From the first post, we know if we take two identical signals, delay one, and then sum the them, there will be frequency cancellation known as comb filtering. As abstract as that sounds, it actually happens in physical environments. If one of your speakers is farther from your ear than the other, two identical signals, one delayed, will reach your ears (the identical signals will be whatever is coming from the center of the stereo image, not the left and right). Another example might be the direct sound from the speakers, and the first reflection off of a desk or wall. Lastly, if you are recording an instrument using multiple microphones, an identical signal could reach the microphones at different times if they are spaced unevenly from the source.
What kinds of distances should we be concerned about? A foot? An inch? Here’s I’ll derive a formula for how distance corresponds to delay and frequency cancellation.
In the previous post, we found which frequencies are cancelled as a function of the delay:
f = [(2n+1)*180°] / [Δ*360°] where Δ is the time delay in seconds
We want to relate distance to delay. The units of speed of sound are feet/ second, so if we multiply (feet/second) by seconds, we get feet, the unit of distance. We also have a handy formula above that relates seconds to frequency. So here’s the math:
v * Δ = d (v feet per second, Δ seconds, d feet)
Δ = [(2n+1)*180°] / [f*360°] (f hertz — from the formula above)
v * [(2n+1)*180°] / [f*360°] = d (by substituting in Δ)
f = [(2n+1)*180°*v] / [d*360°] (frequency cancelled as a function of velocity and distance)
Because we solved out math problem in variables, we can choose the system of units we prefer. Because I’m from the United States, and most of the people who read this will probably be from the U.S., I’m going to use feet/inches.
v = 1100 ft/sec
Frequencies Cancelled:
From the table, we can see that 1 inch is relatively harmless because the frequencies cancelled very quickly go beyond the range of human hearing. If we are setting up our microphones or speakers, we probably don’t need to worry about distances much smaller than an inch. This suggests that, rather than be concerned with minute delays from direct sound, we should be more concerned with reflections. Reflections will be the topic of the next post.
kelsound 