Scattering of cylindrical Gaussian pulse near an absorbing half-plane in a moving fluid
- Amer Bilal Mann^{1},
- Rab Nawaz^{2}Email author and
- Mazhar Hussain Tiwana^{3}
Received: 16 November 2015
Accepted: 19 April 2016
Published: 4 May 2016
Abstract
We investigate the scattering of Gaussian pulse by an absorbing half-plane satisfying Myers’ impedance conditions. The model problem is considered for a subsonic flow in a moving fluid. The Wiener-Hopf technique followed by the spatial and temporal Fourier transforms and method of Steepest descent enables us to develop the far field solution analytically. It is observed that the Myers’ impedance condition found higher-order accuracy of Mach number as compared with the results obtained while using Ingard’s condition. The solution to the underlying problem leads itself to the variety of problems thereby including the effects of Gaussian pulses.
Keywords
1 Introduction
The impedance boundary condition (IBC) was first introduced by Leontovich in attempt to solve the problems of radio wave propagation over the earth. The IBCs are the approximate boundary conditions that relate the field outside the scatterer only, and thus analysis of the related problem is much more simplified [1]. These IBCs have been utilized by many researchers in the field of electromagnetics and acoustics; refer, for instance, to Wang [1], Nawaz et al. [2, 3], Rawlins [4], Ahmad [5], Buyukaksoy et al. [6], Ayub et al. [7], etc. Rawlins [4] used Ingard’s condition [8] to model the impedance conditions that arise in the noise reduction problems by barriers. Ahmad [5] reconsidered Rawlins problem [4] and showed that Myers’ condition [9] gives better results than Ingard’s conditions when the diffraction problems of acoustic waves related to noise reduction by barriers are considered in a moving fluid regime. Myers’ condition [9] contains a correction term and thus allows a straightforward manipulation of the condition into a form that is more convenient to apply than Ingard’s condition. In this paper, we focus ourselves on the diffraction of cylindrical Gaussian pulse by an absorbing half-plane in a moving fluid satisfying Myers’ impedance condition.
Gaussian functions and integrals frequently occur in many problems of mathematics, physics, statistics, and also in other branches of science and technology. To name a few, Gaussian integrals occur in normal distributions (also known as Gaussian distributions), which occupy a central position in statistical inference, sampling distributions and are excellent approximations to several other distributions. In quantum field theory, the Gaussian integrals involve ordinary real or complex variables or the Grassmann variables. Because Gaussian beams have favorable propagation characteristics and represent physically observable entities, these have played a vital role in many modeling schemes; see, for instance, [10]. In particular, Gaussian and comb functions are the best known examples of self-Fourier functions [11]. Keeping in view the importance of Gaussian functions, the diffraction of cylindrical Gaussian pulse near an absorbing half-plane in a moving fluid regime is examined mathematically.
While investigating the diffraction problems of acoustic/electromagnetic/elastic waves, harmonic time variation is assumed and suppressed throughout the analysis. Although time harmonic waves are of great importance, yet there are significant fields whose time variation is nonharmonic. The time-dependant wave phenomenon is also an important aspect in the wave motion theory and gives a more transparent picture of wave motion phenomenon. A good account of transient problems can be found in the books of Friedlander [12] and Jones [13]. The transient wave phenomenon is also important due to its ability to produce short electromagnetic pulses, which are used as a diagnostic tool for identification/location of cracks or other defects, implosion and seismological prospects like bore hole sounding and nondestructive testing [14]. Keeping in view the importance of transient wave motion, many scientists have contributed transient wave problems in diffraction theory, to name a few, for example, Haris [15, 16], Rienstra [17], Kriegsmann et al. [18], Ahmad [19], Ishii and Tanaka [20], Alford et al. [21], and Ayub et al. [22, 23]. Moreover, Marin et al. [24] and Marin [25] have also focused themselves on related studies by considering nonsimple material problems.
The solution to the underlying problem is presented while using spatial and temporal Fourier transforms, the Wiener-Hopf technique [26], and the method of steepest descent [13]. Firstly, temporal Fourier transform is applied to obtain the transfer function in frequency domain, and then following the approach of Sun et al. [27], finally, the inverse transform is used to get the results in time domain. The results for the rigid barrier and still air can be computed as a special case from the given diffracted field.
