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Wave properties . The Doppler Effect. B. A.

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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.


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.




Now: source moves to the right at speed < wave speed.

Each new wave originates from the point farther to the right.

First wavereaches 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.


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


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

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.


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


Doppler effect  Radar guns

When radar waves bounce off a moving object

(echo ) the frequency of the reflected wave changes

by an amount that depends on how fast the object

is moving. The detector senses the frequency

shift 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.



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

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.

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 teacher

demonstrating the

Doppler effect


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

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

How much of each?

That depends on the media and the wave itself.


Reflection of Waves

All waves can be reflected.




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

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

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

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).

Free end – reflected pulse is not inverted.

there is no phase change.

Imagine whip


i = r.

Law of Reflection

The incident and reflected wavefronts.

Angle of reflection is equal to angle of incidence.

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

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



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.


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.

Visualization of refraction

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.


The incident and refracted wavefronts.

light waves

sound waves

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.


A mathematical law which will tell us exactly HOW MUCH

the direction has changed is calledSNELL'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.


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
refraction of light
Refraction of light





Water n= 1.33

Glass (n=1.5)

The refracted ray is refracted more in the glass



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

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



Even 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.

White light

contains all











Total internal reflection





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 whichangle of refraction is 900

When the incident angle is greater than qc, the refracted ray disappears

and the incident ray is totally reflected back.


A Smile in the Sky

  • Rainbows are caused by dispersion of sunlight by water droplets

1. When white sunlight enters droplet

its component colors are refracted

at different angles (dispersion)

2. These colored lights then undergo

total internal reflection.

observer is between the Sun and a rain shower.

3. Second refraction from droplet into air – more dispersion

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


A gemstone's brilliance is caused by total internal reflection

A gemstone's"fire" is caused by dispersion

Maximize brilliance & fire by knowing physics

what does it mean to see something
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.


where is the fish deeper than you think
Where is the fish? Deeper than you think!

Apparent location

of the fish


Is the straw really broken?

refracted ray

perceived straw

incident ray

real straw


Where is the ball? Closer than you think!

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.


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.



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.


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.


Fiber optics

Click me

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 isvery 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 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 …).



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.

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

- into the region of the geometrical shadow


Water waves diffracting through two different sized openings..

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

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

Diffraction by a large object

Almost sharp edges – small diffraction around obstacle

Diffraction by a small object

Strong diffraction effect behind the obstacle


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 reason

why 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!!!

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.

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

dolphins, ocras, whales


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)

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).


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.

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


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.

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


Interference - Superposition

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!

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

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


constructive interference – increased amplitude,

increased energy

(E ~ A2 ) – increased

intensity – brighter light

or loud sound at point

the waves are in phase

  • destructive interference – decreased amplitude,

decreased energy

– decreased intensity

– no light or no sound

the waves are out of phase

Partially destructive interference.

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.


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?

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.

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. 0cm

B. 2cm

C. 4cm

D. -4cm


Standing waves

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

A nodeis a point where the standing wave has minimal amplitude

A antinodeis a point where the standing wave has maximal amplitude

Distance between two nodes is l/2


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.

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.








L = n

Distance between two nodes is l/2


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.


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

Wave with 𝜆n= 2L/n has freq.

n = 1, 2, 3…

the lowest freq. standing wave is called FUNDAMENTAL or the FIRST HARMONICS

The higher freq. standing waves are called HARMONICS (second, third...) or OVERTONES






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

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.


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 untunedfreq of her string?

Beat freq = 4 Hz, so origfreq is either 444 Hz or 436 Hz.

Increasing freq increases beat freq, so makes the difference with concert-A greater. So origfreq 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.


Standing Waves in a Drum Membrane

Standing 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

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

Standing waves in a chain vertically hung


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

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.

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 harmonics

Overtones 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.

see the way the first five harmonics combine to build up the

wave shape created by a clarinet

to see the way the first five harmonics combine to build up the

wave shape created by a violin

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.