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AP Physics – Vibrations, Waves and Sound Waves – Chapter 14-15
Chapter 14 - Reading pp. 421-448 - text HW #1, 7, 8, 13, 20, 23, 25 (a,b,d),
35, 36, 39, 41, 42, 43, 45, 48
#1) A 5 kg block slides down a frictionless incline, collides with and sticks
to a board fastened to a spring with a constant 20 N/m
(a) Find the maximum amount the spring is compressed.
(b) Describe what happens to the block after it sticks to the spring.
The block will oscillate back and forth in SHM – Simple
Harmonic Motion.
Simple Harmonic Motion
A type of motion that results when the net force on a
moving object is a Hooke’s law type of force (a RESTORING
force that is proportional to the displacement but in the
opposite direction
The two most frequently studied harmonic oscillators:
1. The Mass-Spring System
2. The Pendulum
Demo: Mass-Spring Oscillator on air track
2
#2)
A spring is initially at the equilibrium position. It is then pulled down by
an external force and released.
Draw a position vs. time graph that represents this motion.
Pendulum - Simple Harmonic motion applet
Important physical quantities describing SHM:
1) Amplitude (A) - Maximum displacement from equilibrium
position (Unit: meter)
2) Period (T) – Time for one complete oscillation or cycle. (Unit:
sec/cycle)
3) Frequency (f)– the number of cycles per unit of time (Unit:
cycles/s, s-1, or hertz (Hz))
4) Wavelength (λ) – The distance between two consecutive
points on a wave that are in phase (moving in same direction
with same amplitude)
What is the relationship between the frequency of an object and its period?
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An in-depth analysis of the motion of a mass on a spring oscillating horizontally:
From Hooke’s Law and Newton’s Laws:
F = ma
and
F =-kx
Hence -kx = ma
a = + kx
m
Notice: acceleration is not constant and varies from –kx to +kx
m
m
How would the position vs. time, velocity vs. time, and acceleration vs.
time graphs look? Compare mass spring to circular motion here
Equation of curve:
x = A cos(ωt)
(recall ω=θ/t or θ= ωt)
v = - ωA sin(ωt)
a = - ω2A cos(ωt)
(remember: Fnet and a are in same direction)
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#3) The following is a position time graph for an oscillating pendulum:
Find the Amplitude, Period and Frequency.
#4) AP Exam 1998 #38
5
Potential Energy (U), Kinetic Energy (K) and total Mechanical
Energy Graphs for objects in SHM.
#5) A block oscillates along a frictionless, horizontal surface
Graph:
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B) Repeat for a vertical spring. (next page)
(Reference Point: PEg=0 J at the equilibrium position)
Blue – Elastic PE Green – Gravitational PE
Red – Kinetic
Black -Total Mechanical
Now go home and do homework - # 1, 7, 8, 13
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14.4 Period of a Mass-Spring System (Works for both
vertical and horizontal) (Class demo lab) Compare mass spring to circular motion here
What are the factors that effect the period for a mass oscillating on a
vertical spring?
The equation: T  2
m
k
The equation derived:
Since the horizontal projection of an object in circular motion is identical
to simple harmonic motion, we can use the equation for linear velocity as
a starting point:
2r
2r
r
T
v
T

2

or
or
v
v
T
Now consider ¼ of the cycle of the mass spring as it moves from
equilibrium position (max KE), to the fully stretched position (max PE s)
2
1 2 1 2 or
x
m
x
m
or
kx  mv


2
2
v2 k
v
k
Since r=x, combining eq (1) and (2) yields:
T  2
x
m
 2
v
k
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#6) (A) A 2-kg mass oscillates on a mass-spring system with spring
constant 18 N/m. What is the period and frequency?
B) If the period is 6 s for this spring, what mass was used?
#7) (A)When a mass is attached to a spring, the period of oscillation is
approximately 2 s. When the mass attached to the spring is doubled, the
period of the pendulum is nearly
(A) 0.5 s (B) 1.0 s (C) 1.4 s (D) 2.0 s (E) 2.8 s
(B) This new mass is submerged in a liquid, and set into oscillation. The
period of the pendulum is nearly
(A) 0.5 s (B) 1.0 s (C) 1.4 s (D) 2.0 s (E) 2.8 s
Note: This is an example of a damped oscillator. In this case, the period does not change, but it quickly brings
the oscillator to rest, which is important in situations where unwanted vibrations occur. Most common
dampener you utilize every day? The shock absorbers on your car. When your wheel hits a bump, the springs
allow the wheel to move without transferring all the energy to the interior compartment. But, the wheel
would bounce up and down in SHM if the car did not have shock absorbers, which are oil filled pistons (similar
to the piston on a screen door that keeps it from slamming). The shocks limit the spring to 2-3 oscillations.
