In a material medium, the resorting force is provided by intermolecular forces. If a molecule is disturbed, the restoring forces exerted by its neighbors tend to return the molecule to its original position, and it begins to oscillate. In so doing, it affects adjacent molecules, which are in turn set into oscillation. This is propagation of wave.

Medium – the substance or object in which the wave is travelling.

A pulse: a single disturbance that travels through a medium

Why are waves important? è waves carry energy ç Travelling/Continuous/Progressive wave: continuous disturbance transfer energy from one place to another.

without a net motion of the medium through which they travel.

they all involve oscillations – SHM, of one sort or another. The important thing is that when a wave travels in a

medium, parts of the medium do not end up at different places.

The energy of the source of the wave is carried to different parts of the medium by the wave.

Transverse waveThe particles of the medium oscillate perpendicular to the direction of energy transfer/propagation of the wave.

Earthquake secondary waves, waves on a stringed musical instrument, waves on the rope,

EM waves: light, radio waves, microwaves…

Longitudinal/ Compression/ Pressure wave

The particles of the medium oscillate parallel to the direction of energy transfer/propagation of the wave.

Sound waves in any medium, shock waves in an earthquake, compression wave along a spring…

Sound waves in any medium, shock waves in an earthquake,

compression wave along the string

rarefaction – region in a medium with low pressure, low density. compression – region in a medium with high pressure, high density.

Combination wave

There are two types of waves regarding medium.

1. Mechanical waves Ones that need medium to propagate, where particles of

the medium oscillate as the wave passes through.

A mechanical wave is a disturbance that propagates through a medium – solids, liquids or gases, thus transferring energy from one place to another. ● waves on strings ● waves in water ● ocean waves ● sound waves – pressure waves in gas, solid or liquid ● in short, every wave that is NOT EM wave ● As the disturbance moves, the parts of the material (segment of string, air molecules) execute harmonic motion around equilibrium position ● Disturbance travels not the medium

2. Electromagnetic wavesThe other ones, ELECTROMAGNETIC WAVES, do not need medium to propagate. They come to us from faraway stars traveling through a vacuum. Of course, they can travel through a medium, but when they travel through medium, they do definitely not make particles of the medium vibrate at EM frequency. Just imagine window oscillating at frequency of visible light, ~ 1015 Hz. On the other hand when a sound wave (mechanical wave) travels through a window it will make glass vibrate at that frequency. ● EM wave is made up of changing electric and magnetic fields.● The electric and magnetic field components of EM wave are perpendicular to each other and also perpendicular to the direction of wave propagation – hence EM waves are transverse waves.

● They all travel travel through vacuum with the same speed – speed of light c: c = 2.99 792 458 x 108 m / s c ≈ 3 x 108 m/s

Speed of mechanical wavesThe speed of the wave is the speed of energy transfer and is not the same as the speed of the particle of the

medium oscillating around equilibrium position.

water wave

A wave of energy. The electric and magnetic field oscillate (change magnitude and direction)

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The wave speed is determined by: ● the stiffness of the material à more stiff à higher speed each segment of medium is in tighter contact with its neighbor

● density - more difficult to change the velocity of larger masses than smaller ones à greater densityà more inertia à lower speed

Why do waves travel faster in steel than in air? 

As far as waves are concerned, the difference between steel and air is that steel is stiffer and denser than air. But the stiffness of steel is much greater than that of air, even though the density of steel is greater. Consequently, the stiffness factor influences the wave speed more and waves travel much faster in steel than in air.

Speed of sound in:

air: 343 m/s helium: 1005 m/s water: 1500 m/s bone: 3000 m/s glass: 4500 m/s steel rod: 5000 m/sWaves in a violin string: A-string: 288 m/s, G-string: 128 m/s

Definitions associated with wavesAmplitude, A ● is the maximum displacement of a particle from its equilibrium position. ● It is also equal to the maximum displacement of the source that produces the wave. ● energy of a wave ∞ A2.

Period, T ● is the time taken for one complete wave to pass any given point.

Frequency, f ● the number of wavelengths passing by a given point.

Wavelength, λ ● This is the distance along the medium between two successive particles that have the same displacement (that are in phase – e.g. from crest to crest, or from compression to compression)

Wave speed, v ● The speed at which wavefronts pass a stationary observer. ● It is constant, depending on the medium only.

Energy, Power (Energy per time) and Intensity (Power per unit area received by observer) ● Hence for a wave of amplitude A, we have that Energy ∞ A2 , Power ∞ A2, Intensity ∞ A2

Displacement vs. position graph shows the displacement of all points along the wave. A snapshot of a wave at one instant of time.

Transverse wave: Longitudinal wave:

displacement vs. x, density vs. x or pressure vs. x

Displacement vs. time graph shows the oscillations of one point on the wave. All other points will oscillate in a similar manner, but they will not start their oscillations at exactly the same time.

Transverse wave: Longitudinal wave: displacement vs. t, density vs. t or

pressure vs. t

Wave Equation

This applies to all waves à water waves, waves on strings, sound waves, radio, light . .

