http://flickr.com/photos/d10b/1341482466/

The Doppler Effect

is an apparent (observed) change in frequency and wavelength of a wave occurring when the source and observer are in motion relative to each other, with the perceived frequency increasing when the source and observer approach each other and decreasing when they move apart.

WHY?

Let A and B be two stationary observers.

Consider first stationary source (student tapping a desk at a constant pace) : The crests move away from the source at a constant speed. The distance between adjacent crests is one wavelength and is the same toward observer A as toward observer B. The freq of waves reaching both observes is the same and equal to the freq of a wave as it leaves its source.

A B

First wave reaches A and B at the same time. 2nd , 3rd , 4th ,.. wave reaches B sooner than it reaches A. B sees waves coming more frequently i.e. B observes higher frequency and shorter wavelength. Similarly, A observes lower frequency and longer wavelength.

General: If the source and the listener are approaching each other the perceived frequency is higher: if they are moving apart, the perceived frequency is lower.

A B

Now: source moves to the right at speed < wave speed.Each new wave originates from the point farther to the right.

• If a source of sound is moving toward you at constant speed, you hear a higher freq than when it is at rest

• If it is moving at increasing speed you hear higher and higher freq• If a source of sound is moving away from you, you hear a lower freq than

when it is at rest• If it is moving at increasing speed you hear lower and lower freq• You can hear this effect with sirens on fire engines of train whistles• A similar effect occurs with light waves and radar waves

Ultrasound (high-frequency sound waves) are directed into an artery. The waves are reflected by blood cells back to a receiver. The frequency detected at the receiver fr relative to

that emitted by the source f indicates the cell’s speed and the speed of the blood.

Applications

A similar arrangement is used to measure the speed of cars, but microwaves (EM waves) are used instead of ultrasound.

The Doppler effect is the basis of a technique used to measure the speed of flow of blood.

Doppler effect Radar guns

http://auto.howstuffworks.com/radar-detector1.htm

When radar waves bounce off a moving object(echo ) the frequency of the reflected wave changesby an amount that depends on how fast the objectis moving. The detector senses the frequencyshift and translates this into a speed.

Light or in general EM wave is a wave. Doppler effect is the characteristic of these waves too.

Based on calculations using the Doppler effect, it appears that nearby galaxies are moving away from us at speed of about 250,000 m/s. The distant galaxies are moving away at speeds up to 90 percent the speed of light. The universe is moving apart and expanding in all directions.

Freqs. of EM waves coming from stars are very often lower than those obtained in the laboratory emitted from same elements (He, H). Redshift – shift toward lower freq. Red light has the lowest frequency out of all of the visible lights.

Astronomy: the velocities of distant galaxies can be determined from the Doppler shift.

Most distant galaxies are observed to be red-shifted in the color of their light, which indicates that they are moving away from the Earth. Some galaxies, however, are moving toward us, and their light shows a blue shift.

Edwin Hubble discovered the Redshift in the 1920's. His discovery led to him formulating the Big Bang Theory of the Universe's origin.

A science teacherdemonstrating the

Doppler effect

When a wave strikes a boundary between two media some of it is reflected, some is absorbed

and some of it is transmitted.

We now look to see what happens to a wave when it is incident on the boundary between two media.

How much of each?

That depends on the media and the wave

itself.

Reflection of Waves All waves can be reflected.

The end of the rope if fixed – reflected pulse returns inverted. The end of the rope if fixed – reflected pulse returns inverted.

Imagine whipImagine whip

Some of the energy of the pulse will actually be absorbed at the support and as such, the amplitude of the reflected pulse will be less than that of the incident pulse (E ~ A2).

the pulse has undergone a 180° (π) phase change.

there is no phase change.

First of all, we shall look at a single pulse travelling along a string.

Free end – reflected pulse is not inverted. Free end – reflected pulse is not inverted.

fixed end

Law of ReflectionThe incident and reflected wavefronts.

Angle of reflection is equal to angle of incidence.

All waves, including light, sound, water obey this relationship, the law of reflection.

i = r .

(the angles are measured to the normal to the barrier).

RefractionWhen a wave passes from one medium to

another, its velocity changes. The change in speed results in a change in direction of

propagation of the refracted wave.

As a toy car rolls from a hardwood floor onto carpet, it changes direction because the wheel that hits the carpet first is slowed down first.

