Standing Waves

1
Standing Waves

• Physics 202
• Professor Lee Carkner
• Lecture 8

2
PAL 7 Wave Energy

• How do you find linear density?
• Get frequency from function generator or by
  timing oscillator (f 40 Hz)
• Get tension from hanging weights (hanging mass is
  250g so t mg (.250)(9.8) 2.45 N)
• m

3
PAL 7 Wave Energy (cont.)

• Can you maximize P by maximizing input energy and
  wave speed?
• Can maximize t and minimize m to increase wave
  speed
• Since P ½mvw2ym2, this does not maximize P
• Slow waves on a massive string transfer more
  energy than fast waves on a light string

4
Exam 1 Friday

• About 1/3 multiple choice
• Study notes
• Study concept questions of PALs and SuperPALs
• Look at textbook Checkpoint questions
• About 2/3 problems
• Study problem questions of PALs and SuperPALS
• Study old homework
• Do new practice homework questions
• Try to do this with just equation sheet
• Need calculator and pencil

5
Standing Waves

• The two waves will interfere, but if the input
  waves do not change, the resultant wave will be
  constant
• Nodes --
• Antinodes -- places where the amplitude is a
  maximum (only place where string has max or min
  displacement)
• The positions of the nodes and antinodes do not
  change, unlike a traveling wave

6
Standing Wave Amplitudes
7
Equation of a Standing Wave

• If the two waves have equations of the form
• y2 ym sin (kx wt)
• Then the sum is
• yr 2ym sin kx cos wt
• e.g. at places where sin kx 0 the amplitude is
  always 0 (a node)

8
Nodes and Antinodes

• Consider different values of x (where n is an
  integer)
• For kx np, sin kx 0 and y 0
• Node
• Nodes occur every 1/2 wavelength
• Antinode
• Antinodes also occur every 1/2 wavelength, but at
  a spot 1/4 wavelength before and after the nodes

9
Standing Wave on a String
10
Reflecting Waves

• How are standing waves produced?
• The wave will reflect off of the boundary and
  travel back up the string
• The sign of the pulse will be reversed after the
  reflection

11
Reflecting Standing Waves

• After reflection the string then contains 2 waves
  traveling in opposite directions
• The wave bounces back and forth between the
  boundaries reinforcing itself as it goes

12
Allowed Standing Waves
13
Resonance Frequency

• When do you get resonance?
• Since you are folding the wave on to itself
• You need an integer number of half wavelengths to
  fit on the string (length L)
• In order to produce standing waves through
  resonance the wavelength must satisfy
• l 2L/n where n 1,2,3,4,5

14
Standing Wave on Tacoma Narrows Bridge
15
Harmonics

• We can express the resonance condition in terms
  of the frequency (vfl or fv/l)
• f(nv/2L)
• Remember v depends only on t and m
• n1 is the first harmonic, n2 is the second etc.