Now go home and do homework - # 20, 23, 25abd
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14.6 Period of a Pendulum
What are the factors that effect the period for a simple pendulum
oscillating at small amplitudes?
The equation: T  2
l
g
The equation derived:
We start with the restoring force (Ftangent), which remember is:
Ftangent = mgsinθ
Since
θ=s/r
For small angles, (<15°)
Ftan 
sinθ = θ or Ftangent = mgθ
mg
s
l
Since the Ftangent equation above is of the form Fs = -kx, mg/l is equivalent
to k. Thus mg/l can be subbed into the equation for the period of a mass
spring system:
T  2
m
 2
k
m
mg
l
or
T  2
l
g
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#8) A) Three physics students of combined mass of 200 kg oscillate on
the Daredevil Dive at Great Adventure with cable length of 47 m. What is
the period and frequency of this pendulum on Earth?
B) If we moved this pendulum to the moon (at great monetary expense)
gmoon = 1.63m/s2, what is the period?
#9) The pendulum of a Grandfather clock requires a period of 2 s to keep
accurate time for a pendulum on Earth, what length pendulum is
needed? What length would be needed on Jupiter (g= 26 m/s2)
#10) A pendulum with mass M and length L on Earth, oscillates with a
period T. If the length is doubled, the period is
Now go home and do homework - #35, 36, 39
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14.8-14.9 Wave Motion and types of waves
Important definitions:
1)Wave – the periodic motion of a medium due to a
vibrating object. A wave always TRANSFERS THE ENERGY of
the vibrating object but does not transfer mass.
2)MEDIUM – The substance or material through which the
wave travels.
Hmmm…Matter medium vs. empty space: is empty space a medium? Let’s scour
the Web for theories on this.
3)Transverse Wave – a wave in which the particles of the
disturbed medium move perpendicular to the direction of
wave velocity.
Examples of Transverse waves:
1) Water waves (actually a combination of transverse
and longitudinal)
2) Electromagnetic Waves (no matter medium
required)
Check out Physics applet: (transverse waves)
4)Pulse: A single vibratory disturbance
#11 (A) What happens to point p as the pulse passes through it?
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(B) Pulse on a string – fixed vs free-end reflection
Fixed end: Inversion occurs
(180° change of phase)
Free end: no inversion
(Applet pulse on a string – reflected waves)
Which of Newton’s law applies to reflected pulse being inverted?
Newton’s third Law of Action-Reaction. The last molecule in the string is attached
to the pole, which acts as a really BIG molecule. Thus the molecule in the string
pulls up on the pole, while the pole simultaneously pulls down on the molecule in
the string. Since the pole is so massive, it’s motion changes negligibly, while the
strings motion changes greatly and is thus pulled DOWN.
5)Longitudinal Wave – a wave in which the particles of the
disturbed medium move parallel to the direction of the
wave velocity.
Example of Longitudinal waves:
1) SOUND WAVES
DEMO: Use slinky as a Longitudinal Wave (longitudinal wave applet)
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14.10Relationship of Frequency, Wavelength and Speed of a wave
Derive the formula for the speed of a wave in terms of frequency and
wavelength. Remember, that waves travel at constant speed within the
same medium.
Since the length of 1 cycle
and the time to complete 1 cycle
and
v
v = d/t
then:

T
v
which leads to
= λ
= T

1
f
or
v  f
In a uniform medium, the relationship
between f and λ is inverse.
Demo with wave spring: vary f and watch λ change.
#12)Find the amplitude, frequency, period and wavelength
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#13) Find the velocity of the wave in the string below.
Does the answer make sense? Yes if you CHANGE THE MEDIUM!
BIG NOTE:
The VELOCITY of a wave is dependent on the
MEDIUM, ONLY.
In a uniform rope:
Examples:
v
Ft

Will a wave travel faster through a rope stretched tightly or one hanging loosely?
(Tight rope since more tension increases speed)
Will a wave travel faster through a thick rope or a thin rope?
(Thin rope - since linear density is inversely related
to speed
(Applet pulse on a string – thick to thin)
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Let’s test this equation in the hall:
#14 (A) Find the speed of a pulse on a uniform string of mass ____ kg and
length ____ m when subjected to a tension of _____ N.
(B) How long will it take the pulse to travel the _____ m?
If the tension were doubled, the speed would now be_________.
Now go home and do homework - #41,42,43,45,48
When are 2 points on a wave in PHASE?
Name 2 points on the wave that have a
0o phase difference __________
180o phase difference ____________
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Two points on a wave are in phase when they have the
same displacement from the equilibrium position AND are
moving in the same direction.
In terms of wavelengths:
0o phase difference: nλ apart (n=0, 1, 2,…)
180o phase difference: nλ/2 apart (n=1, 2, 3,…)
14.12Superposition and Interference
What happens when 2 or more waves are passing through
the same medium?
Predict what happens:
#15)
Draw the resulting wave at the instant both waves are at pt. p.
This is an example of CONSTRUCTIVE interference.
(Demo: in phase pulse)Applet
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#14)
Draw the resulting wave at the instant both waves are at pt. p
This is an example of DESTRUCTIVE interference.
(Demo: out-of-phase pulse) Applet
The Superposition Principle – When 2 or more wave are
traveling through the same medium, they interfere with
each other. The resulting waveform is found by ADDING
the displacements of the individual waves together, point
by point.
#15) Draw the resultant wave when the dotted lines align at point p
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#16)This diagram shows two theoretical pulses headed towards each other. The
pulses each travel one grid block in one time unit.
Draw the resulting pattern one unit after original diagram.
Draw the resulting pattern two units after original diagram.
Draw the resulting pattern three units after original diagram.
Draw the resulting pattern four units after original diagram.
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15.7 Interference of Sound Waves
#17) Two speakers are in phase and each produce a sound with a
frequency of 120 Hz. The diagram below is the representation of sound
on an oscilloscope screen.
What is the minimum non zero distance “d” that either speaker can be
shifted backwards so that: (Use vsound in air @ STP = 330 m/s)
(A) Maximum constructive interference occurs.
(B) Maximum destructive interference occurs.
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AP Physics – Sound Waves – Chapter 15
Chapter 15 - Reading pp. 458-487 - text HW #4, 21, 22, 24, 31, 37, 38, 39,
40, 44, 47, 48, 50, 53
Sound – the most important example of a longitudinal
wave. The direction of motion of the medium is PARALLEL
to the direction of the wave velocity.
Sound waves require a matter medium!
Sound waves in air consist of a series of:
Compressions or Condensations (high density region of air
molecules) and
Rarefactions (low density region of air molecules).
(Applet: longitudinal waves)
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For a sound wave:
The louder the sound, the greater the wave’s AMPLITUDE.
The higher the pitch the greater the wave’s FREQUENCY.
Speed of sound in air – remember this equation:
v  (331m / s) 1 
T
273
Where T is measured in Celcius!
#18 A) Calculate the speed of sound in air at STP.
B) Calculate the speed of sound in air at room temperature (68o F) or ____oC
#19) Middle C on the piano has a frequency of 264 Hz and a wavelength
of 1.3 m. Find the speed of sound in air. Determine the temperature of
the room that the piano was played.
What is Echo? - A reflected sound wave.
#20) Sonar (originally an acronym for SOund Navigation And Ranging) is used to locate
underwater obstacles. A sonar ping is sounded from a Naval vessel and is
heard by the sonar officer 0.4 sec later. How far away is the obstacle,
given the speed of sound in water as 1500 m/s?
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15.6 The Doppler Effect
The Doppler effect is experienced whenever there is relative
motion between the source of sound and observer. If
relative motion between source and observer is towards
each other, the observed/apparent frequency is greater
than actual. When the relative motion between the source
and observer is away from each other, the observed or
apparent frequency is less than actual.
Doppler Applet Link
General Formula:
Moving Towards = ___
Moving Away = ___
(Don’t use words like closer or approaches)
Case I:
(Doppler Videos)
Case I Formula:
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Case II:
Case II Formula:
Case III:
Case III Formula:
Case IV:
Case IV Formula:
Case V:
Case V Formula:
Case VI:
Case VI Formula:
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#21) A train emits a 400-Hz horn blast. Use the speed of sound in air as 343 m/s.
If the train is moving at 36 m/s, what is the frequency of the sound perceived by an
observer standing beside the track?
(A) as the train approaches.
(B) after the train has passes the observer and is moving away.
#22) A tea kettle sitting on the stove blows its whistle with a frequency of 500 Hz.
Mr. Berbit, a big fan of tea, taps into his old-man strength and races to the stove at
12 m/s to shut it off. (assume room temperature: vsound=343 m/s)
A) What frequency does Mr. Berbit hear?
B) What is the apparent wavelength of the observed frequency?
#23) Responding to a 911 call for massive 3rd degree burns at the old Berbit house,
An ambulance moving at 45 m/s emits a siren with a frequency of 400 Hz (at STP).
What is the frequency observed by a person in a car traveling at 30 m/s when:
(A) Both are moving away from each other?
(B) Both are moving towards each other?
C) What is the frequency observed by the ambulance driver in case A and B?
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#24) The same ambulance moving at 45 m/s north with Mr. Berbit inside emits a
siren with a frequency of 400 Hz. A car ahead of the ambulance is also moving
north at 30 m/s.
A) What is the frequency observed by the person in a car? (Remember, the car
moves away from the ambulance and the ambulance is moving towards the car!)
Key: Draw picture….is the object (whether or not it is the source or observer)
moving towards or away? Towards causes f ' to increase, away causes f ' to
decrease
B) If the car behind the ambulance is moving north at 30 m/s, what frequency
would be observed by a passenger in the car?
C) If the car that was ahead of the ambulance had the same siren, what frequency
would a person in the ambulance hear?
Notice (A) > 400 Hz,(B) <400 Hz,(C) >400 Hz
Now go home and do Homework - #21, 22, 24, 31
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#24) A small vibrating object moves across the surface of a ripple tank
producing the wave fronts as shown. The wave fronts move with speed
v. The object is traveling in what direction and with what speed relative
to the speed of the wave fronts produced?
Direction
Speed
A) To the right
Equal to v
B) To the right
Less than v
C) To the right
Greater than v
D) To the left
Less than v
E) To the left
Greater than v
Check out Physics Applet for answer: Doppler Applet Link
#25) Now draw the wave when
A) vsource = 0
B) vsource < vsound and source moves to right
C) vsource = vsound and source moves to left
D) vsource > vsound and source moves to right
Shock wave? (When vsource > vsound)
Sonic Boom Clip
FYI: Mach Number is the ratio of vsource:vsound
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15.8 What is a Standing Wave (In General)?
When waves are confined in a space, like waves on a guitar string
or in an organ pipe, there are reflections at both ends, leading to
WAVES traveling in OPPOSITE directions with the SAME
AMPLITUDE and FREQUENCY.
These waves combine according to the superposition principle.
For a given length string or pipe, there are specific frequencies for
which this interference results in a stationary vibration pattern
called a Standing Wave.
DEMO: Standing Wave using a slinky
Standing Wave on a String)
Standing Wave on a String “Continuous”
Standing Wave in a guitar string:
1st Harmonic: (the fundamental frequency)
# of nodes
# of antinodes
_____
_____
L = ____λ1
_____
_____
L = ____λ2
2nd Harmonic:
# of nodes
# of antinodes
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rd
3 Harmonic:
# of nodes
# of antinodes
_____
_____
Since velocity of a wave on a string is
v
Ft
v  f
and

What is each frequency in terms of L? In terms of f1?
f1 = v
λ1
f1 =
f2 = v
λ2
f2 =
f3 = v
f3 =
λ3
Hence, General Formula for nth frequency
v
1
fn  n
n
2L
2L
Ft

#26) Given: f48 = 312 Hz find f5.
L = ____λ3
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#27) A piano string is 1.10 m long and has a mass of 9.0 grams.
A) How much tension must the string be under if it is to vibrate at a
fundamental frequency of 131 Hz?
B) Find f2, f3, f4.
C) If the tension is quadrupled, what is the
new fundamental frequency f1? f2? f3?
Interesting aside: Frequency Matching. Why do Tires on cars sometimes
seem to run backwards when watching them on the highway at night or in a
video?
That’s because they are either being illuminated by a vapor lamp hooked up to AC current (which switches on
and off at 60 Hz) or because they are being filmed by a camera taking pictures at a certain rate. In either case,
you are witnessing a stroboscopic effect called frequency matching. Here is a Wiki description:
Special calibrated strobe lights, capable of flashing up to hundreds of times per second, are used in industry to
"stop" the motion of rotating and other repetitively-operating machinery and to measure, or adjust, the
rotation speeds or cycle times. Since this stop is only apparent, a marked point on the rotating body will either
appear to move backward or forward, or not move, depending on the frequency of the strobe-flash. If the
flash occurs equal to the period of rotation (or evenly multiplied or divided, i.e. n*ω or ω/n, where n is an
integer and ω the angular frequency) the marked point will appear to not move. Any non integer flash setting
will make the mark appear to move forward or backward, e.g. a slight increase of the flash frequency will
make the point appear to move backward.
DEMO: Stroboscope and motor
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#28) A frequency generator is hooked up to a string under Tension T
due to mass M (___kg) being hung on the string of linear density μ
(___kg/m).
(A) For the given tension T, determine the fundamental frequency.
B) Predict f2, f3 etc.
C) If the tension is increased, the fundamental frequency __________.
