Superposition and Standing Waves (Cont.)

1
Chapter 18

• Superposition and Standing Waves (Cont.)

2
Outline

• Standing waves
• Standing waves in a string fixed at both ends
• Standing waves in air columns
• Resonance

3
Standing waves

• Standing wave An oscillation pattern with a
  stationary outline that results from the
  superposition of two identical waves traveling in
  opposite directions.
• y1 A sin(kx - ?t), y2 A sin(kx ?t)
• y y1 y2 (2A sinkx) cos ?t
• Nodes The points of zero displacement are called
  nodes Antinodes The positions in the medium at
  which the maximum displacement of 2A occurs.
• The distance between adjacent antinodes ?/2
  The distance between adjacent nodes ?/2 The
  distance between a node and an adjacent antinode
  ?/4.

• Important features for a standing wave
• It is not a traveling wave.
• Every particle of the medium oscillates in
  simple harmonic motion with the same frequency ?.
• The amplitude of a given particle depends on the
  location x of the particle.

4
Problem 14

• Two waves in a long string are given by
• where y1, y2, and x are in meters and t is in
  seconds.
• (a) Determine the positions of the nodes of the
  resulting standing wave.
• (b) What is the maximum transverse position of an
  element of the string at the position x 0.400 m?

5
Standing waves in a string fixed at both ends

• Normal modes Natural patterns of oscillation,
  each of which has a characteristic frequency.
• Wavelengths of the normal modes ?n 2L/n,
  n1,2,3
• Index n Refers to the nth normal mode of
  oscillation.
• Natural frequencies of normal modes
• Fundamental (frequency)
• fn nf1- harmonic series and the normal modes
  are called harmonics.

6
Example 18.4

• The high E string on a guitar measures 64.0cm in
  length and has a fundamental frequency of 330Hz.
  By pressing down on it at the first fret
  (Fug.18.8), the string is shortened so that it
  plays an F note that has a frequency of 350 Hz.
  Haw far is the fret from the neck end of the
  string?

7
Standing waves in air columns

• Standing waves can be set up in a tube of air.
• (1) In a pipe open at both ends, natural
  frequencies of oscillation fnnv/(2L), n1,2,3
• (2) In a pipe closed at one end and open at the
  other, natural frequencies of oscillation fn
  nv/(4L), n1,3,5

8
Resonance

• Resonance If a periodic force is applied to a
  system, the amplitude of the resulting motion is
  greater than normal when the frequency of the
  applied force is equal or nearly equal to one of
  the natural frequencies (resonant frequencies) of
  the system.
• An example of resonance

9
Example 18.6

• A vertical pipe open at both ends is partially
  submerged in water, and a tuning fork vibrating
  at an unknown frequency is placed near the top of
  the pipe. The length L of the air column can be
  adjusted by moving the pipe vertically. The sound
  waves generated by the fork are reinforced when L
  corresponds to one of the resonance frequencies
  of the pipe. For a certain tube, the smallest
  value of L for which a peak occurs in the sound
  intensity is 9.00 cm. What are
• (a) the frequency of the tuning fork?
• (b) the value of L for the next two resonance
  frequencies?

10
Homework

• Ch. 18, Problems 14, 25, 42, 46.
• Hints on 42
• The change in volume ?V and the change in the
  height ?h of a cylinder are related by ?V
  (?r2) ?h or ?h ?V/(?r2).
• Since the volume flow rate is R, i.e. ?V/?t R,
  the height flow rate is ?h/?t R/(?r2).