Mathematics and Vibrations for AVT Dr Ian Drumm

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Mathematics and Vibrations for AVTDr Ian Drumm

• Review Logs
• Series Expansion

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Logarithms Reminder
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Linearisation of Exponential Functions

• Expressing exponential functions as linear
  functions mean they are easier plot and hence
  making easier to extrapolate results

So plot ln(P) against t Gradient is Bln(2) Line
cross y axis at ln(A)
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Logarithms in Acoustics and Audio
Sound intensity level ( sound pressure level
SPL) is function of Intensity
If given SPLs (in dB) you can find new SPL by
first converting to intensities, doing addition
or subtraction and then converting back to SPL
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Binomial Expansion

• Consider the polynomial expansions
• Notice how a and b terms change from left to right

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Pascals Triangle

• Coefficients form a triangle
• There are n1 coefficients
• Coefficients are symmetric
• Individual coefficients given by factorial
  numbers
• Note 0!1

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For example

• For polynomial to order 3 find a given term in
  expansion

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Often given as standard formula

• Binomial expansion of

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Example

• Expand
• Applying formula
• Hence

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• Given expansion such as
• Write as
• Substitute
• Expansions can be given as approximate if

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Non-Polynomials

• Binomial Expansion valid for negative and
  fractional values of n if -1 lt x lt 1
• For example
• Series infinite
• Converges towards zero

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Other series

• Exponential
• Trigonometric
• Small angles approximation