Resonance - a vibration of large amplitude in a mechanical or
electrical system caused by a relatively small periodic stimulus
of the same or nearly the same period as the natural vibration
period of the system.
Requires:
 a system capable to vibrate with a reasonable small
damping;
 an external driving force with a frequency close to that of the
system.
http://www.youtube.com/w
atch?v=YLBt_07-Vek
Question:
why are we discussing resonance now and have not
discussed it while talking about propagating waves?
Propagating waves transport
energy. So, it is continuously carried
away and does not get accumulated
in specific locations.
Propagating waves do not have any
special natural wavelengths and
frequencies.
Both wavelengths and frequencies
can vary continuously.
Standing waves have a particular
set of allowed frequencies, which
are characteristic for the system.
v
f1 
2L
Fundamental mode
f m  mf1 Higher harmonics
Doppler effect – change in the wave frequency and/or wave length due
to motion of the source or the observer (or both).
Case #1 – moving observer.

v
L
The train is moving at a speed v, and the
cars have a length L.
When the observer is at rest there are
v
f 
L
cars passing by every
second
Doppler effect – change in the wave frequency and/or wave length due
to motion of the source or the observer.
Case #1 – moving observer.

v
L

u
The train is moving at a speed v, and the cars
have a length L.
When the observer is moving in the same
direction as the train at a speed u, the relative
speed is v – u and there are
v u
f '
L
cars passing by every
second
Doppler effect – change in the wave frequency and/or wave length due
to motion of the source or the observer.
Case #1 – moving observer.

v
L

u
The train is moving at a speed v, and the cars
have a length L.
When the observer is moving in the direction
opposite to the train at a speed u, the relative
speed is v + u and there are
vu
f '
L
cars passing by every
second

v
l
With “+” and “-” corresponding to the opposite and
same directions of motion, respectively.
+
The wave length does not change (of course!),
but the relative velocity does.
The general equation is

u
The original frequency is
Therefore,
vu
f '
l
v
v
f   l
l
f
vu
f ' f
 f (1  u / v)
v
I love hearing that
lonesome wail of the train
whistle as the magnitude of
the frequency of the wave
changes due to the Doppler
effect…
Doppler effect.
Stationary Sound Source
Car horn
large l
small l
low f
high f
Source moving with
vsource < vsound ( Mach 0.7 )
Case #2 – moving source.
Imagine splashing water with time
intervals T, and making a circular
wave every T seconds.
In the time intervals between two
consecutive splashes, a circle travels
a distance l = Tv, where v is the wave speed.
So the distance between the consecutive circles is l.
Imagine now moving your hand a distance Dx during the time T. Then
the distance between the consecutive circles in front of your hand will be
l’ = l – Dx.
And behind your hand the distance will be l’ = l + Dx.
Your hand is the source of waves here and its speed is u = Dx/T.
Therefore the distance between the circles is
Dx
Dx / T
u
l '  l  Dx  l (1  )  l (1 
)  l (1  )
l
l /T
v
Case #2 – moving source.
Dx
Dx / T
u
l '  l  Dx  l (1  )  l (1 
)  l (1  )
l
l /T
v
u
l '  l (1  )
v
The wavelength changes if the source of the
waves moves.
It decreases if the source and the wave move in
the same direction (approaching source)
The wavelength increases if the source moves in the direction
opposite to the wave motion (receding source), “+” in the equation.
What about the wave speed? does it change?
What about the frequency registered by an observer at rest?
u
l '  l (1  )
v
What about the wave speed? does it change?
What about frequency registered by an observer at rest?
The wave speed does NOT change, since the
circles (wave crests), once generated, loose any
connection with the source, and cannot “know”
about motion of the source. They only care
about the mechanical properties of the medium.
Therefore the wave length and the frequency are
connected by the usual equation
f 
v
l
v
v
1
f
f '  

l' l 1 u / v 1 u / v
Here again “-” is for approaching source – higher frequency.
“+” is for receding source – lower frequency.
Doppler effect.
Stationary Sound Source
Car horn
large l
small l
low f
high f
Source moving with
vsource < vsound ( Mach 0.7 )
Police radar – sends frequency f.
The car moving toward the radar at a speed u receives a
frequency
vu
f '
v
f
It reflects the received frequency f’, but the car is a source
moving toward the radar, so that the radar receives a frequency
f'
1 u / v
f ''
 f
1 u / v
1 u / v
If u/v is small, which is typical for electromagnetic waves.
f ' '  f (1  2u / v)
The frequency of the received signal is higher by
2u
f
v
Why the equations for moving source
u
f
l '  l (1  ) f ' 
1 u / v
v
and moving observer are different ?
f '  f (1  u / v)
Isn’t any motion relative?
There is third player in the game – the medium for the waves.
It makes a lot of difference, whether or not the source of the
wave moves with respect to the material medium.
The mechanical waves (surface waves, string waves, sound etc.)
absolutely need a material medium to propagate.
Any waves that do not need a material medium?
Electromagnetic waves can propagate in vacuum and do not
require any medium.
Does it make any difference?
A lot of difference! The whole theory of relativity can be derived
from it.
For light and other electromagnetic waves only the relative
velocity of the source and the observer matter.
If this velocity is much lower than the speed of light, c (the
normal situation), the equations become:
l '  l  (1  u / c) f '  f  (1  u / c)
Red light has a larger wavelength than green or white light.
The upper sign is for motion away from each other;
the light becomes more red.
The lower sign is for motion toward each other;
the light becomes more violet.
Waves from a source, which is receding from us, are perceived to
have lower frequency and larger wavelength.
Red shift (shift toward larger wavelengths) in the spectra of
distant galaxies is an evidence of expanding universe.