TranslationsTranslations4.24.2

TranslationsTranslations4.24.2

JMerrill, 2009JMerrill, 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 to the 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)

ax min2 2

M M ma

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.

1

Shifts• Vertical Shift• When a graph is

shifted vertically, then we add that number to the equation.

4

Shifts• 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.

2

Axis of the Wave• If the x-axis is not the 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 =

min2

Max

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?

5 13

2a

5 12

2x

3

3

Shifts• 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.

2

Reminder: Critical points

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

• To find the endpoints of any period:

4

PCI

2P phase shift

Write the Equation• Axis of the wave?

• Amplitude?

• Period?4

• So, a = 2, b = ?

4 02

2a

4 02

2x

2 24 2

p bb

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?2 2cos

2y x

Write the 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.

2 26

6 3b b

b

1

6

Write the Equation• a = 1

• Cosine wave• Equation:

• This is how the book does it.

3b

2 cos3 3

y x

2 cos 13

y x

Easier???

Same Graph - 2nd Equation

• Axis of wave = 2• Move the y-axis

to the left ½ unit.• Now it’s a sine

wave!• Equation?

12 sin

3 2y x

Equations of the Graph• Since sine and

cosine are both general sine waves, both equations are correct!

12 sin

3 2y x

2 cos ( 1)3

y x

You Do• Cosine wave:

• Another cosine wave?

3 2cos8y x

3 2cos88

y x

You Do—Part Deux• Sine wave:

3 2sin816

y x