Standing Waves + Reflection
Announcements:
Will discuss reflections of
transverse waves, standing
waves and speed of sound.
We will be covering
material in Chap. 16.
Plan to review material on
Wednesday and Friday.
Web page: http://www.colorado.edu/physics/phys1110/phys1110_sp12/
Announcements
•  Have decided to increase from 1 -> 2 crib
sheets 8.5” x 11” front and back for the final.
•  Smart Physics will be done today – there was a
mix-up on the fluids section which had the
wrong due date and it will be dropped from
counting.
•  Last CAPA assignment is due tomorrow
(Wednesday at 8am). Will put the CAPA
summary on D2L by end of week.
•  Today is the last day that Clicker response will
count.
Energy Transport of Transverse Wave
y(x,t) = y m sin(kx − ωt + φ )
€
Pavg
 dK 
1
= 2   = µv ω 2 y m2
 dt  avg 2
µ is the mass per unit length
€
€
Depends on amplitude squared and angular frequency squared
Wave Speed on a Stretched String
τ
v=
µ
τ is the tension in the string
µ is the mass per unit length
The speed of a wave along a stretched ideal string depends only on the
tension and the linear density of the string and not on the frequency.
€
€
Reflection at a wall
A pulse reaching the end of a
medium becomes inverted
whenever it either
• reflects off a fixed end,
• or is moving in a less dense
medium and reflects off a more
dense medium.
Reflection at an open end
A pulse reaching the end of a
medium does not invert in
reflection if it either
• reflects off an open end,
• or is moving in a more dense
medium and reflects off a less
dense medium.
Reflection Dense to Less Dense
A pulse in a more dense medium is traveling towards the boundary with a less
dense medium.
1. The reflected pulse in medium 1 ________ (will, will not) be inverted.
2. The speed of the transmitted pulse will be ___________ (greater than, less
than, the same as) the speed of the incident pulse.
3. The speed of the reflected pulse will be ______________ (greater than, less
than, the same as) the speed of the incident pulse.
4. The wavelength of the transmitted pulse will be ___________ (greater than,
less than, the same as) the wavelength of the incident pulse.
5. The frequency of the transmitted pulse will be ___________ (greater than,
less than, the same as) the frequency of the incident pulse.
Reflection Less Dense to Dense
A pulse in a less dense medium is traveling towards the boundary with a more
dense medium.
1. The reflected pulse in medium 1 ________ (will, will not) be inverted.
2. The speed of the transmitted pulse will be ___________ (greater than, less
than, the same as) the speed of the incident pulse.
3. The speed of the reflected pulse will be ______________ (greater than, less
than, the same as) the speed of the incident pulse.
4. The wavelength of the transmitted pulse will be ___________ (greater than,
less than, the same as) the wavelength of the incident pulse.
5. The frequency of the transmitted pulse will be ___________ (greater than,
less than, the same as) the frequency of the incident pulse.
Clicker question 1
A
B
C
D
E
Set frequency to BA
Clicker question 1
A
B
C
D
E
Set frequency to BA
Interference (Standing Waves)
Wave moving
left and right
y(x,t) = y m sin(kx − ωt) + y m sin(kx + ωt)
1
1
sin α + sin β = 2sin (α + β )cos (α − β )
2
2
€
€
β = kx + ωt
α = kx − ωt
α − β = −(2ω )t
α + β = (2k)x
so y(x,t) =€[2y m sin kx] cos ωt
€
In a standing wave the amplitude varies
with€position. The place where amplitude
Places where string
doesn’t move are called
nodes. Places where the
string moves the
maximum are called
antinodes.
is zero is when kx = nπ for n = 0, 1, 2,.....
€
k=
2π
λ
€
€
€
so x =
nλ
for n = 0, 1, 2, ... (nodes)
2
1 λ
x = (n + )
for n = 0, 1, 2, ... (antinodes)
2 2
Interference (Standing Waves)
y(x,t) = [2y m sin kx] cosωt
Amplitude
at position x
€
In a standing wave the amplitude varies
with position. The place where amplitude
is zero is when kx = nπ for n = 0, 1, 2,.....
k=
2π
λ
so x =
nλ
for n = 0, 1, 2, ... (nodes)
2
2L
λ€
=
for n = 1, 2, 3,...
n
€ v
v
f = =n
for n = 1, 2, 3,...
λ
2L
€
€
€
Oscillating
Term
n is called the harmonic number
Clicker question 2
Set frequency to BA
A string on an instrument plays an A (440 Hz) when plucked.
