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)=ikρ0cP(x,k)θ1r,
(4a)
Vϕ(x,k)=ikρ0cP(x,k)ϕ1rsinθ.
(4b)
Accordingly, the correct versions of Eqs. (5b), (5c), (9b), and (9c) are expressed, respectively, as

Vθ(x,k)=ikρ0cn=0m=nnαnm(k)jn(kr)rAnmPnm(cosθ)eimϕ,
(5b)
Vϕ(x,k)=ikρ0cn=0m=nnimαnm(k)jn(kr)rAnmP̂nm(cosθ)eimϕ,
(5c)
Iθ(x,k)=n=0m=nnn=0m=nnAnmTnmnm(k,r)rYnm*(θ,ϕ)Pnm(cosθ)eimϕ,
(9b)
Iϕ(x,k)=n=0m=nnn=0m=nnimAnmTnmnm(k,r)rYnm*(θ,ϕ)P̂nm(cosθ)eimϕ,
(9c)

where

P̂nm(cosθ)={Pnm(1),ifθ=0Pnm(1),ifθ=πPnm(cosθ)sinθ,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=0m=nnn=0m=nnAnmAnmApqPnmnmpqEmmqTnmnm(k,r)r,
(14)
Spq(ϕ)(k,r)=n=0m=nnn=0m=nnimAnmAnmApqP̂nmnmpqEmmqTnmnm(k,r)r,
(10b)

where

P̂nmnmpq=H(n,m)H(n,m)H(p,q)G(m+m+q+12,4δm+nδm+nδp+q2;1+mnδm+n2,1+mnδm+n2,1+qpδp+q2;2+m+n2δm+n2,2+m+nδm+n2,2+p+qδp+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
).