are also specified. Corresponding results for spheres and spheroids considering them as geometrically degenerate forms of the ellipsoid are presented.

Recovery of an embedded obstacle and the surrounding medium for Maxwell's system

In this paper, we are concerned with the inverse electromagnetic scattering problem of recovering a complex scatterer by the corresponding electric far-field data. The complex scatterer consists of an inhomogeneous medium and a possibly embedded perfectly electric conducting (PEC) obstacle. The far-field data are collected corresponding to incident plane waves with a fixed incident direction and a fixed polarisation, but frequencies from an open interval. It is shown that the embedded obstacle can be uniquely recovered by the aforementioned far-field data, independent of the surrounding medium. Furthermore, if the surrounding medium is piecewise homogeneous, then the medium can be recovered as well. Those unique recovery results are new to the literature. Our argument is based on low-frequency expansions of the electromagnetic fields and certain harmonic analysis techniques.

Transient heat conduction analysis using the NURBS-enhanced scaled boundary finite element method and modified precise integration method

The Non-uniform rational B-spline (NURBS) enhanced scaled boundary finite element method in combination with the modified precise integration method is proposed for the transient heat conduction problems in this paper. The scaled boundary finite element method is a semi-analytical technique, which weakens the governing differential equations along the circumferential direction and solves those analytically in the radial direction. In this method, only the boundary is discretized in the finite element sense leading to a reduction of the spatial dimension by one with no fundamental solution required. Nevertheless, in case of the complex geometry, a huge number of elements are generally required to properly approximate the exact shape of the domain and distorted meshes are often unavoidable in the conventional finite element approach, which leads to huge computational efforts and loss of accuracy. NURBS are the most popular mathematical tool in CAD industry due to its flexibility to fit any free-form shape. In the proposed methodology, the arbitrary curved boundary of problem domain is exactly represented with NURBS basis functions, while the straight part of the boundary is discretized by the conventional Lagrange shape functions. Both the concepts of isogeometric analysis and scaled boundary finite element method are combined to form the governing equations of transient heat conduction analysis and the solution is obtained using the modified precise integration method. The stiffness matrix is obtained from a standard quadratic eigenvalue problem and the mass matrix is determined from the low-frequency expansion. Finally the governing equations become a system of first-order ordinary differential equations and the time domain response is solved numerically by the modified precise integration method. The accuracy and stability of the proposed method to deal with the transient heat conduction problems are demonstrated by numerical examples.

Substantiation of low frequency ultrasonic cavitation application in the early postoperative period in rhinosurgical patients (review)

The article presents the review of application of low-frequency ultrasonic cavitation in the treatment of certain ENT diseases. The authors discuss physical characteristics of ultrasound, providing explanation of cavitation phenomenon and the mechanisms of impact on the damaged mucous membrane. Based on the analysis of pathophysiological changes (cellular, tissue, functional deviations, etc.) in the postoperative period in persons after rhinosurgical interventions the article substantiates the need for further study of therapeutic possibilities of low-frequency ultrasound cavitation and expansion of indications for its application.

Revisiting the Low-Frequency Dipolar Perturbation by an Impenetrable Ellipsoid in a Conductive Surrounding

This contribution deals with the scattering by a metallic ellipsoidal target, embedded in a homogeneous conductive medium, which is stimulated when a 3D time-harmonic magnetic dipole is operating at the low-frequency realm. The incident, the scattered, and the total three-dimensional electromagnetic fields, which satisfy Maxwell’s equations, yield low-frequency expansions in terms of positive integral powers of the complex-valued wave number of the exterior medium. We preserve the static Rayleigh approximation and the first three dynamic terms, while the additional terms of minor contribution are neglected. The Maxwell-type problem is transformed into intertwined potential-type boundary value problems with impenetrable boundary conditions, whereas the environment of a genuine ellipsoidal coordinate system provides the necessary setting for tackling such problems in anisotropic space. The fields are represented via nonaxisymmetric infinite series expansions in terms of harmonic eigenfunctions, affiliated with the ellipsoidal system, obtaining analytical closed-form solutions in a compact fashion. Until nowadays, such problems were attacked by using the very few ellipsoidal harmonics exhibiting an analytical form. In the present article, we address this issue by incorporating the full series expansion of the potentials and utilizing the entire subspace of ellipsoidal harmonic eigenfunctions.

Unsteadiness in a large turbulent separation bubble

The unsteady behaviour of a massively separated, pressure-induced turbulent separation bubble (TSB) is investigated experimentally using high-speed particle image velocimetry (PIV) and piezo-resistive pressure sensors. The TSB is generated on a flat test surface by a combination of adverse and favourable pressure gradients. The Reynolds number based on the momentum thickness of the incoming boundary layer is 5000 and the free stream velocity is$25~\text{m}~\text{s}^{-1}$. The proper orthogonal decomposition (POD) is used to separate the different unsteady modes in the flow. The first POD mode contains approximately 30 % of the total kinetic energy and is shown to describe a low-frequency contraction and expansion, called ‘breathing’, of the TSB. This breathing is responsible for a variation in TSB size of approximately 90 % of its average length. It also generates low-frequency wall-pressure fluctuations that are mainly felt upstream of the mean detachment and downstream of the mean reattachment. A medium-frequency unsteadiness, which is linked to the convection of large-scale vortices in the shear layer bounding the recirculation zone and their shedding downstream of the TSB, is also observed. When scaled with the vorticity thickness of the shear layer and the convection velocity of the structures, this medium frequency is very close to the characteristic frequency of vortices convected in turbulent mixing layers. The streamwise position of maximum vertical turbulence intensity generated by the convected structures is located downstream of the mean reattachment line and corresponds to the position of maximum wall-pressure fluctuations.

Mathematical and numerical analysis of low-frequency scattering from a PEC ring torus in a conductive medium

The electric and magnetic fields scattered off a non-penetrable ring torus, being characterized as perfect conductor, embedded in a homogeneous conductive medium and illuminated by a low-frequency magnetic dipole of arbitrary orientation and harmonic time-dependence are investigated herein. Upon definition of the complex wave number of the exterior medium k via its skin-depth, the 3-D scattering boundary value problem is handled via convenient low-frequency expansions in terms of powers of (ik)n, n ≥ 0 for the fields. A Maxwell-type problem is transformed into intertwined Laplace's or Poisson's potential-type boundary value problems with impenetrable boundary conditions. Using a toroidal coordinate system attached to the torus, they are solved as infinite series expansions for the fields in terms of toroidal eigenfunctions. In practice, what is accessible to the measurement is the scattered magnetic field. The static term (n=0) provides most of its real (or in-phase) part and the second-order term (n=2) consists of most of its imaginary part (quadrature), where in both cases a small contribution of the third-order term (n=3) is being calculated. For n=1, there exists no field, while the terms for n ≥ 4 and for such kind of applications have been proved to be of minor significance, hence they are neglected. The resulting infinite linear systems can be solved at any accuracy level through a cut-off process or via an analytical technique based on the method of finite continuous fraction solutions. Basics of the far-field approximation and the magnetic polarizability tensor are also included. At implementation stage, simulations are proposed in various situations, where a full-wave, finite-element approach is discussed.