Visualization of refraction

When a wave passes from one medium to another, its velocity changes. The change in speed results in a change in direction of propagation of the refracted wave.

light waves sound waves

The incident and refracted wavefronts.

frequency is determined by the source so it doesn’t change. Only wavelenght changes. Wavelength of the same wave is smaller in the medium with smaller speed.

1 2

1 2

v vf = =

λ λ

θ1

1 1

2 2

sin θ v =

sin θ v

θ2

A mathematical law which will tell us exactly HOW MUCH the direction has changed is called SNELL'S LAW.

Although it can be derived by using little geometry and algebra, it was introduced as experimental law for light in 1621.

For a given pair of media, the ratio

is constant for the given frequency. The Snell’s law is of course valid for all types of waves.

Refraction occurs when a wave enters a medium with different speed of wave.When the wave enters a medium with slower speed the wavelength becomes shorter and the wave speed decreases. The frequency remains the same.

www.le.ac.uk/ua/mjm33/wave2/images/Snell.gif

1 2

1 2

v vf = =

λ λ

The speed of light inside matter

• The speed of light c = 300,000,000 m/s = 3 x 108 m/s• In any other medium such as water or glass, light travels at a lower speed.

• INDEX OF REFRACTION, n, of the medium is the ratio of the speed of light in a vacuum, c, and the speed of light, v, in that medium:

•

no units

As c is greater than v for all media, n will always be > 1.

greater n – smaller speed of light in the medium.

As the speed of light in air is almost equal to c, nair ~ 1

cn =

v

MEDIUM n v (m/s)

Vacuumair

Exactly 11.000293

300,000,000

water 1.33 225,564,000

glass 1.52 197,368,000

diamond 2.42 123,967,000

Refraction of light

Water n= 1.33 Glass (n=1.5)

Incidentray

refractedray

The refracted ray is refracted more in the glass

SNELL'S LAW

Can be written in another form for refraction of light only.

sin𝜃1

sin𝜃2

=𝑣1

𝑣2

𝑣1

𝑣2

=

𝑐𝑛1

𝑐𝑛2

=𝑛2

𝑛1

→

greater n ⇒ smaller speed of light ⇒ stronger refraction smaller angle ⇒

Which of the three drawings (if any) show physically possible refraction?   

Answer: (a) refraction toward normal as it should be

White lightcontains all

wavelengths(colors)

RedOrangeYellowGreenBlueIndigoViolet

DispersionEven though all colors of the visible spectrum travel with the same speed in vacuum, the speed of the colors of the visible spectrum varies when they pass through a transparent medium like glass and water. That is, the refractive index of glass is different for different colours.Different colors are refracted by different amounts.

qc

q2=900

n2

n1

Total internal reflection

21 nn

Angle of refraction is greater than angle of incidence. As the angle of incidence increases, so does angle of refraction. The intensity of refracted light decreases, intensity of reflected light increases until angle of incidence is such that angle of refraction is 900.

Critical angle: qc - angle of incidence for which angle of refraction is 900

When the incident angle is greater than qc , the refracted ray disappearsand the incident ray is totally reflected back.

Critical angle: qc - angle of incidence for which angle of refraction is 900

sin 𝜃𝑐=𝑛2

𝑛1

A Smile in the Sky

1. When white sunlight enters droplet its component colors are refracted at different angles (dispersion)

2. These colored lights then undergo total internal reflection.

Rainbows are caused by dispersion of sunlight by water droplets

3. Second refraction from droplet into air – more dispersion

4. Each droplet produces a complete spectrum, but only one from each is seen by observer – you have your own personal rainbow and I have MINE!

observer is between the Sun and a rain shower.

A gemstone's brilliance is caused by total internal reflectionA gemstone's "fire" is caused by dispersion

Maximize brilliance & fire by knowing physics

What does it mean to “see” something?

• To “see” something, light rays from the object must get into your eyes.• unless the object if a light bulb or some other luminous object, the light

rays from some light source (like the sun) reflect off of the object and enter our eyes.

(root)BEER!

Where is the fish? Deeper than you think!

Apparent location of the fish

even the long legs are pretty short in the water

Is the straw really broken?refracted ray

real straw

incident rayperceived straw

Where is the ball? Closer than you think!

ball

Apparent location of the ball

It looks like a moon setting over a body of water, with the

moon's reflection in the surface. We can even see

floating leaves on the water's surface.

