Limitations of the asymptotic equations for exterior sound waves

Abstract

The determination of exterior sound field due to incident plane waves on a cylinder may be of importance in many areas of acoustics. Applications in acoustics include detecting, locating and classifying objects that generate a scattered sound field excited by the incident sound waves. The exterior sound field in the acoustic domain due to incident plane waves on a cylinder can be determined analytically and numerically. However, it has been observed that for convergence the asymptotic expressions of the analytical models used in the literature required infinite number of terms in the series calculations or are seldom restricted either by low or high frequency limit or the requirement of near-field and far-field approximations. This paper investigates the reasons that caused the nonconformity of the asymptotic analytical expressions. The investigation thus carried out based on the numerically determined exterior sound field using the boundary element method (BEM) and compared with the results obtained using the governing expressions. The comparisons showed good agreement when the governing expressions are not restricted, but nonconformity occurs when they are restricted.

Keywords

Scattering propagation and radiation asymptotic boundary element method

Introduction

The prediction of the exterior sound field caused by a diffracting object is of interest in many fields of acoustics, including the detection, location and classification of objects that generate a scattered sound field as a result of incident sound waves. Prediction of the exterior sound field at any location from the surface of a cylinder can be done analytically using the previously published governing models by Morse,¹ Morse & Ingard² and Junger & Feit.³ In these models, the incident sound pressure is determined at different positions on the surface of the cylinder assuming that no distortion of sound waves occurs by the cylinder of infinite length. Then the scattered sound pressure and intensity can be determined taking into account the distortion of incident sound waves by the cylinder. The total sound pressure can then be determined by superposition of incident and scattered sound pressure. However, the convergence of these models relies upon the infinite number of terms required in the series summation, which makes it difficult to determine exactly how many terms are needed for convergence for a particular frequency or wavelength. Perhaps this is the reason that the authors^1–3 have derived the asymptotic expressions which are widely used in the literature. It has been observed that the asymptotic expressions are seldom restricted either by low or high frequency limit or the requirement of near-field and far-field approximations.

The current research presents a novel work in order to formulate a technique which can determine a finite number of terms needed for convergence of the analytical model with compatible explanations. The converging accuracy of the model was verified using the boundary element technique. The boundary element method (BEM) is a widely used tool within the aeroacoustics community. For instance, Papamoschou⁴ uses the BEM to predict scattering and propagate the acoustic field induced by a ducted fan. Another example might be found in Wu et al.,⁵ where the BEM is used in noisy environments within the context of near-field holography. Moreover, the BEM can be coupled with the partial fields decomposition method in which the pressure fields of supersonic jets are obtained at a radiating surface.⁶

The BEM software used here is written in MATLAB and is publicly available software known as Open BEM, which has been mainly developed by the Acoustic Laboratory, Technical University of Denmark.⁷ Finally, an effort was given to investigate the reason of divergent behaviors of the previously published asymptotic models^1–3 and to compare the previous results^1–3 with the results obtained using the BEM.

Theory

The purpose of the following analysis is to study the existing converging expressions to determine the incident, scattered and total sound fields at any location from the surface in the sound field as a result of scattering from an infinitely long cylinder. The cylinder has been considered as infinitely long so that diffraction effects from both ends of the cylinder can be omitted and thus enable the problem to be tractable. It has also been considered that the cylinder wall is rigid so that all of the incident energy is scattered outward from the surface. To determine the sound pressure at any field location as a result of scattering, it is necessary to first evaluate the incident sound pressure at that location, followed by the scattered pressure. The amplitude and phase of the two pressure fields are taken into account in determining the total sound pressure field.

Consider a plane wave travelling in the positive x direction and impinging normal to a cylinder of radius a as shown in Figure 1. In the figure, r is the radial distance from the centre of the cylinder, which can be any distance of interest from the cylinder. The incident sound pressure at any distance r from the cylinder is^1–3:

P^{i} (r, k, ϕ_{i}) = P^{o} \sum_{m = 0}^{M - 1} ε_{m} i^{m} \cos (m ϕ_{i}) J_{m} (k r),

(1)

where the incident plane waves are traveling from left to right,

P^{o}

is the incident wave amplitude, m and M are the term number and total number of terms required in the series summation respectively,

ε_{m} = 1

if m = 0 and

ε_{m} = 2

if m > 0, and

J_{m}

is a Bessel function of the first kind of mth order for variable coefficients k and r.

