Lecture 29
Goals:
• Chapter 20, Waves
• Final test review on Wednesday.
• Final exam on Monday, Dec 20, at 5:00 pm.
• HW 11 due Wednesday.
Physics 207: Lecture 29, Pg 1
Relationship between wavelength and period
v
D(x,t=0)
x
x0
l
T= l/v
Physics 207: Lecture 29, Pg 2
Exercise Wave Motion

A boat is moored in a fixed location, and waves make it
move up and down. If the spacing between wave crests is
20 meters and the speed of the waves is 5 m/s, how long
Dt does it take the boat to go from the top of a crest to the
bottom of a trough ? (Recall T = l/ v )
(A) 2 sec
(B) 4 sec
(C) 8 sec
t
t + Dt
Physics 207: Lecture 29, Pg 3
Mathematical formalism
D(x=0,t)
t
D(0,t) ~ A cos (wt + f)
w: angular frequency
w=2p/T
T
D(x,t=0)
x
D(x,0) ~ A cos (kx+ f)
k: wave number
k=2p/l
λ
Physics 207: Lecture 29, Pg 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
w : angular frequency
f: phase constant
Physics 207: Lecture 29, Pg 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 + f+p/2)
D(x,t) = A cos [k(x – vt) + f]
Physics 207: Lecture 29, Pg 6
Why the minus sign?
 As time progresses, we need the disturbance to
move towards +x:
at t=0, D(x,t=0) = A cos [k(x-0) + f]
at t=t0, D(x,t=t0) = A cos [k(x-vt0) + f]
vt0
v
Physics 207: Lecture 29, Pg 7
x

Which of the following equations describe a wave
propagating towards -x:
A) D(x,t) = A cos (kx – wt )
B) D(x,t) = A sin (kx – wt )
C) D(x,t) = A cos (-kx + wt )
D)D(x,t) = A cos (kx + wt )
Physics 207: Lecture 29, Pg 8
Speed of waves
 The speed of mechanical waves depend on the
elastic and inertial properties of the medium.

For a string, the speed of the wave can be shown to
be:
v=

Tstring

Tstring: tension in the string
=M/L : mass per unit length
Physics 207: Lecture 29, Pg 9
Waves on a string
v=
Tstring


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.
Physics 207: Lecture 29, Pg 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
Physics 207: Lecture 29, Pg 11
Sound, A special kind of longitudinal wave
λ
Individual molecules undergo harmonic motion with
displacement in same direction as wave motion.
Physics 207: Lecture 29, Pg 12
Waves in two and three dimensions

Waves on the surface of water:
circular waves
wavefront
Physics 207: Lecture 29, Pg 13
Plane waves
 Note that a small portion of a spherical wave
front is well represented as a “plane wave”.
Physics 207: Lecture 29, Pg 14
Intensity (power per unit area)
 A wave can be made more “intense” by focusing
to a smaller area.
I=P/A : J/(s m2)
I=
R
Psource
4pR
2
Physics 207: Lecture 29, Pg 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
Physics 207: Lecture 29, Pg 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
 I 
b = 10 log 10  
I 0 
I0: threshold of human hearing
I0=10-12 W/m2
Physics 207: Lecture 29, Pg 17
Intensity of sounds
 Some examples (1 pascal  10-5 atm) :
Sound
Intensity
Hearing
threshold
Classroom
Pressure
Intensity
(W/m2)
Level (dB)
3  10-5
10-12
0
0.01
10-7
50
Indoor
concert
30
1
120
Jet engine at
30 m
100
10
130
Physics 207: Lecture 29, Pg 18