If you put your finger down in the middle of the string, and
then pluck, you are mostly likely to hear
A: A an octave higher (880 Hz)
B: A, an octave lower (220 Hz)
C: Same tone
D: Some entirely different note
Clicker question 2
Set frequency to BA
A string on an instrument plays an A (440 Hz) when plucked.
If you put your finger down in the middle of the string, and
then pluck, you are mostly likely to hear
A: A an octave higher (880 Hz)
B: A, an octave lower (220 Hz)
C: Same tone
D: Some entirely different note
If you put your finger in the middle, you don't allow the "fundamental"
vibration mode to happen. Putting your finger there requires a NODE
there, and the next higher standing wave, the one with a FULL wave length
fitting into the length, is now allowed. That's half the wavelength, or twice
the frequency, 880 Hz.
Clicker question 3
Set frequency to BA
A string is clamped at both ends and then plucked so that it vibrates in a
standing mode between two extreme positions a and b. Let upward motion
correspond to positive velocities. When the string is in position b, the
instantaneous velocity of points along the string is...
A: zero everywhere.
B: positive everywhere.
C: negative everywhere. D: depends on the position.
a
b
Clicker question 3
Set frequency to BA
A string is clamped at both ends and then plucked so that it vibrates in a
standing mode between two extreme positions a and b. Let upward motion
correspond to positive velocities. When the string is in position b, the
instantaneous velocity of points along the string is...
A: zero everywhere.
B: positive everywhere.
C: negative everywhere. D: depends on the position.
a
b
Zero everywhere! Every point on the string has reached
its "extreme value.” When you reach the end of your
motion, your speed is instantaneously zero.
Clicker question 4
Set frequency to BA
A string is clamped at both ends and then plucked so that it vibrates in a
standing mode between two extreme positions a and c. Let upward motion
correspond to positive velocities. When the string is in position b, the
instantaneous velocity of points along the string is...
a
A: zero everywhere.
B: positive everywhere.
C: negative everywhere. D: depends on position.
b
c
Clicker question 4
Set frequency to BA
A string is clamped at both ends and then plucked so that it vibrates in a
standing mode between two extreme positions a and c. Let upward motion
correspond to positive velocities. When the string is in position b, the
instantaneous velocity of points along the string is...
a
A: zero everywhere.
B: positive everywhere.
C: negative everywhere. D: depends on position.
b
c
It depends on position. At the point above "b", the string is on its way from a to c
and thus moving DOWN. But 1/2 wavelength to the right, it's on its way up! node: never moves.
The Speed of Sound
τ
elastic property
v=
=
µ
inertial property
Speed of any wave depends:
On the inertial property of the medium (to store
€
kinetic
energy)
On the elastic property of the medium (to store
potential energy)
For sound it is
v=
B
Bulk Modulus
=
ρ
density
The bulk modulus determines the extent to which an
element of the medium changes in volume when the
Δp
B
=
−
pressure on€it changes.
(Units: Pascals)
ΔV /V
Table of Speeds of Sound
Molecules in a solid are much closer together than those in a liquid or gas,
allowing sound waves to travel more quickly through it. Sound waves travel 17
times faster through steel than through air. The speed of sound in steel is 5,960
meters per second (13,332 mph)!
So why does sound travel slower in cold air than hot air?
The speed of sound depends on wind conditions, temperature, and
humidity, but not on frequency. All notes travel at the same speed.
Water vapor slightly increases the speed. Sound travels faster through warm
air than through cold air because the faster-moving molecules in warm air
bump into each other more often and transmit a pulse in less time.
For each degree rise in temperature above 0 degrees C, the speed of
sound in air increases by 0.6 meters per second.
Clicker question 5
Set frequency to BA
A source of frequency of 500 Hz emits waves of wavelength 0.4 m,
how long does the waves take to travel 600 m?
A 3s
B 6s
C 9s
D 12 s
Clicker question 5
Set frequency to BA
A source of frequency of 500 Hz emits waves of wavelength 0.4 m,
how long does the waves take to travel 600 m?
A 3s
B 6s
C 9s
D 12 s
v = fλ = (500Hz)(0.4m) = 200m /s
d
600 m
t= =
=3s
v 200 m /s
€
€
Clicker question 6
A. λ < λ' and f < f '
B.
C.
D.
E.
λ = λ' and f
λ > λ' and f
λ < λ' and f
λ > λ ' and f
= f'
= f'
= f'
> f'
Set frequency to BA
€
Clicker question 6
Set frequency to BA
A. λ < λ' and f < f '
B.
C.
D.
E.
λ = λ' and f
λ > λ' and f
λ < λ' and f
λ > λ ' and f
= f'
= f'
= f'
> f'
The time interval between two successive
wave fronts remains the same. The distance
between two successive wave fronts
decreases since the waves travel slower.