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Lecture 19
Waves
Types of waves
Transverse waves
Longitudinal waves
Periodic Motion
Superposition
Standing waves
Beats
Waves
Waves:
•Transmit energy and information
•Originate from: source oscillating
Mechanical waves
Require a medium for their transmission
Involve mechanical displacement
•Sound waves
•Water waves (tsunami)
•Earthquakes
•Wave on a stretched string
Non-mechanical waves
Can propagate in a vacuum
Electromagnetic waves
Involve electric & magnetic fields
•Light,
• X-rays
•Gamma waves,
• radio waves
•microwaves, etc
Waves
Mechanical waves
•Need a source of disturbance
•Medium
•Mechanism with which adjacent sections
of medium can influence each other
Consider a stone
dropped into water.
Produces water waves
which move away from
the point of impact
An object on the surface
of the water nearby moves up and down and
back and forth about its original position
Object does not undergo any net displacement
“Water wave” will move but the water itself will
not be carried along.
Mexican wave
Waves
Transverse and Longitudinal waves
Transverse Waves
Particles of the disturbed medium through
which the wave passes move in a direction
perpendicular to the direction of wave propagation
Wave on a stretched string
Electromagnetic waves
•Light,
• X-rays etc
Longitudinal Waves
Particles of the disturbed medium move back
and forth in a direction along the direction of
wave propagation.
Mechanical waves
•Sound waves
Waves
Transverse waves
Pulse (wave) moves left to right
Particles of rope move in a direction perpendicular
to the direction of the wave
Rope never moves in the direction of the wave
Energy and not matter is transported
by the wave
Waves
Transverse and Longitudinal waves
Transverse Waves
Motion of disturbed medium is in a direction
perpendicular to the direction of wave propagation
Longitudinal Waves
Particles of the disturbed medium move in a direction
along the direction of wave propagation.
Waves
Vibrational motion
Object attached to spring. Spring compressed
or stretched a small distance (x) and then
released ; a force (Fr) is exerted on the object
by the spring
Motion of mass as a function
of time traces out a sine wave
Displacement
m
Fr
x
m
time
x
Fr
m
Net restoring force Fr
Fr  x
proportional to x: object undergoes
Fr  kx
simple harmonic motion
Waves
Fr  kx
Hooke’s law
Not only applies to springs
Waves
Displacement
Object vibrating with single frequency
Single frequency wave characteristics
Displacement versus time (or distance)
l
crest
amplitude
Time or
distance
l
Trough
Wavelength : Distance between two successive
identical points on the wave (λ)
Amplitude : Max height of a crest or depth of
a trough relative to the normal level
Wave Velocity : Velocity at which the wave
s
crests move.
v
1
l  v 
f 
t
s  vt l  vT
v fl
Waves
Sound
A plucked string will vibrate at its natural
frequency and alternately compresses and
rarefies the air alongside it.
compression
Density of Air
rarefaction
Compressed air [increased pressure]
Rarefied air
[reduced pressure]
Air molecules move away from high pressure
region >>>>>> setting up longitudinal wave
organised vibrations of air molecules>> sound
Waves
Sound waves-(variation in air pressure)
can cause objects to oscillate
Example: ear drum is forced to vibrate in
response to the air pressure variation
Waves
Wave characteristics
Frequency of waves
• Frequency (f) of a wave is independent of
the medium through which the wave travels.
–it is determined by the frequency of the
oscillator that is the source of the waves.
Speed of waves
•The speed of the wave is dependent on
the characteristics of the medium
through which the wave is traveling.
Wavelength
•The wavelength (l) is a function of both
the oscillator frequency and the speed (v)
of the wave such that
v fl
Waves
Superposition
Two or more waves travelling through same
part of medium at the same time
What happens?
