143
where e r is a unit vector in the radial direction so the time-averaged intensity is
given by
I r
v
e r
( ) =
( )=
æ
è
ç
ö
ø
÷
1
2
2
0
2
0
2
Re ˆ ˆ
ˆ
*
p
p
c
r
r
r
(7)
The time-averaged intensity always points in the radial direction, i.e. away from the
source. The fish, however, measures particle velocity in its own body-centered coordinate system. If we assume the positive direction in the fish’s coordinate system is
towards its head, the time-average value of the product of p and v will be positive if
the source is behind the fish and negative if it is in front of the fish. In the near field
(kr ≪ 1), the term in ˆ ˆ *
pv involving 1/ikr which would be much larger than the
other term but it does not contribute to I r
( ) because it is imaginary. The timeaveraged intensity, I r
( ) , is proportional to 1/r
2 so the total power passing through
a spherical surface centered on the source, at any range from the source is constant
as is required by conservation of energy. For the field of a point monopole which has
arbitrary time dependence
p t
r p t
r
c
r
t
p t
r
c
c
p t
r
c
d
r
v r
,
a nd
,
( )=
-
æ
è
ç
ö
ø
÷
( )=
-
æ
è
ç
ö
ø
÷
-
-
æ
è
ç
ö
ø
÷
ò
0
r
t t
r
r
r
é
ë
ê
ê
ê
ê
ù
û
ú
ú
ú
ú
e
(8)
From Eq. (3), the time-averaged intensity is
I r
r
r
r
( ) =
( )
-
( ) ( )
(
)
é
ë
ê
ê
ê
ê
ê
ù
û
ú
ú
ú
ò
ò
ò
0
2
0
t
t
p t dt
cT
p t p t dt dt
r T
,
,
,
r
r
ú ú
ú
=
( )
-
( )
(
)
é
ë
ê
ê
ê
ê
ê
ù
û
ú
ú
ú
ú
ú
=
ò
ò
ò
0
2
2
0
0
t
T
t
p t dt
cT
p t dt
r T
p
r
r
e r
,
,
r
r
r r
e r
, t dt
cT
( )
2
r
(9)
The second term in the brackets is zero because the pressure is zero before the wave
arrives and after it leaves and in any case decreases with increasing T. The first term
in the brackets is always positive so, again, if the time-average value of the product
Directional Hearing and Sound Source Localization in Fishes
where e r is a unit vector in the radial direction so the time-averaged intensity is
given by
I r
v
e r
( ) =
( )=
æ
è
ç
ö
ø
÷
1
2
2
0
2
0
2
Re ˆ ˆ
ˆ
*
p
p
c
r
r
r
(7)
The time-averaged intensity always points in the radial direction, i.e. away from the
source. The fish, however, measures particle velocity in its own body-centered coordinate system. If we assume the positive direction in the fish’s coordinate system is
towards its head, the time-average value of the product of p and v will be positive if
the source is behind the fish and negative if it is in front of the fish. In the near field
(kr ≪ 1), the term in ˆ ˆ *
pv involving 1/ikr which would be much larger than the
other term but it does not contribute to I r
( ) because it is imaginary. The timeaveraged intensity, I r
( ) , is proportional to 1/r
2 so the total power passing through
a spherical surface centered on the source, at any range from the source is constant
as is required by conservation of energy. For the field of a point monopole which has
arbitrary time dependence
p t
r p t
r
c
r
t
p t
r
c
c
p t
r
c
d
r
v r
,
a nd
,
( )=
-
æ
è
ç
ö
ø
÷
( )=
-
æ
è
ç
ö
ø
÷
-
-
æ
è
ç
ö
ø
÷
ò
0
r
t t
r
r
r
é
ë
ê
ê
ê
ê
ù
û
ú
ú
ú
ú
e
(8)
From Eq. (3), the time-averaged intensity is
I r
r
r
r
( ) =
( )
-
( ) ( )
(
)
é
ë
ê
ê
ê
ê
ê
ù
û
ú
ú
ú
ò
ò
ò
0
2
0
t
t
p t dt
cT
p t p t dt dt
r T
,
,
,
r
r
ú ú
ú
=
( )
-
( )
(
)
é
ë
ê
ê
ê
ê
ê
ù
û
ú
ú
ú
ú
ú
=
ò
ò
ò
0
2
2
0
0
t
T
t
p t dt
cT
p t dt
r T
p
r
r
e r
,
,
r
r
r r
e r
, t dt
cT
( )
2
r
(9)
The second term in the brackets is zero because the pressure is zero before the wave
arrives and after it leaves and in any case decreases with increasing T. The first term
in the brackets is always positive so, again, if the time-average value of the product
Directional Hearing and Sound Source Localization in Fishes