Mathematics and Vibrations for AVT

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Mathematics and Vibrations for AVT

• Waves, Coordinate Systems
• Acoustic Applications

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Trigonometry and waves

• Represent simple harmonic motion as circle or
  sinusoid
• Amplitude is R
• If Period is T
• Angular frequency is

In rads s-1
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Phase

• Consider the wave at a point further on in its
  cycle as represented by the phase difference a

We can say the phase lead in seconds is
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Phase difference

• Given two waves
• cos leads sin by a phase difference given by in
  radians of p/2
• Whats the phase difference in seconds?

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Another example

• What the phase lag between

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Combining Sinusoids

• Consider
• How do we get in form ?
• Using addition formula
• Use trig identities to get C a

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Combining two waves
Beware give angle a for correct quadrant,
knowing that tan is periodic repeating every p
or 1800
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Cartesian Coordinates
In 2D (x,y) In 3D (x,y,z)
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Polar Coordinates
(r,?)
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Converting between Cartesian and Polar

• Converting to Cartesian
• Converting to polar

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Some Polar Patterns for Microphones
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Some Polar Patterns for Loudspeakers
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Longitude Azimuth
Latitude Elevation
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Spherical Coordinates
(r,?,?)
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Conversion from Polar to Cartesian Coordinates
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Conversion from Cartesian to Polar Coordinates
If -x