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/

watch?v=YLBt_07-Vek

Question:why are we discussing resonance now and have not discussed it while talking about propagating waves?

L

vf

21

1mffm

Fundamental mode

Higher harmonics

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.

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.

L

v

The train is moving at a speed v, and the

cars have a length L.When the observer is at rest there are

L

vf 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.

L

v

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

L

uvf

' cars passing by every

second

u

Doppler effect – change in the wave frequency and/or wave length due to motion of the source or the observer.

Case #1 – moving observer.

L

v

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

L

uvf

' cars passing by every

second

u

The general equation is

uv

f

'

u

v

The original frequency is

)/1(' vufv

uvff

Therefore,

f

vvf

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.+

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 Source moving with vsource < vsound ( Mach 0.7 )

Car horn

small

high f

large

low f

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 = Tv, where v is the wave speed.

So the distance between the consecutive circles is

Imagine now moving your hand a distance x during the time T. Then the distance between the consecutive circles in front of your hand will be

’=– x.

And behind your hand the distance will be ’=+ x.

Your hand is the source of waves here and its speed is u = x/T.

Therefore the distance between the circles is

)1()/

/1()1('

v

u

T

Txxx

Case #2 – moving source.

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)

)1()/

/1()1('

v

u

T

Txxx

)1('v

u

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?

)1('v

u

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

What about the wave speed? does it change?

What about frequency registered by an observer at rest?

v

f vu

f

vu

vvf

/1/1

1

''

Here again “-” is for approaching source – higher frequency.

“+” is for receding source – lower frequency.

Doppler effect.

Stationary Sound Source Source moving with vsource < vsound ( Mach 0.7 )

Car horn

small

high f

large

low f

Police radar – sends frequency f.

The car moving toward the radar at a speed u receives a frequency

fv

uvf

'

It reflects the received frequency f’, but the car is a source moving toward the radar, so that the radar receives a frequency

vu

vuf

vu

ff

/1

/1

/1

'''

If u/v is small, which is typical for electromagnetic waves.

)/21('' vuff

The frequency of the received signal is higher byv

uf

2

Why the equations for moving source

and moving observer are different ?

Isn’t any motion relative?

)1('v

u

vu

ff

/1'

)/1(' vuff

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:

)/1(' cu )/1(' cuff

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.

Red light has a larger wavelength than green or white light.

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.