Figure 1.

Incident plane waves travelling normal to the cylinder axis z (z axis is out of page).

For the scattered sound pressure, the scattered velocity potential $ψ_{m}^{s}$ as a function of cylinder radius a, wave number k and circumferential angle $ϕ_{i}$ can be used. The scattered velocity potential and the scattered pressure corresponding to that velocity potential at any distance r from the cylinder origin shown to be^1–3

ψ_{m}^{s} (a, k, ϕ_{i}) = ε_{m} i^{m + 1} e^{- i γ_{m}} \sin γ_{m} \cos (m ϕ_{i}),

(2)

P^{s} (r, k, ϕ_{i}) = - P^{o} \sum_{m = 0}^{M - 1} ψ_{m}^{s} H_{m} (k r),

(3)

where

H_{m} (k r)

is the Hankel function (Bessel function of the third kind) of mth order. In equation (2) the sine and

e^{- i γ_{m}}

term have been chosen to ensure the correct combination of the incident and scattered waves and the phase angles,

γ_{m}

, can be defined depending on the values of m as follows^1,2:

\tan γ_{0} = - J_{1} (k a) / N_{1} (k a) when m = 0

\tan γ_{m} = - J_{m}^{'} (k a) / N_{m}^{'} (k a) w h e n m > 0 .

(4)

The scattered intensity corresponding to the scattered sound pressure can be written as

I^{s} (r, k, ϕ_{i}) = 〈 P^{s^{2}} 〉 / ρ_{o} c,

(5)

where

ρ_{o}

is the equilibrium density in the fluid and c is the speed of sound. Finally, the total sound pressure at any position in the sound field from a cylinder due to both the incident and scattered waves can be expressed by summing equations (1) and (3) in the following form^1-3

P_{r}^{t} (r, k, ϕ_{i}) = P^{i} (r, k, ϕ_{i}) + P^{s} (r, k, ϕ_{i}) = P^{o} \sum_{m = 0}^{M - 1} ε_{m} i^{m} \cos (m ϕ_{i}) [J_{m} (k r) - i e^{- i γ_{m}} \sin γ_{m} H_{m} (k r)] .

(6)

It is important to note that the value of M required in the sum in equation (6) will determine how accurate the analytical solution will be and it depends on the value of kr.

Boundary element method

The numerical formulation technique using the BEM for the exterior boundary problem has been extensively discussed by Morshed et al.^8–11 This technique has been used in the current work to include an incident plane wave.

For the exterior problem where the observation point $\vec{P}$ is located on the boundary $Γ$ , the Kirchhoff-Helmohltz (boundary) integral solution for the time harmonic sound pressure p can be determined as^12–18

C (\vec{P}) p (\vec{P}) = \int_{Γ} \frac{\partial G (\vec{P}, \vec{Q})}{\partial n (\vec{Q})} p (\vec{Q}) d Γ (\vec{Q}) + p^{i} .

(7)

Equation (7) can be reduced to a matrix formulation as^8–10

[C] {p_{s}} = {p_{s}} [H] + p^{i},

(8)

where [C] is equal to

2 π I_{N}

( I _N is an N × N identity matrix of the N calculation points on the N nodes) for the surface formulation, and [H] is an

H \times H

global matrix of the H calculation points on the H elements. Here both [C] and [H] are known but the surface pressure

{p_{s}}

is unknown. Therefore to find

{p_{s}}

, equation (8) reduces to

[A] {p_{s}} = - p^{i},

(9)

where [A] = [H – C]. Equation (9) gives the total sound pressure on the surface of a cylinder, which can be used to calculate the sound pressure on the field points outside the surface, that is when

r_{P} > r_{Q}

. The matrix formulation for the total sound pressure on the field points becomes

{p_{f}} = [A_{f}] {p_{s}} - p_{f}^{i},

(10)

where the subscript f denotes the field points.

[A_{f}]

is the coefficient matrix for the field points ,

p_{f}^{i}

is the incident sound pressure on the field points, and

{p_{s}}

is the pressure associated with each node on the surface. On each field node the incident sound pressure

p_{f}^{i}

of equation (10) can be calculated for plane waves as follows:

p_{f}^{i} = P^{o} e^{i k x y},

(11)

where the subscripts x and y represent the field node position along the X and Y coordinates respectively. To calculate only the scattered sound pressure at the field points, the incident sound pressure term given in equation (10) can be omitted.

