Wave Energy and Superposition

1
Wave Energy and Superposition

• Physics 202
• Professor Lee Carkner
• Lecture 7

2
PAL 6 Waves

• A y2sin(2x-2t), B y4sin(4x-6t), C
  y6sin(6x-8t)

• T 2p/w so TA p, TB 1/3p, TC 1/4p
• Which wave has largest transverse velocity?
• Largest wave speed?
• v lf l/T, vA 1, vB 1.5, vC 1.3

3
PAL 6 Waves (cont.)

• Wave with y 2 sin (2x-2t), find time when x
  5.2 cm has max a
• Happens when y ym 2
• 1 sin (2x-2t)
• p/2 (2x - 2t)
• t 2x-(p/2)/2
• Maximum velocity when y 0
• 2x -2t arcsin 0 0
• t x

4
Velocity and the Medium

• If you send a pulse down a string what properties
  of the string will affect the wave motion?
• Tension (t)
• If you force the string up, tension brings it
  back down
• Linear density (m m/l mass/length)
• You have to convert the PE to KE to have the
  string move

5
Wave Tension in a String
6
Force Balance on a String Element

• Consider a small piece of string Dl of linear
  density m with a tension t pulling on each end
  moving in a very small arc a distance R from rest
• F (t Dl)/R
• F (m Dl) (v2/R)
• Solving for v,
• v (t/m)½
• This is also equal to our previous expression for
  v

7
String Properties

• How do we affect wave speed?
• v (t/m)½ lf
• We set the tension by how hard we pull on the
  string
• The wavelength of a wave on a string depends on
  how fast you move it and the string properties

8
Tension and Frequency
9
Energy

• A wave on a string has both kinetic and elastic
  potential energy
• Every time we shake the string up and down we add
  a little more energy
• This energy is transmitted down the string
• The energy of a given piece of string changes
  with time as the string stretches and relaxes
• Assuming no energy dissipation

10
Power Dependency

• P½mvw2ym2
• If we want to move a lot of energy fast, we want
  to add a lot of energy to the string and then
  have it move on a high velocity wave
• ym and w depend on the wave generation process

11
Superposition

• When 2 waves overlap each other they add
  algebraically
• Traveling waves only add up as they overlap and
  then continue on
• Waves can pass right through each other with no
  lasting effect

12
Pulse Collision
13
Interference

• Consider 2 waves of equal wavelength, amplitude
  and speed traveling down a string
• The waves may be offset by a phase constant f
• y2 ym sin (kx - wt f)
• From the principle of superposition the resulting
  wave yr is the sum of y1 and y2
• Is it greater or less than ym?

14
Interference and Phase

• The amplitude of the resultant wave (ymr) depends
  on the phase constant of the initial waves
• The phase constant can be expressed in degrees,
  radians or wavelengths
• Example

15
Resultant Equation
16
Combining Waves
17
Types of Interference

• Constructive Interference -- when the resultant
  has a larger amplitude than the originals
• No offset or offset by a full wavelength
• Destructive Interference -- when the resultant
  has a smaller amplitude than the originals
• Offset by 1/2 wavelength