Sound, Sinusoids, and Spectrum

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Sound, Sinusoids, and Spectrum

• Striking a Tine
• Strings
• Tubes
• Analog vs. Digital

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A program you may be interested in...
Goldwave audio editor www.goldwave.com
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What is sound?
How does it get made? How is it heard?
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What is sound?

• Sound is a wave of vibrating pressure variations
  created in a medium
• Alternatively compressing and decompressing air

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Let us begin with...

• How do some simple sound makers work?

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A Tine
x
x is a function of time t x(t)
A metal rod that is flexible and springy
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Forces on a Tine
x
This is pushing back with force F
Well say F -kx K is the springiness
factor As an equation F(t) -kx(t)
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Newtons Law
This is pushing back with force F
Newtons Law
F ma F -kx Hence ma-kx
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What is x?
Solving for a
Any suggestions as to a function for x?
What is acceleration?
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Sine waves (Sinusoids)
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Solution
TT
1,2
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Tines Vibrate as a Sinusoid
alpha

• Pattern of vibration is asin(wt), where a is the
  amplitude and w is the frequency.

omega
Play tine audio
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Important properties of sinusoids
f(t)asin(wt)

• Frequency
• Number of radians per second w
• Number of cycles per second (Hz) fw/2p
• Frequency ranges
• Infrasonic 0 20Hz
• Audiosonic 20Hz 20KHz (what we hear)
• Ultrasonic 20KHz on up...
• Piano keyboard 29Hz 4186Hz

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Important properties of sinusoids
f(t)asin(wt)

• Amplitude (a)
• What we perceive as loudness
• Phase
• Relative left-to-right placement of the wave
• f(t)asin(wtf)
• Period (in seconds)
• 1/f or 2p/w

phi
TT
3,4
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Tines in Summary

• A tine vibrates with a sinusoidal pattern
• The sound it emits is a sine wave
• An observation
• To halve the frequency, make the tine 4 times as
  massive (why?)

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How does the sound get to you?

• Tine causes alternate compression and
  decompression of the air
• Air causes what in your ear?

• How do we capture the sound electronically?

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What happens when I pluck a string?
Fact Frequency is inverse proportional to length
Play guitar audio
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Damping the string
node
What if I hold the string in the middle?
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Modes
mode1
mode2
mode3
And it can do all of these at the same time.
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Plucking

• Plucking is a complex deformation of the string
• Many modes become active simultaneously
• Each mode is a sinusoidal signal
• Each mode has its own phase
• Does how I pluck affect the sound?

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Phase cancellation?

• Why do we hear mode 2 (or any even mode)?

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The sound of a string

• The actual sound is a sum of sine waves.

Real world other junk is included weight
changes over time
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Harmonics

• Fundamental
• h1
• Harmonics
• hgt1
• We refer to the second, third, etc. harmonics
• No such thing as a first harmonic
• Harmonics are typically at lower amplitudes as
  frequency increases

Play guitar audio
TT
5, 6
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Amazing facts dept.

• Think of sine waves as the pure colors of sound
• Any signal (sound) can be expressed as a sum of
  sine waves!
• Okay, a potentially infinite number, of course.

• Dynamic properties
• The components usually vary over time

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Sine Waves

• Sine waves
• f(t) asin(wt)
• a Amplitude
• w Frequency (radians per second)

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Tubes

• Imagine a sound going down a long tube with a
  closed end

Play clarinet audio
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Phase of the reflection

• What will be the phase of a reflected pulse?

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Reflections in a tube

• The reflection is delayed and the phase is
  inverted
• Some frequencies will cancel, but some will be
  boosted
• Certain frequencies will be favored by the tube
  because they add, reflect again, add again, etc.
• Resonance!

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What are the resonant frequencies?

• Delay ½ cycle and invert phase
• Delay 1 ½ cycle and invert phase (3x)
• Delay 2 ½ cycle and invert phase (5x)

These are modes
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Whats the frequency?

• Let d be the round trip travel time
• ½ cycle delay in d is 1 cycle delay in 2d
• Period of 2d is frequency of 1/2d
• Example
• For a pipe organ to play C1 32.7 Hz,
• f1/2d implies d1/2f, so d0.01529 seconds
• Speed of sound at 22?C is 344.92 m/s
• L0.01529 344.92 / 2 2.636 meters (7.7 ft)

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Resonant frequencies for a tube

• Same as for a string except ch0 for all even h.
• Only the odd harmonics!

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Percussive Instruments

• What about round items like a drum head?
• Membranophones

Play snare audio
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Types of instruments

• Tines
• Fundamental only
• Strings
• All harmonics
• Tubes
• Odd harmonics only
• Percussive
• Non-harmonic