Mathematics and Vibrations for AVT

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Mathematics and Vibrations for AVT

• Dr Ian Drumm
• Logarithmic Functions

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Logarithms

• Reverse of raising a base a to a power n

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Natural Log Function

• The index of the exponential term can be found
  using the natural log function
• Conversely
• For example given
• Log both sides to solve for x

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Log to the base 10

• Base for Sound Pressure Level (in decibels)
• Where is the intensity of the faintest
  audible sound at 1kHz (i.e. 10-12 Wm-2)
• For most simple cases we can assume sound
  intensity level and sound pressure level are the
  same thing (though there is a subtle difference
  due to the directional nature by which intensity
  is defined)

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Transposing to get intensity

• Important you can do this!

Here we lose log term knowing
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Laws of indices

• Useful

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Acoustic Example

• A sound source gives a maximum SPL of 80dB ref
  10-12Wm-2. What is the SPL from two such
  incoherent sources?
• Two sources give twice the intensity

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Another important identity
For example
Why do you think ?
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Useful

• We know

• Hence..

• For example

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Useful

• We know
• Similarly
• So why the 20µP in
  ?

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Changing Bases

• Useful if theres no function for log to required
  base, or for simplifying equations
• i.e. divide by log of base, for example

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The logarithmic function
log(1)0 log(x) where xlt1 is negative Log(0) is
infinity Log(x) where xlt0 doesnt exist
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Solving equations involving logarithms and
exponentials
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Solving equations

• Sometimes you need to express logs to same base,
  for example solve for