Microphones with a Cardioid Pickup Pattern
A microphone with a cardioid pickup pattern is often the choice of sound engineers recording
live performances. The cardioid pattern offers good frontal sensitivity, while suppressing
audience noise in back of the microphone. Consider the figure below. Find the area on the stage
that lies within the optimal pickup range of the microphone, if the boundaries of this region are
given by the cardioid r = 20 + 20 cos θ and the vertical line x = 15 (the microphone has been
placed 15 feet from the edge of the stage).
y
stage
audience
x
microphone
15 ft
r = 20 + 20 cos θ
x = 15
© 2006, Jim Trefzger, Parkland College. Used with permission.
Solution:
First, rewrite x = 15 in polar coordinates.
x = 15
r cos θ = 15
15
r=
cos θ
r = 15 sec θ
Next, find the θ values of the points of intersection.
15
cos θ
2
20 cos θ + 20 cos θ = 15
20 + 20 cos θ =
4 cos 2 θ + 4 cos θ − 3 = 0
(2 cos θ + 3)(2 cos θ − 1) = 0
cos θ = −
3
1
or cos θ =
2
2
θ =±
π
3
The integral for the area of the region is then:
[
]
π 3
1
(20 + 20 cos θ )2 − (15 secθ )2 dθ
∫
2 −π 3
1
= 2⋅
2
π 3
=
π 3
∫ [400 + 800 cosθ + 400 cos
2
θ − 225 sec 2 θ ]dθ
0
⎡
∫ ⎢⎣400 + 800 cosθ + 400
0
1 + cos 2θ
⎤
− 225 sec 2 θ ⎥ dθ
2
⎦
π 3
=
∫ [600 + 800 cosθ + 200 cos 2θ − 225 sec θ ]dθ
2
0
= (600θ + 800 sin θ + 100 sin 2θ + 225 tan θ ) 0
π 3
⎛ 3⎞
⎛ 3⎞
⎛π ⎞
⎟ + 100⎜
⎟
= 600⎜ ⎟ + 800⎜⎜
⎟
⎜ 2 ⎟ − 225 3 − 0
2
⎝3⎠
⎝
⎠
⎝
⎠
= 200π + 225 3
≈ 1018 ft 2
© 2006, Jim Trefzger, Parkland College. Used with permission.