Comparison between analytical and numerical results

For this comparison 80 field points were placed circumferentially at a distance of 5a (five times the cylinder radius) from the origin of a cylinder of radius a. The cylinder surface was divided into 40 elements. Each element had two end nodes and one mid node, and there were in total 80 nodes on the surface. It was assumed that the incident plane waves travel from left to right and impinge normal to the circumference of the cylinder. The comparisons between the results obtained using the analytical and BEM techniques for the scattered and total sound pressure at a distance of 5a in the sound field from the origin of the cylinder of radius a are shown in Figures 2 and 3, for $k a = 5$ . Figure 4 shows the total sound pressure field when the observation points are placed at the surface of the cylinder of radius a, for ka = 5. The results show very good agreement between the analytical and BEM techniques. The comparisons also show that the equations (3) and (6) can converge completely and are capable of predicting the sound pressure field at any location in the sound field including the cylinder surface.

Figure 2.

Scattered sound pressure comparison between the analytical and BEM results at a distance of 5a (five times the cylinder radius) from the origin of a cylinder of radius a, for plane waves incident normal to node number 41, for ka = 5. [Incident pressure amplitude $P^{ο} = 1 Pa$ ].

Figure 3.

Total sound pressure comparison between the analytical and BEM results at a distance of 5a (five times the cylinder radius) from the origin of a cylinder of radius a, for plane waves incident normal to node number 41, for ka = 5. [Reference pressure 20 μPa and incident pressure amplitude $P^{ο} = 1 Pa$ ].

Figure 4.

Total sound pressure comparison between the analytical and BEM results at the surface of a cylinder of radius a, for plane waves incident normal to node number 41, for ka = 5. [Reference pressure 20 μPa and incident pressure amplitude $P^{ο} = 1 Pa$ ].

The directivity patterns of the scattered and total sound pressure fields are also shown in Figures 5 and 6 for $k a = 5$ . From the results it can be seen that the scattered and total pressure amplitudes vary substantially in the regions from approximately 60^o to 150^o and 210^o to 300^o, and more or less uniformly over all other directions. Also a predominantly higher scattered sound pressure peak occurs at the back ( $φ_{i}$ = 0^o) of the cylinder. The reason for this variation in amplitude is that waves are diffracted around the two sides of the cylinder in the clockwise and anti-clockwise directions. When one wave travels a distance which is an odd number of half wavelengths more than the distance traveled by the other, there is destructive interference. When they travel the same distance or an integer number of wavelengths, there is reinforcement and a peak generates.

Figure 5.

Directivity pattern of the scattered sound pressure obtained using the analytical and BEM results at a distance of 5a (five times the cylinder radius) in the sound field from the origin of a cylinder of radius a, for $k a = 5$ . [Incident pressure magnitude $P^{o} = 1 Pa$ ].

Figure 6.

Directivity pattern of the total sound pressure obtained using the analytical and BEM results at a distance 5a (five times the cylinder radius) in the sound field from the origin of a cylinder of radius a, for $k a = 5$ . [Reference pressure 20 μPa and incident pressure magnitude $P^{o} = 1 Pa$ ].

It has been found that the phase angles, $γ_{m}$ , and the velocity potential, $ψ_{m}^{s}$ , which determine the total circumferential sound pressure, and tend to zero by trial and error (relative error of less than 1 × 10⁻⁷) as the value of m increases as shown in Figures 7 and 8 for different values of ka. This trend means that after a certain value of M in equation (6) there will be negligible contribution from additional terms to the total circumferential sound pressure at the surface of a cylinder. Therefore, for the case $k a = 5$ , the results presented in Figures 2 to 6 correspond to values of M up to M = 12.

Figure 7.

Characteristic behaviour of $γ_{m}$ as a function of m, for values of ka = 1, 3 and 5 respectively.

Figure 8.

Characteristic behaviour of $ψ_{m}^{s}$ as a function of m, for values of ka = 1, 3 and 5 respectively.