2 Statement of the problem
3 Problem in frequency domain
4 Nondimensional form
5 Analytic solution
6 Discussion
Declarations
Acknowledgements
The authors sincerely thank the reviewers for their painstaking review and useful comments.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
- Wang, D: Limits and validity of the impedance boundary condition on penetrable surface. IEEE Trans. Antennas Propag. 163, 453-457 (2005) Google Scholar
- Nawaz, R, Ayub, M, Javaid, A: Plane wave diffraction by a finite plate with impedance boundary conditions. PLoS ONE 9(4), e92566 (2014) MathSciNetView ArticleGoogle Scholar
- Nawaz, R, Ayub, M: Closed form solution of electromagnetic wave diffraction problem in a homogeneous bi-isotropic medium. Math. Methods Appl. Sci. 38(1), 176-187 (2014) MathSciNetView ArticleMATHGoogle Scholar
- Rawlins, AD: Acoustic diffraction by an absorbing semi-infinite half plane in a moving fluid. Proc. R. Soc. Edinb. A 72, 337-357 (1975) MathSciNetMATHGoogle Scholar
- Ahmad, B: An improved model for noise barriers in a moving fluid. J. Math. Anal. Appl. 321(2), 609-620 (2006) MathSciNetView ArticleMATHGoogle Scholar
- Buyukaksoy, A, Cinar, G: Solution of a matrix Wiener-Hopf equation connected with the plane wave diffraction by an impedance loaded parallel plate waveguide. Math. Methods Appl. Sci. 28, 1633-1645 (2005) MathSciNetView ArticleMATHGoogle Scholar
- Ayub, M, Nawaz, R, Naeem, A: Line source diffraction by a slit in a moving fluid. Can. J. Phys. 87(11), 1139-1149 (2009) View ArticleMATHGoogle Scholar
- Ingard, U: Influence of fluid motion past a plane boundary on sound reflection, absorption, and transmission. J. Acoust. Soc. Am. 31, 1035-1036 (1959) View ArticleGoogle Scholar
- Myers, MK: On the acoustic boundary condition in the presence of flow. J. Sound Vib. 71, 429-434 (1980) View ArticleMATHGoogle Scholar
- Felson, LB, Klosner, JM, Lu, IT, Grossfeld, Z: Source field modeling by self-consistent Gaussian beam superposition (two-dimensional). J. Acoust. Soc. Am. 89(1), 63-72 (1991) View ArticleGoogle Scholar
- Lakhtakia, A: Fractal self-Fourier functions. Optik 94(1), 51-52 (1993) Google Scholar
- Friedlander, FG: Sound Pulses. Cambridge University Press, Cambridge (1958) MATHGoogle Scholar
- Jones, DS: Acoustic and Electromagnetic Waves. Clarendon, Oxford (1986) Google Scholar
- Harris, JG: Diffraction by a crack of a cylindrical longitudinal pulse. Z. Angew. Math. Phys. 31, 367-383 (1980) MathSciNetView ArticleMATHGoogle Scholar
- Harris, JG: Uniform approximations to pulses diffraction by a crack. Z. Angew. Math. Phys. 31, 771-775 (1980) View ArticleMATHGoogle Scholar
- Harris, JG, Pott, J: Surface motion excited by acoustic emission from a buried crack. J. Appl. Mech. 51, 77-83 (1984) View ArticleGoogle Scholar
- Rienstra, SW: Sound diffraction at a trailing edge. J. Fluid Mech. 108, 443-460 (1981) MathSciNetView ArticleMATHGoogle Scholar
- Kriegsmann, GA, Norris, AN, Reiss, EL: Acoustic pulse scattering by baffled membranes. J. Acoust. Soc. Am. 79(1), 1-8 (1986) MathSciNetView ArticleMATHGoogle Scholar
- Ahmad, B: Sub mach-1 sound due to an arbitrary time dependent source near an absorbing half plane. Appl. Math. Comput. 163, 39-50 (2005) MathSciNetMATHGoogle Scholar
- Tanaka, K, Ishii, S: Acoustic radiation from a moving line source. J. Sound Vib. 77(3), 397-401 (1981) View ArticleMATHGoogle Scholar
- Alford, RM, Kelly, KR, Boore, DM: Accuracy of finite difference modeling of the acoustic wave equation. In: The 43rd Annual International Society of Exploration Geophysicists 834-843 (1974) Google Scholar
- Ayub, M, Tiwana, MH, Mann, AB, Ramzan, M: Diffraction of waves by an oscillating source and an oscillating half plane. J. Mod. Opt. 56(12), 1335-1340 (2009) View ArticleMATHGoogle Scholar
- Ayub, M, Naeem, A, Nawaz, R: Sound due to an impulsive line source. Comput. Math. Appl. 60, 3123-3129 (2010) MathSciNetView ArticleMATHGoogle Scholar
- Marin, MI, Agarwal, RP, Mahmoud, SR: Non-simple material problems addressed by the Lagrange’s identity. Bound. Value Probl. 2013, 135 (2013) MathSciNetView ArticleMATHGoogle Scholar
- Marin, M: A temporally evolutionary equation in elasticity of micropolar bodies with voids. Bull. Ser. Appl. Math. Phys. 60, 3-12 (1998) MathSciNetMATHGoogle Scholar
- Noble, B: Methods Based on the Wiener-Hopf Technique. Pergamon, London (1958) MATHGoogle Scholar
- Sun, Z, Gimenez, G, Vray, D, Denis, F: Calculation of the impulse response of a rigid sphere using the physical optic method and modal method jointly. J. Acoust. Soc. Am. 89(1), 10-18 (1991) View ArticleGoogle Scholar
- Campbell, GA, Foster, RM: Fourier Integrals for Practical Applications. Van Nostrand Company, Princeton (1948) MATHGoogle Scholar
- Abramowitz, M, Stegun, IA: Handbook of Mathematical Functions. Dover, New York (1965) MATHGoogle Scholar