That is not what this picture is.

Instead we are looking at a light in the swimming pool, and

the light's reflection on the surface of the water.

Where in the world is fish?

A fish’s eye view result of total internal reflection

Looking upward from beneath the water a person would see the outside world squeezed into a into a cone with an angle of 980.

Formation of mirages: On a hot day (dessert – every day is hot) there is a layer of very hot air just above the ground. There are actually layers of less and less hot air farther above the surface. Hot air is less dense then the cooler air. The light of the distant object originally slanted downward is refracted away from the normal at every interface between two layers of different density until eventually the angle of incidence is greater then the critical angle for interface between two layers of different densities.

MIRAGES

A mirage is caused not by a loss of mental facility on a hot, dry dessert, but by refracted light that appears to be reflected from the smooth surface of a pond in front of an object which is image of the sky.

cooler

hot

So, when you see something that appears to be a wet spot on a distant part of a hot highway, you are OK. This is just light that originates in the sky and travels toward the highway. Before reaching the pavement, it is bent, reflected and then again refracted up into your eye by the hot air above the road. You are seeing light from the blue sky.

A fiber optic cable is a bunch (thousandths) of very fine (less than the diameter of a hair) glass strands clad together made of material with high index of refraction.

Critical angle is very small almost everything is totally internally reflected.

The light is guided through the cable by successive internal reflections with almost no loss (a little escapes).

Trick is in sending the light at just right angle initially.

Even if the light pipe is bent into a complicated shape (tied into knots), light is transmitted practically undiminished to the other end.

Fiber optics

Click me

fiber optic communications

can carry more info with less distortion over long distances not affected by atmospheric conditions or lightning and does not corrode Used to transmit telephone calls and other communication signals. One single optical fiber can transmit several TV programs and tens of thousands of telephone conversations, all at the same time. copper can carry 32 telephone calls, fiber optics can carry 32,000 calls takes 300 lbs of copper to carry same info as 1 lb of fiber optics downside expensive

Used to illuminate difficult places to reach, such as inside the human body (endoscope – bronchoscope – colonoscope …).

The spreading of a wave into a region behind an obstruction is called diffraction.

Diffraction

When waves pass through a small opening, or pass the edge of a obstacle, they always spread out to some extent into the region that is not directly in the path of the waves.

- into the region of the geometrical shadow

Water waves diffracting through two different sized openings..

diffraction effects are small when slit is much larger than the incident λ.

The waves are diffracted more through the narrower opening, when wavelength is larger than the opening.

Diffraction by a small objectStrong diffraction effect behind the obstacle

Diffraction by a large object Almost sharp edges – small diffraction around obstacle

remember: big wavelength

big diffraction effects

For example, if two rooms are connected by an open doorway and a sound is produced in a remote corner of one of them, a person in the other room will hear the sound as if it originated at the doorway.

Diffraction provides the reasonwhy we can hear something even if we can not see it.

Lower-frequency (longer-wavelength) waves can diffract around larger obstacles, while high-frequency waves are simply stopped by the same obstacles. This is why AM radio waves (~1 MHz, 300 m wavelength) signals can diffract around a building, mountain still producing a usable signal on the other side, while FM (~100 MHz, 3 m wavelength) signals essentially require a line-of-sight path between transmitter and receiver.

Ultrasound is used for echolocation: dolphins, bats, sonar, sonograms Sonar appeared in the animal kingdom long before it was developed by human engineers.

But why ultrasound? Because of diffraction!!! Or should we say because of no difraction!!!

So when ultrasound is emitted toward obstacle it will be reflected back rather then spread behind the obstacle.

dolphins, ocras, whales

Low frequency sound has longer wavelength, so they will be diffracted, so not being able to detect the prey. High frequency sound has smaller wavelength, so it will be reflected back from the prey. That’s how bat “sees” its prey.

The ultrasound bats typically chirp is ~ 50 000 Hz. What is wavelength of that sound? The speed of sound wave in air is ~ 340 m/s.

v = lf so l = v/f l = 0.0068 m = 0.7 cm

So, bats use ultrasonic waves with l smaller than the dimensions of their prey (moth – couple of centimeters).