Comparison with asymptotic equations

The asymptotic equations for the scattered sound pressure, velocity potential and intensity at a very large distance from a cylinder ( $k r ≫ 1$ ) are given by Morse & Ingard1^1,2 as:

p_{s} ≅ - \sqrt{\frac{2 ρ_{o} c I^{o} a}{π r}} ψ_{s} (ϕ_{i}) e^{i k (r - c t)},

(12)

where

I^{o} = (P^{o^{2}} / ρ_{o} c)

. Equation (12) can be expressed as

p_{s} = - P^{o} \sqrt{\frac{2 a}{π r}} ψ_{s} (ϕ_{i}) e^{i k (r - c t)},

(13)

and velocity potential

ψ_{s} (ϕ_{i}) = \frac{1}{\sqrt{k a}} \sum_{m = 0}^{\infty} ε_{m} \sin (γ_{m}) e^{- i γ_{m}} \cos (m ϕ_{i}) .

(14)

The scattered intensity is

Υ_{s} ≅ (\frac{2 I^{o} a}{π r}) {| ψ_{s} |}^{2},

(15)

where

{| ψ_{s} |}^{2} = \frac{1}{k a} \sum_{m, n = 0}^{\infty} ε_{m} ε_{n} \sin γ_{m} \sin γ_{n} \cos (γ_{m} - γ_{n}) \cos (m ϕ_{i}) \cos (n ϕ_{i}) .

(16)

Sound pressure field

For the far-field scattered sound pressure it was observed by trial and error that at a distance of 57a from the origin of a cylinder of radius a the results obtained using equations (3) and (13) agree well, as shown in Figure 9. If the distance decreases, the error increases as shown in Figures 10 to 12 for distances of 5a, 2a and 1.2a from the cylinder respectively. Particularly significant differences have been observed in the near field region of the cylinder, as shown in Figures 11 and 12. These differences are due to the fact that equation (13) is not converging completely compared with equation (3). It has been found that in equation (13), the asymptotic form of the Hankel function is being used, whereas equation (3) uses the exact form of Hankel function. Unfortunately, there are significant differences found between the asymptotic values and the computational values of Hankel function for finite values of kr. Substituting equation (14) into equation (13), one can find that the asymptotic form of Hankel function used by Morse & Ingard² in equation (13) is:

H_{m} (k r) = J_{m} (k r) + i N_{m} (k r) \underset{k r \to \infty}{\to} i^{- (m + 1)} \sqrt{2 / π k r} e^{i k r}

(17)

Figure 9.

Scattered sound pressure comparison between results obtained using equations (3) and (13), for ka = 5 and r = 57a. [Incident pressure magnitude $P^{o} = 1 Pa$ ].

Figure 10.

Scattered sound pressure comparison between results obtained using equations (3) and (13), for ka = 5 and r = 5a. [Incident pressure magnitude $P^{o} = 1 Pa$ ].

Figure 11.

Scattered sound pressure comparison between results obtained using equations (3) and (13), for ka = 5 and r = 2a. [Incident pressure magnitude $P^{o} = 1 Pa$ ].

Figure 12.

Scattered sound pressure comparison between results obtained using equations (3) and (13), for ka = 5 and r = 1.2a. [Incident pressure magnitude $P^{o} = 1 Pa$ ].

In equation (17), the time dependency part

e^{- i c t}

has been omitted for simplicity. Comparison between the values of the Hankel function obtained using this asymptotic form and the computational values obtained using MATLAB are given in Table 1 (the computational values are considered as exact). The table shows that the asymptotic form provides no variation in amplitude as a function of m for a particular value of kr, but the computational values vary significantly for small values of kr. However, for large values of kr the error is small enough to be acceptable. It is worth noting that the asymptotic form gives a value that is approximately equal to the true computational value of the 0^th order of Hankel function. This effectively means that equation (13) only considers

m = 0

component for the outgoing Hankel function, which is why it only gives good results when the distance from a cylinder is very large. How large the distance needs to be is suggested in the work presented by Friot & Bordier,¹⁹ which uses kr = 1000a to obtain convergence. Therefore, it may be concluded that equation (13) only converges for the Fraunhofer diffraction region, where to calculate the scattered sound pressure or intensity (the intensity is discussed in next section), the distance considered needs to be so large (>57a) compared with the cylinder circumference that the diffraction angle subtended by the cylinder is negligible. However, equation (13) do not converge in the Fresnel diffraction region or the near-field region. To ensure convergence of the expression for acoustic pressure at any location from a cylinder, the full expansion of the Hankel function must be used as shown in equation (3).

