Objectives:
1.
To model simple harmonic motion
2.
To model sound signals with sinusoids
Assignment:
• P. 361: 43, 48
• P. 362: 50
• P. 362: 51-54
• P. 362: 55-58 S
• P. 362: 59, 60, 61
• P. 363: 66
• HW Supplement: 2-7

Harmonic motion is movement that is periodic; that is, it involves a pattern that repeats at a fixed interval called the period.
• Pendulums, mass on a spring

Simple harmonic motion is the type of oscillating movement that is not damped.
• This means that the amplitude remains constant
• No friction or air resistance
• Can be modeled by a sinusoid where the independent variable is time and the dependent variable is displacement
Damped Simple

Damped
Click for GSP Demo

Simple

• Amplitude =
|Maximum
Displacement|
• Period = time it takes to complete one cycle (from one point back to that point
• Frequency = Number of cycles (periods) per second

What is the relationship between the period and the frequency of an object moving in simple harmonic motion?
If it takes 4 seconds to complete one cycle, then how many cycles occur in one second?

A point that moves on a coordinate line is said to be in simple harmonic motion if its distance d from equilibrium at time t is given by either d
 a sin
 
or d
 a cos
 
• a and b are real numbers, b > 0
• Amplitude of motion = | a |
• Period = 2π/ b
• Frequency = b /(2π)

Write the equation for the simple harmonic motion of the ball shown, where the period is 5 seconds. What is the frequency?

Given the equation for simple harmonic motion d = 4 cos 6π t , find each of the following:
1.
Period
2.
Frequency
3.
Maximum displacement
4.
The value of d when t = 4
5.
The least positive value of t for which d = 0

You will be able to model sound signals with sinusoids

There are two basic types of waves.
Longitudinal:
Displacement is parallel to the direction of the wave
Transverse: Displacement is perpendicular to the direction of the wave
© Dan Russell

A sound wave is longitudinal.
Longitudinal:
Displacement is parallel to the direction of the wave
Sound in air is created by a disturbance causing air molecules to move back and forth. Sound is basically oscillating air pressure.
© Dan Russell

Since sound is a type of oscillation, it can be represented mathematically by a sinusoid.

Actual Sound:
Mathematical
Representation:

Pictured are two different notes from the same piano.
• A period from each note looks the same

• Digitally, to change notes, you just have to scale the independent
(time) variable.

• The second note is more compressed over time: it oscillates faster
• The second note will sound higher; that is, it will have a higher pitch than the first note

• Digitally, to change volume, you just have to scale the amplitude.
• A phase shift just changes when the sound starts.

493.88 Hz
Click for GSP Demo
The pitch of a note, how high or low it sounds, is determined by the period:
• The shorter the period = higher pitch
• The longer the period
= lower pitch

493.88 Hz
Click for Frequency Generator
The pitch of a note can be described mathematically by the frequency of the wave.
• Frequency = period −1
• Units of frequency:
Hertz (cycles per sec)

Human hearing ranges from 20 Hz to 20 kHz.
What is the period of each of those signals?
Species
Human
Dog
Cat
Mouse
Hedgehog
Bat
Beluga Whale
Elephant
Porpoise
Approx. Hearing
Range
20-20,000
67-45,000
45-64,000
1,000-91,000
250-45,000
2,000-110,000
1,000-123,000
16-12,000
75-150,000

Where was this sound recorded?
• This came from the telephone which eliminates frequencies above
3800 Hz.

Sound signals can be represented mathematically with a sinusoid:
( )
 a sin
 
1.
What’s the period?
2.
We would want our sound wave to be periodic with some time interval T, not
2π. Rewrite the function so that it has a period of T.

Sound signals can be represented mathematically with a sinusoid:
( )
 a sin
 
3.
Finally, since frequency is the reciprocal of period, rewrite the function with 𝑓 .

A sound signal with an amplitude of | a | and a frequency of f Hertz can be approximated mathematically with the sinusoid
( )
 a
  
• This function could also be written with cosine
• You could also have a phase shift to change the time at which the sound starts

A sound signal with an amplitude of | a | and a frequency of f Hertz can be approximated mathematically with the sinusoid
( )
 a
  
• This is a very basic sound wave. Most sounds are far more complex and can be approximated by the sum of sinusoids of different frequencies and amplitudes: Click here for a demo.

Write a the function of a sound signal with a frequency of 440 Hz and an amplitude of 5.

Objectives:
1.
To model simple harmonic motion
2.
To model sound signals with sinusoids
Assignment:
• P. 361: 43, 48
• P. 362: 50
• P. 362: 51-54
• P. 362: 55-58 S
• P. 362: 59, 60, 61
• P. 363: 66
• HW Supplement: 2-7