To echolocate an object one must have both emitter and detector. If the wavelength of an emitted wave is smaller than the obstacle which it encounters, the wave is not able to diffract around the obstacle, instead the wave reflects off the obstacle. Reflected wave is caught by detector giving it information on how far (2d = vt) and how big is the object (reflection from different directions)

Keep in mind that wavelengths of the audible sound are < 1m, of the visible light ~ 10-7 m, and water waves you can see for yourself. Now you can understand that diffraction in the case of sound or water can be very obvious, but for light is not so.

Light waves (red light: λ ~ 500 nm = 0.0005 mm) do not diffract very much. Obstacle should be very small. a Shadow!!! (No light behind the obstacle!)

2. Suggest one reason why ships at sea use a very low frequency sound for their foghorn.

Another reason and maybe even better explanation is that the method of generating the sound involves the production of a very strong pressure pulse. The fog horn is loud so that it can be heard far away.

And low frequency sounds do propagate much further than high-frequency ones – due to diffraction.

Elephants also use these deep sounds to communicate over long distance.

Interference - SuperpositionInterference - Superposition

Property that distinguishes waves from particles: waves can superpose when overlapping and as the result a lot of possible craziness can happen.

and when they meet they interfere, superimpose and then carry on living happily ever after as they never met each other

two objects can not be at the same place at the same time!

but two waves can be at the same place at the same time!

Principle of superposition: When two or more waves overlap, the resultant displacement at any point is the sum of the displacements of the individual waves at that point.

constructive interference – increased amplitude,

increased energy (E ~ A2 ) – increased intensity – brighter light or loud sound at point

destructive interference – decreased amplitude,

decreased energy– decreased intensity – no light or no sound

the waves are in phase

the waves are out of phase

Partially destructive interference.

The waves from the boys will interfere when they meet, if the girl is 4 m from the first boy, then she must be 6 m from the other. This is a path difference of 2 m, one whole wavelength. The waves are therefore in phase and will add.

Two boys playing in a pool make identical waves that travel towards each other.The boys are 10 m apart and the waves have a wavelength 2 m. Their little sister is swimming from one boy to the other. When she is 4 m from the first boy, will she big wave or a small wave?

d1 - d2 = 2 m = λ constructive interference

X and Y are coherent sources of 2cm waves.

Will they interfere constructively or destructively at:

(a) A(b) B(c) C

Real-world examples of interference

So where in this world do we observe two sources interference? Where can we experience the phenomenon that sound or light taking two paths from two locations to the same point in space can undergo constructive and destructive interference?

This is relatively common for homes located near mountain cliffs. Waves are taking two different paths from the source to the antenna - a direct path and a reflected path. If the top of the house (antenna) is the point of destructive interference for some wavelength, that wavelength is not received. To fix this sell the house.

While the interference is momentary (the plane does not remain in a stationary location), it is nonetheless observable.

The wavelength of a transverse wave train is 4cm. At some point on the wave the displacement is -4cm. At the same instant, at another point 50cm away in the direction of propagation of the wave, the displacement is

A. 0cmB. 2cmC. 4cmD. -4cm

Standing wavesStanding waves

are the result of the interference of two identical waves with the same frequency and the same amplitude traveling in opposite direction. are the result of the interference of two identical waves with the same frequency and the same amplitude traveling in opposite direction.

A node is a point where the standing wave has minimal amplitude

A antinode is a point where the standing wave has maximal amplitude

Distance between two nodes is l/2

L

Let’s consider a string of length L.Let both ends be fixed (someone is holding each end). One person jiggles one end. We still consider this end to be fixed.

Let us see what shapes can be formed on that string.

They can occur when a wave reflects back from a boundary along the route that it came. Incident and reflected wave have the same A, the same f, and of course the same .l

L

L

L

n = 1, 2, 3…L = n l2

Distance between two nodes is l/2

L𝜆1

2

L𝜆3

23

L𝜆2

22

L𝜆4

24

ln = 2Ln

only standing wave that has wavelength

can be formed on the string of length L.

Wave with 𝜆1 = 2L has freq.

Wave with 𝜆2 = L has freq. …

Wave with 𝜆n = 2L/n has freq.

The frequencies at which standing waves are produced are called natural frequencies or resonant frequencies of the string or pipe or...

the lowest freq. standing wave is called FUNDAMENTAL or the FIRST HARMONICSThe higher freq. standing waves are called HARMONICS (second, third...) or OVERTONES

Beats are a periodic variation in loudness (amplitude) – throbbing - due to interference of two tones of slightly different frequency.

