Physics 5B
Lecture 9, February 1, 2012
Chapter 15, Waves
Example Periodic Wave
T is the period of the upand-down oscillation of
an individual point on the
string.
v

T
v
 f
Harmonic Periodic Wave
vf
v
EEvery point
i on the
h string
i is
i making
ki a harmonic
h
i
oscillation up and down.
http://phet.colorado.edu/en/simulation/wave-on-a-string
Harmonic Periodic Waves
D( x, t )  A sin(k x   t )
Harmonic Oscillation of the
point on the string at x=0
T
2
2
and  
k

T
Shape of the string at time 0

Some animations of harmonic waves:
vf 
http://scipp.ucsc.edu/~johnson/applets/Wave_SHM.htm
http://scipp.ucsc.edu/~johnson/applets/wavelength_period.htm

k
The graph below shows a snapshot of a wave on a string
th t iis ttraveling
li to
t th
i ht Th
i a bit off paint
i t on th
that
the right.
There is
the
string at point P.
y
vwave
P
Q
x
At the instant
shown, the velocity
of paint point P has
hi h di
ti ?
which
direction?
A.
B.
C.
D
D.
Up
Down
Left
Ri ht
Right
The graph below shows a snapshot of a wave on a string
that is traveling to the right. There is a bit of paint on the
string at point P.
y
vwave
P
Q
x
At the instant shown,
the acceleration of
paint point P has
hi h di
ti ?
which
direction?
A.
B.
C.
D
D.
Up
Down
Left
Ri ht
Right
Problem 15
15--25
D( x, t )  A sin( k x   t   )
A  0.026 m; k  45 m 1;   1570 rad/s;   0.66 rad
Consider the point x=1.00 m on the cord of Example
15-5. Determine
a) The maximum velocity of the point,
point and
b) The maximum acceleration.
c)) What are the velocityy and acceleration at t=2.50 s?
d) What is the phase velocity of the wave?
Power Transported by a Wave
The most important
p
point
p
to remember is
that the power (energy per unit time)
transported
p
byy a wave is proportional
p p
to
the square of the amplitude and the square
q
y
of the frequency.
2 2
I  1 v  A
2
(15-7)
For a wave moving through a volume (e.g.
(e g
sound) intensity is power per unit area. We
will come back to this in the next chapter.
chapter
Superposition
A tremendously important subject in physics (Waves, Optics,
Electricity and Magnetism, Quantum Mechanics, etc.)!
Property of linear differential equations (e.g. the wave equation).
http://phet.colorado.edu/en/simulation/fourier
MathCad Fourier Examples
The red curve
i a simple
is
i l sum
of the cosine
waves shown
above
MathCad Fourier Examples
The red curve
is a simple sum
of the sine
waves shown
above
b
MathCad Fourier Examples
The red curve
is a simple sum
of the sine
waves shown
above
b
A string has two pulses approaching each other moving 1 m/s.
1m
What will the string
look like 3 sec later?
A
B
C
D
1m
1m
1m
1m
E
1m