Goals:

1
Lecture 29

• Goals

• Chapter 20, Waves

• No labs this week.
• HW 11 is due tomorrow night.
• HW 12 (a short one) is due Thursday night.

2
Relationship between wavelength and period
v
D(x,t0)
x
x0
?
T?/v
3
Mathematical formalism
D(x0,t)

• D(x0,t) A cos (wt f)
• w angular frequency
• w2p/T

t
4
Mathematical formalism

• The two dimensional displacement function for a
  sinusoidal wave traveling along x direction

D(x,t) A cos (kx - wt f)
A Amplitude k wave number ? angular
frequency ? phase constant
5
Mathematical formalism

• Note that there are equivalent ways of describing
  a wave propagating in x direction

D(x,t) A cos (kx - wt f) D(x,t) A
sin (kx - wt fp/2) D(x,t) A cos k(x
vt) f
6
Why the minus sign?

• As time progresses, we need the disturbance to
  move towards x

at t0, D(x,t0) A cos k(x-0) f at tt0,
D(x,tt0) A cos k(x-vt0) f
v
x
7

• Which of the following equations describe a wave
  propagating towards -x

• D(x,t) A cos (kx wt )
• D(x,t) A sin (kx wt )
• C) D(x,t) A cos (-kx wt )
• D) D(x,t) A cos (kx wt )

8
Speed of waves

• The speed of mechanical waves depend on the
  elastic and inertial properties of the medium.

9
Waves on a string

• Making the tension bigger increases the speed.
• Making the string heavier decreases the speed.
• The speed depends only on the nature of the
  medium, not on amplitude, frequency etc of the
  wave.

10
Exercise Wave Motion

• A heavy rope hangs from the ceiling, and a small
  amplitude transverse wave is started by jiggling
  the rope at the bottom.
• As the wave travels up the rope, its speed will

v
(a) increase (b) decrease (c) stay the same
11
Sound, A special kind of longitudinal wave
Individual molecules undergo harmonic motion with
displacement in same direction as wave motion.
12
Waves in two and three dimensions

• Waves on the surface of water

13
Plane waves

• Note that a small portion of a spherical wave
  front is well represented as a plane wave.

14
Intensity (power per unit area)

• A wave can be made more intense by focusing to
  a smaller area.

IP/A J/(s m2)
15
Exercise Spherical Waves

• You are standing 10 m away from a very loud,
  small speaker. The noise hurts your ears. In
  order to reduce the intensity to 1/4 its original
  value, how far away do you need to stand?

(A) 14 m (B) 20 m (C) 30 m (D) 40 m
16
Intensity of sounds

• The range of intensities detectible by the human
  ear is very large
• It is convenient to use a logarithmic scale to
  determine the intensity level, b

I0 threshold of human hearing I010-12 W/m2
17
Intensity of sounds

• Some examples (1 pascal ? 10-5 atm)

Sound Intensity Pressure Intensity (W/m2) Level (dB)
Hearing threshold 3 ? 10-5 10-12 0
Classroom 0.01 10-7 50
Indoor concert 30 1 120
Jet engine at 30 m 100 10 130
18
The Doppler effect

• The frequency of the wave that is observed
  depends on the relative speed between the
  observer and the source.

observer
19
vs
observer
20
The Doppler effect

• Approaching source

ff0/(1-vs/v)

• Receding source

ff0/(1vs/v)