CLICK

Two waves with slightly different frequencies are travelling to the right. The resulting wave travels in the same direction and with the same speed as the two component waves.

Producing beats:When two sound waves of different frequency approach your ear, the alternating constructive and destructive interference results in alternating soft and loud sound.

The beat frequency is equal to the absolute value of the difference in frequencies of the two waves.

𝑓 =|𝑓 1− 𝑓 2|

Useful for tuning musical instruments – listen for beats to disappear (when frequency of instrument is identical to a tuning fork)

Beats produced when incident wave interferes with a reflected wave from a moving object: reflected wave has Doppler-shifted frequency, so the two waves differ slightly in freq. Hear beats.

That underlies how any instrument that measures speed using ultrasound work – measures beat freq. – gets Doppler shift in frequency which is related to speed of the object.

Also underlies how dolphins (and others) use beats to sense motions

(1) A violinist tuning her violin, plays her A-string while sounding a tuning fork at concert-A 440 Hz, and hears 4 beats per second. When she tightens the string (so increasing its freq), the beat frequency increases.

What should she do to tune the string to concert-A, and what was the original untuned freq of her string?

Beat freq = 4 Hz, so orig freq is either 444 Hz or 436 Hz. Increasing freq increases beat freq, so makes the difference with

concert-A greater. So orig freq must have been 444 Hz, and she should loosen the string to tune it to concert-A.

(2) A human cannot hear sound at freqs above 20 000 Hz. But if you walk into a room in which two sources are emitting sound waves at 100 kHz and 102 kHz you will hear sound. Why?

You are hearing the (much lower) beat frequency, 2 kHz = 2000 Hz.

applet that has everything – standing waves – comparison of transverse and longitudinal waves

Standing Waves in a Drum MembraneStanding waves in a drum membrane are complicated and satisfactory analysis requires knowledge of Bessel functions. So, this is just for fun.Mrs. Radja’s fun

Standing waves in a chain vertically hung

Mathematicians Pythagoras, Leibniz, Riemann, Fourier are just few names in a long humankind's attempt to understand the mysteries of the music.There is a beautiful simple logic together with aesthetic similarities that are shared by mathematics and music.

Why a tuning fork, a violin and a clarinet sound very different, even when they are all playing an A, say?

The reason the violin doesn't look and sound like the tuning fork is that it is playing, not just an A, but also a combination of different frequencies called the harmonics.

Overtones – higher harmonicsOvertones are the other frequencies besides the fundamental that exist in musical instruments. Instruments of different shapes and actions produce different overtones. The overtones combine to form the characteristic sound of the instrument. For example, both the waves below are the same frequency, and therefore the same note. But their overtones are different, and therefore their sounds are different. Note that the violin's jagged waveform produces a sharper sound, while the smooth waveform of the piano produces a purer sound, closer to a sine wave. Click on each instrument to hear what it sounds like. Keep in mind that

all are playing the same note. 

to see the way the first five harmonics combine to build up the wave shape created by a violin

see the way the first five harmonics combine to build up the wave shape created by a clarinet

it is important to notice that although these sounds have the same fundamental frequency/pitch each sound sounds different because it is a combination (mixture) of harmonics at different intensities.

Stringed instruments

Three types Plucked: guitar, bass, harp, harpsichord Bowed: violin, viola, cello, bass Struck: piano

All use strings that are fixed at both ends Use different diameter strings (mass per unit length is different) The string tension is adjustable - tuning

Vibration frequencies

In general, f = v / , where v is the propagation speed of the string The propagation speed depends on the diameter (mass per unit length) and tension Modes

Fundamental: f1 = v / 2L

First harmonic: f2 = v / L = 2 f1

The effective length can be changed by the musician “fingering” the strings

Sounds may be generally characterized by pitch, loudness (amplitude) and quality.

Pitch is perceived freq – determined by fundamental freq.

Timbre is that unique combination of fundamental freq and overtones (harmonics) that gives each voice, musical instrument, and sound effect its unique coloring and character. 

The greater the number of harmonics, the more interesting is the sound that is produced.  

Sound "quality“, “color” or "timbre" describes those characteristics of sound which allow the ear to distinguish sounds which have the same pitch and loudness.