Transcript Physics 131: Lecture 14 Notes

Physics 151: Lecture 35
Today’s Agenda

Topics
Waves on a string
Superposition
Power
Physics 151: Lecture 35, Pg 1
Review: Wave Properties...

The speed of a wave (v) is a constant and depends
only on the medium, not on amplitude (A),
wavelength () or period (T).
v
 and T are related !


F
v

T
remember : T = 1/ f and T = 2 / )


Travleing 1-D wave: y(x,t):
yx,t )  Acoskx  t   )
k
2

Physics 151: Lecture 35, Pg 2
Example

Bats can detect small objects such as insects
that are of a size on the order of a wavelength. If
bats emit a chirp at a frequency of 60 kHz and
the speed of soundwaves in air is 330 m/s, what
is the smallest size insect they can detect ?
a.
b.
c.
d.
e.
f.
1.5 cm
5.5 cm
1.5 mm
5.5 mm
1.5 um
5.5 um
Physics 151: Lecture 35, Pg 3
Example

Write the equation of a wave, traveling along the
+x axis with an amplitude of 0.02 m, a frequency
of 440 Hz, and a speed of 330 m/sec.
A.
b.
c.
d.
e.
y = 0.02 sin [880 (x/330 – t)]
y = 0.02 cos [880 x/330 – 440t]
y = 0.02 sin [880(x/330 + t)]
y = 0.02 sin [2(x/330 + 440t)]
y = 0.02 cos [2(x/330 - 440t)]
Physics 151: Lecture 35, Pg 4
Example

For the transverse wave described by
y = 0.15 sin [ (2x - 64 t)/16] (in SI units),
determine the maximum transverse speed of
the particles of the medium.
a.
b.
c.
d.
e.
0.192 m/s
0.6 m/s
9.6 m/s
4 m/s
2 m/s
Physics 151: Lecture 35, Pg 5
Lecture 34, Act 4
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

Can you calcuate how long will it take for a pulse
travels a rope of length L and mass m ?
Physics 151: Lecture 35, Pg 6
See text: 16.4
Superposition

Q: What happens when two waves “collide” ?

A: They ADD together!
We say the waves are “superposed”.
Animation-1
Animation-2
see Figure 16.8
Physics 151: Lecture 35, Pg 7
Aside: Why superposition works

It can be shown that the equation governing waves (a.k.a.
“the wave equation”) is linear.
It has no terms where variables are squared.

For linear equations, if we have two (or more) separate
solutions, f1 and f2 , then Bf1 + Cf2 is also a solution !

You have already seen this in the case of simple harmonic
motion:
d2x
2



x
linear in x !
2
dt
x = Bsin(t)+ Ccos(t)
Physics 151: Lecture 35, Pg 8
See text: 16.4
Superposition & Interference


We have seen that when colliding waves combine (add) the
result can either be bigger or smaller than the original waves.
We say the waves add “constructively” or “destructively”
depending on the relative sign of each wave.
will add constructively
will add destructively

In general, we will have both happening
see Figure 16.8
Physics 151: Lecture 35, Pg 9
Superposition & Interference


Consider two harmonic waves A and B meeting.
Same frequency and amplitudes, but phases differ.
The displacement versus time for each is shown below:
A(t)
B(t)
What does C(t) = A(t) + B(t) look like ??
Physics 151: Lecture 35, Pg 10
Superposition & Interference

Add the two curves,
A = A0 cos(kx – t)
B = A0 cos (kx – t - )

Easy,
C = A + B
C = A0 (cos(kx – t) + cos (kx – t + ))
formula cos(a)+cos(b) = 2 cos[ 1/2(a+b)] cos[1/2(a-b)]
Doing the algebra gives,
C = 2 A0 cos(/2) cos(kx – t - /2)
Physics 151: Lecture 35, Pg 11
Superposition & Interference

Consider,
C = 2 A0 cos(/2) cos(kx – t - /2)
A(t)
B(t)
Amp = 2 A0 cos(/2)
C(kx-t)
Phase shift = /2
Physics 151: Lecture 35, Pg 12
Lecture 35, Act 1
Superposition

A)
You have two continuous harmonic waves with the
same frequency and amplitude but a phase
difference of 170° meet. Which of the following best
represents the resultant wave?
Original wave
(other has different phase)
B)
D)
C)
E)
Physics 151: Lecture 35, Pg 13
Lecture 35, Act 1
Superposition




The equation for adding two waves with different
frequencies, C = 2 A0 cos(/2) cos(kx – t - /2).
The wavelength (2/k) does not change.
The amplitude becomes 2Aocos(/2). With =170, we have
cos(85°) which is very small, but not quite zero.
Our choice has same as original, but small amplitude.
D)
Physics 151: Lecture 35, Pg 14
See text: 16.8
Wave Power

A wave propagates because each part of the medium
communicates its motion to adjacent parts.
Energy is transferred since work is done !

