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Translations 4.2. JMerrill, 2009. Sine Waves. Recorded sounds, that are periodic, are some kind of sine wave. Most of these sine waves are not of the “parent function” type. They are either not lying nicely on the x-axis or are a combination of waves (don’t worry—we’ll get to that!). YEA!!.

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Translations 4 2

Translations4.2

JMerrill, 2009


Sine waves
Sine Waves

  • Recorded sounds, that are periodic, are some kind of sine wave. Most of these sine waves are not of the “parent function” type. They are either not lying nicely on the x-axis or are a combination of waves (don’t worry—we’ll get to that!)

YEA!!


Signals sound
Signals: Sound

  • These terms are used to describe sine waves and other waveforms precisely:

  • Period: The period is the time taken for one complete cycle of a repeating waveform.

  • Frequency: This is the number of cycles completed per second. The measurement unit for frequency is the hertz, Hz. 1 Hz = 1 cycle/second.

  • The pitch of a musical note is the same as its frequency (which relates tothe period).

  • The intensity (loudness) of a musical note is the same as its amplitude


Radio waves
Radio Waves

  • Radio transmissions are a combination of two kinds of waves: audio frequency waves that represent the sounds being transmitted and radio frequency waves that "carry" the audio information. All waves have a wavelength, an amplitude and a frequency. These properties allow the wave to be modified to carry sound information.


Am radio waves
AM Radio Waves

  • In AM (amplitude modulation) radio transmissions, the amplitude of the combined audio frequency and radio frequency waves varies to match the audio signal. AM radio is subject to problems with static interference. Electromagnetic waves (like radio waves) are produced by the spark discharges in car ignition systems, brushes of electric motors and in all sorts of electrical appliances, as well as in thunderstorms. Background noise changes the amplitude of the radio wave signal adds random crackling noises called static.


Fm radio waves
FM Radio Waves

  • In FM (frequency modulation) radio transmissions, the frequency of the combined waves changes to reproduce the audio signal. For example, higher frequency is associated with the peak amplitude in the audio wave. FM waves do not have a problem with interference because the background noise does not modify the radio wave frequency. In addition FM waves give better sound reproduction.


General form
General Form

  • Equations will be of the form

  • y = k + A sin (Bx + C), or

  • y = k + A cos (Bx + C), where

    • k = vertical shift

    • A = amplitude

    • B = frequency—the way to find the period

    • C = phase (phase shift)—our new “origin”, found by –C/B. However…


General form1
General Form

  • Use my way and the phase shift finds itself:

  • y = k + A sin B(x + C)


Amplitude revisited
Amplitude revisited

  • If the center of the wave is not at the x-axis, then amplitude can be found by

  • Amplitude can still be measured by the vertical distance between the center of the wave to the peak (and/or valley)


Shifts

Phase Shift

When a graph is shifted h units horizontally, then x is replaced with (x+c)

Remember that a phase shifts acts in the opposite direction—just like all other functions.

Shifts

1


Shifts1
Shifts

  • Vertical Shift

  • When a graph is shifted vertically, then we add that number to the equation.

4


Shifts2

When the sine wave is shifted units to the left, what is the result?

A cosine wave!

So, sine and cosine curves are referred to as general sine waves.

Shifts


Axis of the wave
Axis of the Wave is the result?

  • If the x-axis is notthe center of the wave, then you need to find the center. The center is the average of the peak and the valley points

  • Axis of the wave: x =


Example
Example is the result?

  • To find the axis of the wave

  • To find the amplitude

  • To verify the amplitude, what is the vertical distance from the axis of the wave to the peak or valley?

3

3


Shifts3

When the sine wave is shifted units to the left, what is the result?

A cosine wave!

So, sine and cosine curves are referred to as general sine waves.

Shifts


Reminder critical points
Reminder: Critical points is the result?

  • To find the critical intervals (max/min, intercepts)

  • To find the endpoints of any period:


Write the equation
Write the Equation is the result?

  • Axis of the wave?

  • Amplitude?

  • Period?

    4

  • So, a = 2, b = ?


Write the equation1
Write the Equation is the result?

  • To write the equation, look at the new x/y-axis (forget the old). Here, we changed the x-axis, but not the y-axis.

  • Sine or cosine?

    Cosine

  • The vertical shift is the amount we raised the x-axis.

  • Equation?


Write the equation2

Axis of the wave? is the result?

x = 2

Amplitude?

Period?

B?

Sine or cosine?

Can’t tell? Move the y-axis. Sine or cosine?

Now, use the new set of axes and write the equation.

Write the Equation


Write the equation3
Write the Equation is the result?

  • a = 1

  • Cosine wave

  • Equation:

  • This is how the book does it.

Easier???


Same graph 2 nd equation

Axis of wave = 2 is the result?

Move the y-axis to the left ½ unit.

Now it’s a sine wave!

Equation?

Same Graph - 2nd Equation


Equations of the graph
Equations of the Graph is the result?

  • Since sine and cosine are both general sine waves, both equations are correct!


You do
You Do is the result?

  • Cosine wave:

  • Another cosine wave?


You do part deux
You Do—Part Deux is the result?

  • Sine wave: