Vibrations and Waves
Chapter 14
Vibrations and oscillations
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Periodic motions ( like: uniform circular motion )
usually motions where restoring forces are present (
restoring forces: forces acting in opposite direction to
the distortion )
example 1 : mass hanging on a spring ( restoring force-->
elastic )
example 2: simple pendulum ( restoring force -->
gravitational )
example 3: vibrating membrane or strings ( restoring
force--> elastic)
1. Mass hanging on a spring
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describing the motion (case with no friction)
new notions
-equilibrium position (where there is no net force)
-amplitude (maximal deviation from equilibrium,
units: m)
-one cycle (the object reaches its original position and
momentum)
-period , T (the length of time for completing one cycle,
units: s)
-frequency, f (number of cycles per unit time,
units: 1/s )
period of the system
- does NOT depend on amplitude !
- stiffness of the spring ( spring constant, k,
units: N/m )
-mass of the object , m (units: kg)
f 
1
T
m
T  2
k
2. Pendulum
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Describing the motion (case with no
friction)
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period
- does NOT depend on the amount of
swing (amplitude)
- does NOT depend on the mass !
- depends on its length, l (units: m)
-depends on the strength of gravity, g
(units: m/s2)
l
T  2
g
Vibrations and oscillations: key to the clocks!
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Early clocks:
- motion of the heavens: Sun, Moon, stars
- flow of substance: water, sand
pendulum clocks:
- Christian Huygens, 1656
spring clocks:
- John Harrison, 1756
today’s instruments:
-electronic clocks (electronic oscillators)
-atomic clocks --> using the frequency of atomic transitions
Resonance
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each oscillating system has its distinctive, natural frequency : fn
driving an oscillator with a frequency which is strongly different of
this is usually not effective
driving the system with a frequency close to its natural frequency
increases strongly the amplitude--> this phenomenon is called
RESONANCE
resonance type increases can be achieved also by driving with with
frequencies with integer parts of fn
complex systems have all natural frequencies (depends on the
stiffness of material, the mass, size and form of the objects)
good effects of resonance: tuning the radio, tuning musical
instruments, etc ….
bad effects of resonance: safety of buildings, bridges, airplanes etc.
Waves: Vibrations that Propagate
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Simplest waves: perturbations (or pulses) that propagates
example: domino waves (just one pulse propagation possible, no
mechanism for restoring the dominoes)
better example: a chain of balls connected by springs (existence of
restoring forces)
other examples: tidal waves, earthquake, light, sound
although the waves travel, the individual particles vibrate around
their equilibrium position
waves transport energy rather than matter
in real medium a part of the energy dissipates by friction
two basic type of waves: longitudinal and transverse waves
One dimensional waves in a rope
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We study simple cases, but the results will be general ones
wave pulse on a rope
- the speed can be changed by the tension in the rope
- the speed depends on the linear density of the rope
- amplitude of the pulse have no effect on the pulse speed
when a pulse hits the end that is attached to the post, it “bounces” off
and heads back (reflection of the pulse)
if the incident pulse is an “up” pulse (crest), the reflected pulse is
“down” pulse (trough)
Superposition of waves
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we send a crest and when it reflects as a
trough we send another crest to meet it
the two waves pass each other as if the
other one were not there !
- very strange word if it would not be like
this!
during the time the waves pass through
each other the resulting disturbance is a
combination (superposition) of the two
pulses
the displacement is the algebraic sum of
the displacements of the two pulses
Periodic waves
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Moving the rope up and down with a steady frequency and
amplitude generates periodic waves
properties of periodic waves
- frequency, f: : oscillation frequency of any piece in the medium
- wavelength, l: the smallest distance for which the wave pattern
repeats (distance between two adjacent crest or troughs) units: m
- speed of the waves, v: traveling speed of a particular crest, units: m/s
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v
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T
v  f
Standing waves
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When a periodic wave is confined --> new effects
superposition of reflected waves with the original one-->
standing waves
oscillating pattern that does not travel
understanding it by using the superposition principle
portions of the rope do not move at all : nodes
positions on the rope that have largest amplitudes:
antinodes
different possible standing waves
- all have the same speed but different frequency and wavelength
- fundamental mode and harmonics
- longest wavelength: fundamental mode
Interference
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Scientific term for the superposition of waves
we consider the 2D case, example: surface of liquids
we assume that there are two sources with the same
frequency which oscillate in phase (both sources
produce crests at the same time)
superposition--> creates interference patterns
bright regions produced by crests
dark regions produced by troughs
regions with large amplitude (where crest meet
crest, or trough meets trough) --> antinodes
regions with little or no amplitude (crest meet trough)-> nodes
the amplitude at a given point depends on the
difference in the path lengths from the two sources
if the difference is a an integer multiple of l we
have antinode
if the difference is an odd multiple of l/2 we
have antinodes
nodes and antinodes form a complex pattern in space
nodal and antinodal lines
Diffraction
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waves can spread out behind the barrier
this bending of waves around obstacles is called
diffraction
the amount of diffraction depends on the relative sizes
of the opening and the wavelength
if the wavelength is much smaller than the opening,
very little diffraction is evident (harder to see it with
light)
as the wavelength gets closer to the size of the opening,
the amount of diffraction gets bigger
diffraction through many obstacles can produce again
interference patterns
Home-Work assignments
 Vibrations and Oscillations
358/1, 3-18; 361/1-10; 362/11-12
 Waves
358/20-21; 359/22, 24-33; 360/34-51;
361/55-56, 58-60; 362/15-24
Summary
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Vibrations and oscillations are described by: period, T (time required for one cycle); frequency, f
(oscillations per unit time); f=1/T ; amplitude (maximum distance from the equilibrium)
- examples: mass hanging on a spring and pendulum
all systems have a distinctive natural frequency
l
when a system is excited at a natural frequency --> resonance T  2
T  2
g
waves are vibrations moving through the medium
waves can be transverse or longitudinal one
waves are characterized by: speed v, frequency f ,and wavelength l;
v=Fl
waves pass through each other, when overlap the total displacement is given by the
superposition (sum) of the individual waves
when periodic wave is confined resonant patterns -- standing waves - can be produced
nodes (portions in the standing wave that do not move), antinodes (moves with the largest
amplitude) ;
fundamental standing wave and harmonics
two identical periodic wave sources with a constant phase difference produce an interference
pattern which contains nodal and antinodal regions
waves do not go straight through openings or around barriers --> they suffer diffraction. The
diffraction pattern depends on the relative sizes of the openings and the wavelength
m
k