Table 1.

Comparison between the asymptotic values using equation (17) and the computational values using MATLAB of the outgoing Hankel function, $H_{m} (k r)$ .

kr	m	Asymptotic values		Computational values		Error	Error (%)
kr	m	Complex	Absolute	Complex	Absolute	Error	Error (%)
1	0	0.6714 + 0.4311i	0.7979	0.7652–0.0883i	0.7703	−0.0276	3.58
	1	−0.4311 + 0.6714i	0.7979	0.4401+ 0.7812i	0.8966	0.0987	11.00
	2	−0.6714–0.4311i	0.7979	0.1149+ 1.6507i	1.6547	0.8568	51.78
3	0	0.0650–0.4560i	0.4607	−0.2601–0.3769i	0.4579	−0.0028	0.61
	1	0.4560 + 0.0650i	0.4607	0.3391–0.3247i	0.4694	0.0088	1.87
	2	−0.0650 + 0.4560i	0.4607	0.4861 + 0.1604i	0.5119	0.0512	10.00
5	0	−0.3422 + 0.1012i	0.3568	−0.1776 + 0.3085i	0.3560	−0.0008	0.22
	1	−0.1012–0.3422i	0.3568	−0.3276–0.1479i	0.3594	0.0026	0.72
	2	0.3422–0.1012i	0.3568	0.0466–0.3677i	0.3706	0.0138	3.72
10	0	−0.1373–0.2117i	0.2523	−0.2459–0.0557i	0.2522	−0.0002	0.08
	1	0.2117–0.1373i	0.2523	0.0435–0.2490i	0.2528	0.0005	0.20
	2	0.1373 + 0.2117i	0.2523	0.2546 + 0.0059i	0.2547	0.0024	0.94
15	0	0.1340–0.1565i	0.2060	−0.0142–0.2055i	0.2060	0.00	0.00
	1	0.1565 + 0.1340i	0.2060	0.2051–0.0211i	0.2062	0.0002	0.09
	2	−0.1340 + 0.1565i	0.2060	0.0416 + 0.2027i	0.2069	0.0009	0.43

Sound intensity field

Here, comparisons are made between the results obtained using equation (5) and results presented in Morse¹ and Morse & Ingard² for three particular wavelengths, λ = 2πa/5, λ = 2πa/3 and λ = 2πa, respectively. It has been shown in the preceding section that the limitation of the previous derivations is the use of asymptotic form of the Hankel function, which gives incorrect results for small values of kr.

For comparisons of the intensity values, the distance

r = 57 a

from the origin of a cylinder of radius a has been chosen because, as has been shown earlier, at that distance the former equations of Morse¹ and Morse & Ingard² converge, so that a comparison of far-field intensity can be made with the previous results at that distance. The parameters for the three cases examined are given in Table 2.

Table 2.

Total number of terms required in the series calculation in equation (5), for scattered sound intensity.

Case	Wavelength, λ (m)	ka	Total number of terms, M, needed to achieve a relative error < 1 × 10⁻⁷
I	$2 π a / 5$	5	12
II	$2 π a / 3$	3	8
III	$2 π a$	1	6

Case I (short wavelength)

Consider the case where the wavelength is very small compared with the circumference of the cylinder, that is when λ = 2πa/5. The comparison of scattered intensity with the previous result presented by Morse¹ and Morse & Ingard² for this particular wavelength is shown in Figure 13. In the figure, the result has been scanned from Morse & Ingard² and the current result overlaid. Since equation (5) is a finite series summation, it has been observed from trial and error that, in this particular case, using only the first five scattered waves (M = 5) gives good agreement with the previous result at a distance of 57a from the origin of a cylinder of radius a. That is, the pattern of the scattered sound intensity is almost the same as the result presented by Morse¹ and Morse & Ingard.² However, this series has not fully converged for M = 5 and consideration needs to be given to more scattered waves in the series. It has been found by trial and error that for this case, 12 terms are required before the use of additional terms has a negligible effect (relative error of less than 1 × 10⁻⁷) on the phase angles, $γ_{m}$ , and the velocity potential, $ψ_{m}^{s}$ , as shown in Figures 7 and 8. The intensity result using 12 terms is shown in Figure 14. The figure shows that the intensity distribution is almost flat at the front of the cylinder, which shows significant difference to the result shown by Morse¹ and Morse & Ingard² presented in Figure 13. This is due to the influence of the additional terms that have been added in the series. It should be noted that decreasing the tolerance does not noticeably improve the results.

Figure 13.

Directivity patterns of the scattered sound intensity at a distance of r = $57 a$ from the origin of a cylinder of radius a, for $k a = 5$ and M = 5. The red (dash) and black (solid) lines show the current and the previous (Morse¹; Morse & Ingard²) results respectively. [Incident pressure magnitude $P^{o} = 1 Pa$ ].

Figure 14.

Directivity pattern of the scattered sound intensity at a distance of r = 57a from the origin of a cylinder of radius a, for ka = 5 and M = 12. [Incident pressure magnitude $P^{o} = 1 Pa$ ].

Case II and III (medium and large wavelength)

For the cases when λ = 2πa/3 and λ = 2πa, the comparisons are shown in Figures 15 and 16. For each case, the scattered waves series required eight and six terms respectively, until $γ_{m}$ and $ψ_{m}^{s}$ became sufficiently small (relative error of less than 1 × 10⁻⁷). Again the relative size of the various terms is shown in Figures 7 and 8. For case II, the comparison shows quite a different pattern to that of Morse¹ and Morse & Ingard.² The scattered waves do not create a sharp peak at the rear of the cylinder, in contrast with the Morse & Ingard² result. The reason of this is due to only a few number of terms used earlier in the series summation of equation (16).

Figure 15.

Directivity patterns of the scattered sound intensity at a distance of r = 57a from the origin of a cylinder of radius a, for ka = 3 and M = 8. The red (dash) and black (solid) lines show the current and the previous (Morse¹; Morse & Ingard²) results respectively. [Incident pressure magnitude $P^{o} = 1 Pa$ ].

Figure 16.

Directivity patterns of the scattered sound intensity at a distance of r = 57a from the origin of a cylinder of radius a, for ka = 1 and M = 6. The red (dash) and black (solid) lines show the current and the previous (Morse¹; Morse & Ingard²) results respectively. [Incident pressure magnitude $P^{o} = 1 Pa$ ].

For case III, there is comparatively good agreement between the results. The intensity is uniformly distributed at the front of the cylinder and there is very little scattering at the rear of the cylinder. This result indicates that the analysis of Morse¹ and Morse & Ingard² may be useful for small values of ka, since only a few terms in the series are required for convergence. The total number of terms required for each case examined are summarised in Table 2. It should be taken into account that for accurate results the number of terms should not be less than that required for $γ_{m}$ and $ψ_{m}^{s}$ to become sufficiently small (relative error of less than 1 × 10⁻⁷) as shown in Figures 7 and 8.

Comparison with Wronskian equation

Junger & Feit³ have used instead the Wronskian relation to simplify the equation (6). The simplified equation presented by Junger & Feit³ is

p (a, φ_{i}) = \frac{2 P^{o}}{π k a} \sum_{m = 0} \frac{ε_{m} i^{m + 1}}{H_{m}^{'} (k a)} \cos (m φ_{i}),

(18)

where according to Wronskian³

J_{m} (k a) H_{m}^{'} (k a) - J_{m}^{'} (k a) H_{m} (k a) = \frac{2 i}{π k a} .

(19)

Equation (18) gives good agreement with the result obtained from equation (6) as shown in Figure 17. But the limitation of equation (18) is that it is unable to determine the sound pressure at locations in the sound field away from the cylinder surface, because there is no scope in that equation to consider the radial distance in the sound field from the cylinder.

Figure 17.

Total sound pressure comparison between results obtained at the surface of a cylinder of radius a using equations (6) and (18), for ka = 5. [Reference pressure 20 μPa and incident pressure magnitude $P^{o} = 1 Pa$ ].

Conclusions

The governing equations presented by Morse,¹ Morse & Ingard² and Junger & Feit³ for predicting the exterior sound field converge fully compared with the results obtained using the BEM. It was proved that a finite number of terms are required for each frequency of interest for convergence of those models. In other word, the analytical expressions do not require any low or high frequency or near-field and far-field limit for convergence.

