Erratum: Spatial sound intensity vectors in spherical harmonic domain [J. Acoust. Soc. Am. 145(2), EL149–EL155 (2019)]

Zuo, Huanyu; Samarasinghe, Prasanga N.; Abhayapala, Thushara D.; Dickins, Glenn

doi:10.1121/1.5116565

This erratum concerns the expression of acoustic particle velocity [Eq. (4)] in the original paper.¹ We have mistakenly omitted a coordinate transformation term for the expression. The expression in the original paper is correct for the direction r. However, for the direction θ and ϕ, the correct equations should be as follows:

(4)

V_{θ} (x, k) = \frac{i}{k ρ_{0} c} \frac{\partial P (x, k)}{\partial θ} \frac{1}{r},

(4a)

V_{ϕ} (x, k) = \frac{i}{k ρ_{0} c} \frac{\partial P (x, k)}{\partial ϕ} \frac{1}{r sin θ} .

(4b)

Accordingly, the correct versions of Eqs. , , , and are expressed, respectively, as

V_{θ} (x, k) = \frac{i}{k ρ_{0} c} \sum_{n = 0}^{\infty} \sum_{m = - n}^{n} α_{nm} (k) \frac{j_{n} (kr)}{r} A_{nm} P_{nm}^{'} (cos θ) e^{im ϕ},

(5b)

V_{ϕ} (x, k) = \frac{i}{k ρ_{0} c} \sum_{n = 0}^{\infty} \sum_{m = - n}^{n} i m α_{nm} (k) \frac{j_{n} (kr)}{r} A_{nm} {\hat{P}}_{nm} (cos θ) e^{im ϕ},

(5c)

I_{θ} (x, k) = \sum_{n = 0}^{\infty} \sum_{m = - n}^{n} \sum_{n^{'} = 0}^{\infty} \sum_{m^{'} = - n^{'}}^{n^{'}} A_{n^{'} m^{'}} \frac{T_{nm n^{'} m^{'}} (k, r)}{r} Y_{nm}^{*} (θ, ϕ) P_{n^{'} m^{'}}^{'} (cos θ) e^{i m^{'} ϕ},

(9b)

I_{ϕ} (x, k) = \sum_{n = 0}^{\infty} \sum_{m = - n}^{n} \sum_{n^{'} = 0}^{\infty} \sum_{m^{'} = - n^{'}}^{n^{'}} i m^{'} A_{n^{'} m^{'}} \frac{T_{nm n^{'} m^{'}} (k, r)}{r} Y_{nm}^{*} (θ, ϕ) {\hat{P}}_{n^{'} m^{'}} (cos θ) e^{i m^{'} ϕ},

(9c)

where

{\hat{P}}_{nm} (cos θ) = {\begin{matrix} P_{nm}^{'} (1), & if θ = 0 \\ - P_{nm}^{'} (- 1), & if θ = π \\ \frac{P_{nm} (cos θ)}{sin θ}, & otherwise . \end{matrix}

Therefore, the expressions of intensity coefficients for the θ [Eq. (14)] and ϕ [Eq. (10b)] direction in the original paper should be replaced, respectively, with

S_{pq}^{(θ)} (k, r) = \sum_{n = 0}^{\infty} \sum_{m = - n}^{n} \sum_{n^{'} = 0}^{\infty} \sum_{m^{'} = - n^{'}}^{n^{'}} A_{nm} A_{n^{'} m^{'}} A_{pq} P_{nm n^{'} m^{'} p q} E_{m m^{'} q} \frac{T_{nm n^{'} m^{'}} (k, r)}{r},

(14)

S_{pq}^{(ϕ)} (k, r) = \sum_{n = 0}^{\infty} \sum_{m = - n}^{n} \sum_{n^{'} = 0}^{\infty} \sum_{m^{'} = - n^{'}}^{n'} i m^{'} A_{nm} A_{n^{'} m^{'}} A_{pq} {\hat{P}}_{nm n^{'} m^{'} p q} E_{m m^{'} q} \frac{T_{nm n^{'} m^{'}} (k, r)}{r},

(10b)

where

\begin{matrix} {\hat{P}}_{nm n^{'} m^{'} p q} = H (n, m) H (n^{'}, m^{'}) H (p, q) G (\frac{m + m^{'} + q + 1}{2}, \frac{4 - δ_{m + n} - δ_{m^{'} + n^{'}} - δ_{p + q}}{2}; \\ \frac{1 + m - n - δ_{m + n}}{2}, \frac{1 + m^{'} - n^{'} - δ_{m^{'} + n^{'}}}{2}, \frac{1 + q - p - δ_{p + q}}{2}; \frac{2 + m + n}{2} - \frac{δ_{m + n}}{2}, \\ \frac{2 + m^{'} + n^{'} - δ_{m^{'} + n^{'}}}{2}, \frac{2 + p + q - δ_{p + q}}{2}; m + 1, m^{'} + 1, q + 1) . \end{matrix}

Note that the proof for $S_{pq}^{(ϕ)} (k, r)$ here is similar to the proof for $S_{pq}^{(θ)} (k, r)$ in the original paper, which is correct. Without the Wigner 3-j symbols in the expression of $S_{pq}^{(ϕ)} (k, r)$ ⁠, the active order is not 2 N anymore. However, the truncation error also falls to an acceptable value as the truncation order increases. Figure 2 from the original paper must also be replaced with the figure given here. Note that the performance of sound intensity in the ϕ direction, similar to the θ direction, is slightly worse than that in the r direction as well due to the truncation error.

FIG. 2.

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Sound intensity on a sphere with radius of 0.05 m, generated by a plane wave from (3π/4, 5π/6), with frequency 600 Hz. (a)–(c) Sound intensity in the r, θ, and ϕ directions, separately, calculated using the proposed theory, (d)–(f) sound intensity in the r, θ, and ϕ directions, separately, obtained from point by point measurement.

ACKNOWLEDGMENTS

The authors very much appreciate Byeongho Jo of Korea Advanced Institute of Science and Technology for discovering this problem.

References

1.

H.

Zuo

,

P. N.

Samarasinghe

,

T. D.

Abhayapala

, and

G.

Dickins

, “

Spatial sound intensity vectors in spherical harmonic domain

,”

J. Acoust. Soc. Am.

145

(

2

),

EL149

–

EL155

(

2019

).

https://doi.org/10.1121/1.5090197

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Acoustical Society of America

Erratum: Spatial sound intensity vectors in spherical harmonic domain [J. Acoust. Soc. Am. 145(2), EL149–EL155 (2019)]

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Erratum: Spatial sound intensity vectors in spherical harmonic domain [J. Acoust. Soc. Am. 145(2), EL149–EL155 (2019)]

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