Waves

1
Waves

• Phys 2101
• Gabriela González

2
Superposition of waves

• If we have two waves traveling in the same
  string, they will overlap and add up to a
  resultant wave.
• Overlapping waves do not in any way alter the
  travel of each other !!

3
Interference of waves

• Two waves traveling in the same string, in the
  same direction, with the same frequency, will add
  up to a single resultant wave with the same
  frequency, but different amplitude and phase
• y(x,t) y1 sin(kx-wtf1) y2 sin(kx-wtf2)
• ym sin(kx-wtf)
• with ym2y12y222y1y2cos(f1-f2)
• and tan f (y1 sin f1 y2 sin f2)/ (y1 cos
  f1 y2 cos f2)

Interference term!
4
Interference

• Total amplitude ym2y12y222y1y2cos(f1-f2)
• If f1-f20, 2p,, ymy1y2 constructive
  interference
• If f1-f2p, 3p,, ymy1-y2 destructive
  interference

5
Standing waves

• Waves traveling in opposite directions can also
  be added. If they have the same frequency, they
  will produce a standing wave.

Waves in strings reaching the end of the string
bounce back traveling in the opposite direction.
If the wave can fit some integer number of half
wavelengths in the string, there is a standing
wave, with l2L/n and fv/?n v/2L
6
Example

• Strings A and B have identical lengths and linear
  densities, but string B is under greater tension
  than is string A. The figure shows four
  situations in which standing wave patterns exist
  on the two strings. In which situations is there
  the possibility that strings A and B are
  oscillating at the same resonant frequency?

f nv/2L v2 ?/?
7
Example

• A nylon guitar string has a linear density of 7.0
  g/m and is under a tension of 149 N. The fixed
  supports are 90 cm apart. The string is
  oscillating in the standing wave pattern shown in
  the figure.
• What is this waves frequency?
• What would be the frequency of the open string?
• Which string is this one in the guitar?
• Is the guitar tuned?

8
Musical scales tempered scale

• Major scale
• 264 HzCdo
• 297 HzDre
• 330 Hz Emi
• 352 Hz Ffa
• 396 Hz Gsol
• 440 Hz Ala
• 495 Hz Bti
• 528 Hz Cdo

• 82.4 E - open 6th string
• 87.3 F
• 92.5 F
• 98.0 G
• 103.8 G
• 110.0 A - open 5th
• 116.5 A
• 123.5 B
• 130.8 C
• 138.6 C
• 146.8 D - open 4th
• 155.6 D
• 164.8 E
• 174.6 F
• 185.0 F
• 196.0 G - open 3rd
• 207.6 G
• 220.0 A
• 233.1 A

• 246.9 B - open 2nd string
• 261.6 C - "middle C"
• 277.2 C
• 293.6 D
• 311.1 D
• 329.6 E - open 1st string
• 349.2 F
• 370.0 F
• 392.0 G
• 415.3 G
• 440.0 A - 5th fret on 1st string
• 466.1 A
• 493.8 B
• 523.2 C
• 554.3 C
• 587.3 D
• 622.2 D
• 659.2 E - 12th fret on 1st string

440Hz/21/12
440Hz x 21/12
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