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### Translations4.2

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!!

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 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

- 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

- 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

- 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 Form

- Use my way and the phase shift finds itself:
- y = k + A sin B(x + C)

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)

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.

Shifts1

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.

ShiftsAxis of the Wave

- 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

- 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

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.

ShiftsReminder: Critical points

- To find the critical intervals (max/min, intercepts)
- To find the endpoints of any period:

Write the Equation

- 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?

Axis of the wave?

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 EquationAxis of wave = 2

Move the y-axis to the left ½ unit.

Now it’s a sine wave!

Equation?

Same Graph - 2nd EquationEquations of the Graph

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

You Do

- Cosine wave:
- Another cosine wave?

You Do—Part Deux

- Sine wave:

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