It has been found that the asymptotic expressions presented by Morse¹ and Morse & Ingard² only converge for far-field calculations or when kr >> 1 or in the Fraunhofer diffraction region and do not converge in the near-field or Fresnel diffraction region. It was shown that the previous intensity results were also not correct because of the use of the asymptotic form of the Hankel function. The reason for this is that asymptotic expressions are restricting the expansion of the Hankel function. Also, it has been noticed that the asymptotic equation to determine the sound pressure field using wronskian presented by Junger & Feit³ can only predict the sound pressure at the surface of the cylinder and is unable to determine the sound pressure field away from the cylinder.

In essence, for convergence, one should use the full expansion of Hankel function in the governing analytical equations and abandon the use of asymptotic expressions.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Mir MM Morshed

Appendix

References

Morse

. Vibration and sound. New York: McGraw-Hill Inc., 1936. Chap. 7.

Morse

Ingard

. Theoretical acoustics. New York: McGraw-Hill Inc., 1986. Chap. 8.

Junger

Feit

. Sound, structures and their interaction. New York: Acoustical Society of America, 1993. Chap. 10.

Papamoschou

. Modeling of aft-emitted tonal fan noise in isolated and installed configurations. In AIAA Paper 2021-0009, AIAA Scitech 2021, 11–15 & 19–21 January 2021. DOI: 10.2514/6.2021-0009

, et al. A boundary element method based near field acoustic holography in noisy environments. J Acoust Soc America 2020; 147: 3360. DOI: 10.1121/10.0001225

Morata

Papamoschou

. Extension of traditional beamforming methods to the continuous-scan Ppradigm. In AIAA Paper 2022-1154, AIAA Scitech 2022, San Diego, CA, 3-7 January, 2022. DOI: 10.2514/6.2022-1154

Juhl

. The boundary element method for sound field calculations. In The acoustic laboratory. Technical University of Denmark, 1993.

Morshed

MMM

Hansen

Zander

, Prediction of acoustics loads on a launch vehicle fairing during lift-off, J Spacecrafts Rockets 2013; 50(1): 159–168. DOI: 10.2514/1.A32324

Morshed

MMM

Zander

Hansen

, Two-dimensional and three-dimensional acoustic loadings on cylinders due to a point source, J Am Inst Aeronautics Astronautics (AIAA) 2011; 49(11): 2421–2429. DOI: 10.2514/1.A33204

10.

Morshed

MMM

Hansen

Zander

. Prediction of acoustics loads on a launch vehicle: nonunique source allocation method. J Spacecrafts Rockets 2015; 52(5): 1478–1485. DOI: 10.2514/1.A33204

11.

Morshed

MMM

. Prediction of exterior sound pressure field of a cylinder using the boundary element method. Proced Eng 2013; 56: 857–863. DOI: 10.1016/j.proeng.2013.03.207

12.

Marburg

Nolte

. Computational acoustics of noise propagation in fluids- finite and boundary element methods. 1st ed. German: Springer, 2008.

13.

Marburg

, A review of the coupling parameter of the Burton and Miller boundary element method. In Inter-noise, Melbourne, Australia, 16-19 November, 2014, pp. 1–6.

14.

Marburg

Amini

. Cat’s eye radiation with boundary elements: comparative study on treatment of irregular frequencies. J Comput Acoust 2005; 13(1): 21–45. DOI: 10.1142/S0218396X05002566

15.

Seybert

Soenarko

Rizzo

, et al. A special integral equation formulation for acoustic radiation and scattering for axisymmetric bodies and boundary conditions. J Acoust Soc Am 1986; 80(4): 1241–1247. DOI: 10.1121/1.393817

16.

Seybert

Casey

. An integral equation method for coupled fluid/fluid scattering in three dimensions. J Acoust Soc Am 1988; 84(1): 379–384. DOI: 10.1121/1.396940

17.

Seybert

Cheng

CYR

. The solution of coupled interior/exterior acoustic problems using the boundary element method. J Acoust Soc Am 1990; 88(3): 1612–1618. DOI: 10.1121/1.400320

18.

Seybert

Soenarko

Rizzo

, et al. An advanced computational method for radiation and scattering of acoustic waves in three dimensions. J Acoust Soc Am 1985; 77(2): 362–368. DOI: 10.1121/1.391908

19.

Friot

Bordier

. Real-time active supression of scattered acoustic radiation. J Sound Vibrat 2004; 278: 563–580. DOI: 10.1016/j.jsv.2003.10.064