Elsevier

Applied Acoustics

Volume 180, September 2021, 108080
Applied Acoustics

Technical note
Review on Klinaku-Berisha equation via numerical modelling for acoustical Doppler effect

https://doi.org/10.1016/j.apacoust.2021.108080Get rights and content

Abstract

Klinaku and Berisha [https://doi.org/10.1016/j.rinp.2018.12.024] conducted an interesting study and proposed a generic equation (named as Klinaku-Berisha Equation – KBE thereafter) to replace the conventional equation for the calculation of apparent frequency due to Doppler effect. However, KBE is not validated yet with any real case investigation. The paper aims to review the validity of KBE by conducting a detailed numerical simulation. Finite difference based Delfim-Soares algorithm was implemented to analyse the wave structure and apparent frequencies due to moving wave source. Simulation results uncovered that KBE has underestimated the apparent frequency at inclined angles, and further correction is required. Corrected KBE is therefore proposed in the study using Gaussian approximation, which could become a more suitable general equation for Doppler effect.

Introduction

Doppler effect can be defined as the apparent shift of frequency due to the moving of wave emitter. Although doppler effect is applicable for all types of waves, the paper focuses only on the acoustical waves. Perhaps, acoustical doppler effect has been utilised in a very wide range of engineering measurement. These comprise ultrasound-assisted visualisation for various medical complications [1], [2], [3], [4], [5], [6], [7], design of laser-doppler velocimetry for fluid flow measurement [8], detection of vibrations in automobile components [9], determination of object velocity through GPS [10], improvement of electric signal in antenna [11], dynamic checking on railway components [12], [13], enhancement on underwater acoustics [14], [15], [16], aircraft tracking [17], and improvement of multiphoton ionization [18]. Moreover, some advanced innovative technologies are put forward based on Doppler effect. For example, Zhang et al. [10] reconciliated the erroneous doppler shift between rotating GPS satellite and moving receiver for a more accurate positioning. Iwaya and Suzuki [19] incorporated Doppler effect into digital rendering for better sound detection. Meanwhile, Huang et al. [20] introduced Doppler effect for early detection of car collision.

Despite much research have been done on the abovementioned applications, most of the works dwelled with the signal processing of the electrical devices for digital detection. The underlying physics of Doppler effect does not receive much attention so far. The improvement of mathematical description for Doppler effect will greatly enhance the engineering applications, and thus more investigation on it is required [21]. In general, the famous general Doppler effect formula can be expressed as in Eq. (1):f=c+vlc+vsfin which f', f, c, vl and vs represents apparent frequency, emission frequency, speed of wave, velocity of receiver, and velocity of wave source, respectively. Notwithstanding the simplicity, Eq. (1) is not able to predict the apparent frequency at the signal receiver at varied angles. Most recently, Klinaku and Berisha [22] have filled the gap by deriving a more inclusive equation to explain the Doppler effect for arbitrary angle, which is named as Klinaku-Berisha Equation in this paper thereafter, by assuming receiver to be stationary. Klinaku-Berisha Equation (KBE) can be described as in Eq. (2):f=cc2-vs2sin2θ+vscosθfwhere θ is the angle between the source and the receiver, as shown in Fig. 1. The summary of the derivation procedure for Eq. (2) can be referred in the Appendix 1.

Due to its new introduction, the purpose of the paper is to verify and discuss KBE via numerical modelling for doppler effect. The numerical simulation includes all three acoustical regions: subsonic, sonic, and supersonic. The investigation aims to reappraise KBE as a general equation to explain Doppler effect, instead of conventional Doppler effect formula.

Section snippets

Numerical methodology

The governing equation for acoustical wave propagation can be described as in Eq. (3). The details of the derivation can be found in many literatures [23], [24].2Pt2=c22Px2+2Py2

P represents the local scalar for the wave propagation. The second order linear wave equation is discretised using central finite differencing Euler explicit method, which can be illustrated as in Eq. (4):tt+Δt2Pt2dt=c2tt+ΔtPi+1,jn-2Pi,jn+Pi-1,jnΔx2+Pi,j+1n-2Pi,jn+Pi,j-1nΔy2dtwhere (i,j), n + 1, n, and n − 1

Results and discussion

The wave structure due to different ξ has been illustrated as in Fig. 5. There is an interesting phenomenon which is named as “perturbed wave” in this paper, can be observed at the wave structure. Perturbed wave is a high frequency and high amplitude intermediate wave generated within the wavelength of primary wave when the wave source is moving, as illustrated as in Fig. 6. It can be further observed that the wave emission region can be further divided into smooth wave region and perturbed

Concluding remarks

Although Klinaku and Berisha [22] claimed that their proposed equation is the correct equation to explain Doppler effect for arbritrary angle between moving wave source and static receiver, a detailed numerical simulation revealed that the equation requires further correction. This is because the apparent frequencies within the quarters have been underestimated using KBE due to convolution or narrowing of wavelength, which is a result of wave piling. Gaussian approximation with assumed Gaussian

CRediT authorship contribution statement

Wah YenTey: Conceptualization, Investigation, Methodology, Validation, Visualization, Writing - original draft, Writing - review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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