Boundary Microphone
Uncategorized August 7th. 2009, 9:12amBoundary Microphone
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Using Shure Boundary Mics with a Canon HV 30
Calculus, help is very much appreciated!?
When recording live performances, sound engineers often use a microphone with a cardioid pickup pattern because it suppresses noise from the audience. Suppose the microphone is placed 4 m from the front of the stage and the boundary of the optimal pickup region is given by the cardioid r = 8 + 8sin(theta), where r is measured in meters and the microphone is at the pole. The musicians want to know the area they will have on stage within the optimal pickup range of the microphone. Answer their question.
Any help is very much appreciated!
If you have a function r = f(θ), then the area between the origin and the curve between a <= θ <= b is
(1/2) ∫( f(θ) )^2 dθ, integrated from a to b
So you just have to integrate this:
(1/2) ∫(8 + 8sin(θ) )^2 dθ
(1/2) ∫ 64(1 + sin(θ) )^2 dθ
32 ∫ (1 + sin(θ) )^2 dθ
32 ∫ 1 + 2sin(θ) + sin^2(θ) dθ
Since cos(2x) = cos^2(x) - sin^2(x) = 1 - 2sin^2(x), then sin^2(θ) = (cos(2θ) - 1)/2
32 ∫ 1 + 2sin(θ) + (cos(2θ) - 1)/2 dθ
32 [ θ - 2cos(θ) + (1/4)sin(2θ) - (1/2)θ ]
Try the full range of 0 <= θ <= 2pi
Nice question. I do a podcast and I was fiddling around with microphone settings just last night. I've worked with cardioid mics before, but never thought of applying polar equations to it.