D) If the length of string is shortened (tension remains the same), the
fundamental frequency ___________.
Use the Stroboscope on the standing wave
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#29) A cord of fixed length and uniform density, when held between
two fixed points under tension T vibrates with a fundamental frequency f.
If the tension is doubled the 2nd harmonic is:
#30) A wire fixed at both ends
is set into vibration. If the
frequency is 60 vibrations
per second, the velocity of
propagation of the transverse
wave along the wire is
A) 30 m/s
B) 60 m/s
C) 120 m/s D) 180 m/s E) 240 m/s
Now go home and do homework #37, 38, 39, 40, 44
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15.10 Standing Waves in Air Columns
Standing LONGITUDINAL waves can be set up in a tube of air, open at
both ends (such as an organ pipe or flute), or closed at one end (such as a
saxomophone or glass bottle). This is the result of two waves traveling in
opposite directions, with the same frequency.
See website: http://www.phys.unsw.edu.au/jw/flutes.v.clarinets.html
(scroll down to see how sound is reflected at the closed end, and even open end)
Case I (Pipe Open at Both Ends)
If the end of the pipe is open, the pipe
vibrates, and thus, the air molecules have
complete freedom of motion and a(n)
___________ exists at the end of the pipe.
Draw the first 4 Harmonics:
1st Harmonic: (Fundamental Frequency)
# Nodes
_____
# Antinodes _____
L = ____λ1
____________________________________________
nd
2 Harmonic:
# Nodes
_____
# Antinodes _____
L = _____ λ2
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rd
3 Harmonic:
# Nodes
_____
# Antinodes _____
L = _____ λ3
THUS, in general, for a pipe open at both ends, the formula
is:
fn  n
v
for n = 1, 2, 3, …
2L
Case II (Pipe Closed at One End)
If the end of the pipe is closed, the pipe
vibrates, but the air molecules do NOT have
complete freedom of motion and a(n)
_________ exists at the closed end of the pipe.
Draw the first 4 Harmonics:
1st Harmonic:(Fundamental Frequency)
# Nodes
_____
# Antinodes _____
L = ____λ1
Pipe organ applet:
34
___Harmonic:
# Nodes
_____
# Antinodes _____
L = ____λ
____________________________________
___ Harmonic:
# Nodes
_____
# Antinodes _____
L = ____λ
Thus, for a pipe closed at one end, the general formula is:
v
fn  n
4L
for n = 1, 3, 5, …
Pipe organ applet:
Demo: Metal and cardboard pipes open at both ends/Bunsen burner
#31) A pipe has a length of 3m (open at both ends). Determine the
frequencies of the first three harmonics if the speed of sound in air is 343
m/s.
#32) A pipe has a length of 3m (closed at one end). Determine the
frequencies of the first three harmonics if the speed of sound in air is 343
m/s.
Now go home and do homework- #47,48,50,53
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#33) 1995 AP Part B #6
A hollow tube of length L is open at both ends as
shown and is held in midair. A tuning fork with a
frequency fo vibrates at one end of the tube and
causes the air in the tube to vibrate at its
fundamental frequency. Express your answers in
terms of L and fo.
A) Determine the wavelength of the sound.
B) Determine the speed of sound in the air inside the tube.
C) Determine the next higher frequency at which this air column
would resonate.
#33 is continued on next page.
36
The tube is submerged in a large, graduated
cylinder filled with water. The tube is slowly
raised out of the water and the same tuning
fork, vibrating at fo is held a fixed distance from
the top of the tube.
D) Determine the height h of the tube above the
water when the air column resonates for the
first time. Express your answer in terms of l.
This Brings us to…
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15.9 Resonance
Resonance is the tendency of a system to oscillate at maximum
amplitude at a certain frequency. This frequency is known as the
system's natural frequency of vibration or resonant frequency.
Note: If two objects have the same natural frequency of vibration,
vibrating one object will cause the other to vibrate, even if they
are not touching. This vibration will lead to an increase in
___________.
(This is called sympathetic vibrations.)
Resonance demos: Singing Pipe, Singing wine glass, Chladni Plates, etc.
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15.11 What are beats?
2 waves traveling in the same direction with slightly
different frequencies.
As time passes, waves A and B move in and out of phase
with each other periodically, as shown in the diagram
below.
So what does the observer hear? (Demonstrate with two tuning forks)
A listener at some fixed point will hear an alternation in
loudness known as BEATS.
The beat frequency equals the difference in frequency
between the two sources.
#34) Two tuning forks, each with a frequency of 100 Hz and 97 Hz
respectively. Calculate the beat frequency.
Chapter 14 and 15. Done.