How much energy is moving down the string per unit
time. (i.e. how much power ?)
P
Physics 151: Lecture 35, Pg 15
See text: 16.8
Wave Power...



Think about grabbing the left side of the string and
pulling it up and down in the y direction.
You are clearly doing work since F.dr > 0 as your
hand moves up and down.
This energy must be moving away from your hand (to
the right) since the kinetic energy (motion) of the
string stays the same.
P
Physics 151: Lecture 35, Pg 16
See text: 16.8

How is the energy moving?
Consider any position x on the string. The string to the left
of x does work on the string to the right of x, just as your
hand did:
x

Power P = F.v
F
see Figure 16-15
x
v
Physics 151: Lecture 35, Pg 17
See text: 16.8
Power along the string.

Since v is along the y axis only, to evaluate Power = F.v
we only need to find Fy = -Fsin   -F  if  is small.

We can easily figure out both the
velocity v and the angle  at any
point on the string:
y

x
vy

If
y ( x, t )  A cos( kx  t )
dy
v y  x, t ) 
 A sin kx  t )
dt
tan  
dy
 kA sin kx  t )  
dx
Fv
 dy
dx
Recall
sin   
cos   1
tan   
for small 
Physics 151: Lecture 35, Pg 18
See text: 16.8
v y  x, t )  Asin kx  t )
  kAsin kx  t )
Power...


So: P(x, t)  Fv y  kFA 2sin 2 (kx  t )
But last time we showed that v 

k
and F 
v 2
P x, t )  v 2 A2 sin 2 kx  t )
cos kx  t )
sin 2 kx  t )
Physics 151: Lecture 35, Pg 19
See text: 16.8
Average Power

We just found that the power flowing past location x on the
string at time t is given by:
P  x , t )  v 2 A2 sin 2 kx  t )

We are often just interested in the average power moving
down the string. To find this we recall that the average
value of the function sin2(kx - t) is 1/2 and find that:
P 

1
v 2 A2
2
It is generally true that wave power is proportional to the
speed of the wave v and its amplitude squared A2.
Physics 151: Lecture 35, Pg 20
Recap & Useful Formulas:
y

A
x

General harmonic waves
y  x , t )  A cos kx  t )
2
  2 f 
T

v  f 
k
2
k


Waves on a string
v
F

tension
mass / length
1
v 2 A2
2
dE 1
  2 A2
dx
2
P 
Physics 151: Lecture 35, Pg 21
Lecture 35, Act 2
Wave Power

A wave propagates on a string. If both the amplitude
and the wavelength are doubled, by what factor will
the average power carried by the wave change ?
i.e. Pfinal/Pinit = X
(a) 1/4
(b) 1/2
(c) 1
(d) 2
(e) 4
initial
final
Physics 151: Lecture 35, Pg 22
Waves, Wavefronts, and Rays



Up to now we have only considered waves in 1-D but we
live in a 3-D world.
The 1-D equations are applicable for a 3-D plane wave.
A plane wave travels in the +x direction (for example) and
has no dependence on y or z,
3-D Representation
RAYS
Wave Fronts
Physics 151: Lecture 35, Pg 23
Waves, Wavefronts, and Rays



Sound radiates away from a source in all directions.
A small source of sound produces a spherical wave.
Note any sound source is small if you are far enough away
from it.
3d representation
Shading represents
density
wave fronts
rays
Physics 151: Lecture 35, Pg 24
Waves, Wavefronts, and Rays

Note that a small portion of a spherical wave front is well
represented as a plane wave.
Physics 151: Lecture 35, Pg 25


Waves, Wavefronts, and Rays
If the power output of a source is constant, the total power
of any wave front is constant.
The Intensity at any point depends on the type of wave.
I
Pav
P
 av 2
A 4R
Pav
Pav
I

A const
Physics 151: Lecture 35, Pg 26
Lecture 35, Act 3
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 1/2 its original value, how far away do
you need to stand?
(a) 14 m
(b) 20 m
(c) 30 m
(d) 40 m
Physics 151: Lecture 35, Pg 27
Lecture 35, Act 4
Traveling Waves
Two ropes are spliced
together as shown.
A short time after the
incident pulse shown in
the diagram reaches the
splice, the ropes
appearance will be that in
• Can you determine the relative amplitudes of the
transmitted and reflected waves ?
Physics 151: Lecture 35, Pg 28
Lecture 35, Act 3b
Plane Waves
You are standing 0.5 m away from a very large wall
hanging speaker. The noise hurts your ears. In order to
reduce the intensity you walk back to 1 m away. What is
the ratio of the new sound intensity to the original?
(a) 1
(b) 1/2
speaker

(c) 1/4
(d) 1/8
1m
Physics 151: Lecture 35, Pg 29
Recap of today’s lecture

Chapter 16
Waves on a string
Superposition
Power
Physics 151: Lecture 35, Pg 30