Adding waves
sum of the disturbances of the combined waves
If amplitude increases: constructive interference
If amplitude decreases: destructive interference
Vocal sounds
combination of waves of different frequencies
Voice individually recognisable
Waves
Superposition
Simple case: Addition of two waves with
same wavelength and amplitude
displacement
In step: Added, crest to crest (trough to trough)
wave 1
wave 2
time
resultant
Out of step: Added, crest to trough
wave 1
wave 2
resultant
Waves
Standing waves
Two waves (same frequency) travelling in
opposite directions
Waves reflected back from a fixed position
Fundamental
1st Overtone
2nd Overtone
3rd Overtone
Nodes; positions of no displacement
Antinodes; positions of maximum displacement
Distance between successive nodes (antinodes)
= l
Applications
2
Microwave ovens
Musical instruments
Waves
Standing waves
Fundamental
1st Overtone
2nd Overtone
3rd Overtone
String held tightly at both ends
Only certain modes of vibration allowed
Only certain wavelengths allowed
Standing wave must have node at either end
Length of string may be changed to get other
wavelengths
Example: guitar fingering
Changing the vibrating length
Standing waves: organ pipes
Waves
Waves on a stretched string
Consider a vibrating string;
Wave speed is a function of
•tension of the string
•Mass per unit length
Wave speed
T
v
m/ L
T is the tension
m is the mass of the string
L is the length of the string
Waves
Example
What is the frequency of the fundamental mode
of vibration of a wire of length 400mm and
mass 3.00 g with a tension of 300N.
T
Wave speed v  f l 
m/ L
300 N  (400 103 m)
v
3 103 kg
v  (4 104 )m2 s 2  200ms 1
v fl
l= 2L
v
200ms 1
f 

 250 Hz
2 L 2  0.4m
Waves
Superposition
Simple case: Addition of two waves with
same frequency and amplitude
Beats
If the two waves interfering have slightly
different frequencies (wavelengths), beats occur.
In step (in phase)
In step (in phase)
Out of step (out of phase)
Waves
Beats
If the two waves interfering have slightly
different frequencies (wavelengths), beats occur.
Wave 1
Wave 2
resultant
Waves get in and out of step as time progresses
Result• constructive and destructive interference occurs
alternately
•Amplitude changes periodically at the beat
frequency
Beat frequency = fb = f1-f2
Absolute value: beat frequency always positive
Waves
Beats
fb = f1-f2
If frequency difference = zero
No beats occur
Wave 1
Wave 2
resultant
Waves
Beats
Beats can happen with any type of waves
Sound waves
Beats perceived as a modulated sound:
loudness varies periodically at the beat frequency
Application
Accurate determination of frequency
Example
Piano tuning
Adjust tension in wire and listen for beats
between it and a tuning fork of known frequency
The two frequencies are equal when the beats
cease.
Easier to determine than when listening to
individual sounds of nearly equal frequencies
f1 = 264Hz
f2 = 266 Hz
Beat frequency 2Hz
Waves
Multiple frequencies of different amplitudes
added together
•complex resultant
Sound waves
Resultant tone
•Particular musical instrument
•Person’s voice
Waves
Question
Tuning a guitar by comparing sound of the
string with that of a standard tuning fork. A beat
frequency of 5 Hz is heard when both sounds
are present. The guitar string is tighten the and
the beat frequency rises to 8Hz. To tune the
string exactly to the frequency of the tuning
fork what should be done?
• a) continue to tighten the string
• b) loosen the string
• c) it is impossible to determine
Resonance
Most objects have a natural frequency:
Determined by
• size
• shape
•composition
System is in resonance if the frequency
of the driving force equals the natural
frequency of the system
Resonance: examples
child being pushed on a swing.
Opera singer -breaking glass
Voice
Air passages of the mouth, larynx and nasal
cavity together form an acoustic resonator.
Voiced sound depend on
•resonant frequencies of the total system
------depends on system’s volume and shape
Resonance: examples
Tacoma narrows Bridge,
Washington
US,
ElectricalState,
Resonance:
1940
Example: Tuning in radio station
Adjust resonant frequency of the electrical circuit
to the broadcast frequency of the radio station
To “pick up” signal
Waves
Question
Frequency is constant. Its is determined by
the source of the wave.
Since
f 
1
period
Period in constant
Propagation speed depends on properties of string
v fl 
T
m/ L