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)
$Vθ(x,k)=ikρ0c∂P(x,k)∂θ1r,$
(4a)
$Vϕ(x,k)=ikρ0c∂P(x,k)∂ϕ1r sin θ.$
(4b)
Accordingly, the correct versions of Eqs. (5b), (5c), (9b), and (9c) are expressed, respectively, as

$Vθ(x,k)=ikρ0c∑n=0∞∑m=−nnαnm(k)jn(kr)rAnmPnm′(cos θ)eimϕ,$
(5b)
$Vϕ(x,k)=ikρ0c∑n=0∞∑m=−nnimαnm(k)jn(kr)rAnmP̂nm(cos θ)eimϕ,$
(5c)
$Iθ(x,k)=∑n=0∞∑m=−nn∑n′=0∞∑m′=−n′n′An′m′Tnmn′m′(k,r)rYnm*(θ,ϕ)Pn′m′′(cos θ)eim′ϕ,$
(9b)
$Iϕ(x,k)=∑n=0∞∑m=−nn∑n′=0∞∑m′=−n′n′im′An′m′Tnmn′m′(k,r)rYnm*(θ,ϕ)P̂n′m′(cos θ)eim′ϕ,$
(9c)

where

$P̂nm(cos θ)={Pnm′(1),if θ=0−Pnm′(−1),if θ=πPnm(cos θ) sin θ,otherwise.$

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

$Spq(θ)(k,r)=∑n=0∞∑m=−nn∑n′=0∞∑m′=−n′n′AnmAn′m′ApqPnmn′m′pqEmm′qTnmn′m′(k,r)r,$
(14)
$Spq(ϕ)(k,r)=∑n=0∞∑m=−nn∑n′=0∞∑m′=−n′n′im′AnmAn′m′ApqP̂nmn′m′pqEmm′qTnmn′m′(k,r)r,$
(10b)

where

$P̂nmn′m′pq=H(n,m)H(n′,m′)H(p,q)G(m+m′+q+12,4−δm+n−δm′+n′−δp+q2;1+m−n−δm+n2,1+m′−n′−δm′+n′2,1+q−p−δp+q2;2+m+n2−δm+n2,2+m′+n′−δm′+n′2,2+p+q−δp+q2;m+1,m′+1,q+1).$

Note that the proof for $Spq(ϕ)(k,r)$ here is similar to the proof for $Spq(θ)(k,r)$ in the original paper, which is correct. Without the Wigner 3-j symbols in the expression of $Spq(ϕ)(k,r)$, 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.

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.

FIG. 2.

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.

Close modal

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

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
).