Waves with different frequencies and wavelength will have the same speed in one medium, determined by that medium. If you shake the string faster (greater freq.) the wavelength will be smaller and vice versa

Wave fronts propagating from a point source

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Spherical wave – The center of the circle is the source of the oscillations. If there is 3-D medium the wave will spread out in all directions. And if the medium is uniform these waves are spherical.

Ray shows direction of wave/ energy propagation Wavefront is the set of crests at the same distance from

the source. Plane waves: far away from the source circular

wavefronts can be approximated with straight parallel lines / planes in 3-D. These are known as plane waves.

EM waves striking the earth are plane waves

Sound Sound is mechanical, longitudinal wave. Can be spread in

gases, liquids and solids.The wave consists of compressed regions alternating with rarefied regions.

The maximum (minimum) pressure during normal

conversation is 3×10−5% higher (lower) than normal

pressure. Ear can detect such small changes. Just like a speed of a wave on a string, the speed of

sound is determined by the properties of the medium through which it propagates. In air, under normal atmospheric pressure and temperature, the speed of sound is approximately 343 m/s.

Frequency of the sound determines the pitch of the sound. The pitch is perceived frequency of the sound.

Humans’ audible range is 20 Hz – 20 kHz Infrasonic sound – frequencies below 20 Hz Ultrarasonic sound – frequencies above 20 kHz Dogs can detect frequencies as low as 50 Hz and as high

as 45,000 Hz while cats detect frequencies between 45 Hz and 85,000 Hz. Bats who rely on reflection of sounds that they emit for navigation can detect frequencies as high as 120,000 Hz. Dolphins can detect frequencies as high as 200,000 Hz. Infrasound in a range of 5 Hz to 10,000 Hz can be detected by elephants.

Infrasound is used in the nature for communication: elephants (~ 15Hz) couple of kilometers, whales – as sound travels faster in water (v ~ 1500 m/s) than in air, the call can be heard over distances of thousands kilometers.

Sources of infrasonic waves include earthquakes, thunder, volcanoes, and waves produced by vibrating heavy machinery. This last source can be particularly troublesome to workers, for infrasonic waves – even though inaudible – can cause damage to the human body. These low freq waves act in a resonant fashion, causing considerable motion and irritation of internal organs of the body.

Ultrasound is used for echolocation: dolphins, bats, sonar, sonograms ....

Sound needs a medium – won’t travel in a vacuum since nothing to compress and expand

Frequency is determined by the source of oscilations, so when guitar string plays a note, the air (or water in the case of underwater concert) vibrate at the that frequency.

As the speed is different in string and air, wavelengths are too.

For the given medium low and high freq have the same speed – higher freq waves have smaller wavelength, and lower freq waves have longer wavelengths since product λf = v is same. Concert – all frequencies played at the same time will reach your ear simultaneously because they have the same speed. MUSIC.

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 observed frequency increasing when the source and observer approach each other and decreasing when they move apart.

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

The same effect occurs with light waves and radar waves

ApplicationsThe 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.When radar waves reflect from 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.Doppler effect is the characteristic of EM 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.

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Astronomy: the velocities of distant galaxies can be determined from the Doppler shift.If a star had a fixed position relative to the earth, the light of one particular frequency would look like this:

As the star moves toward us the observed frequency increases, we say it shifts toward the higher frequency . That’s why we call it a blue shift.

As the star moves away from us the observed frequency decreases, we say it shifts toward the lower frequency. That’s why we call it a red shift.

Red light has the lowest frequency and blue has the highest out of all of the visible lights. The EM spectrum coming from stars/galaxies is compared to one obtained in the laboratory emitted from same elements (He, H), or to one coming from our Sun.

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 red shift in the 1920's. His discovery led to him formulating the Big Bang Theory of the Universe's origin.

Reflection and refraction of wavesWe 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 wavesAll 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.

Free end – reflected pulse is not inverted.

Law of reflectionThe incident and reflected wavefronts.

Angle of reflection is equal to angle of incidence. (the angles are measured to the normal to the interface).

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

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.

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.

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

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

Law of refraction – Snell’s law

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.

greater v → greater angle

The speed of light inside matter

The speed of light in vacuum is:

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.

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

Refraction of light

The refracted ray is refracted more in the medium with greater n / slower speed of light

Where is the fish? Deeper than you think!

Where is the ball? Closer than you think!

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.

Total internal reflectionAngle 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: θc - angle of incidence for which angle of refraction is 900

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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

The spreading of a wave into a region behind an obstruction is called diffraction.Diffraction effects depend on λ of the waves compared to the size of the opening or an object in the path of the waves.

Small diffraction by a large The waves are diffracted opening more through a narrowSmall diffraction around opening, when wavelengthlarge object is larger than the opening. Strong diffraction around small object.

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 (light waves have very small wavelength, so they do not diffract around big object).

Ultrasound (f > 20 kHz, λ < 1.7 cm) is used for echolocation: dolphins, bats, sonar. But why ultrasound? Because of diffraction!!! Or should we say because of no diffraction!!!

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 a bat “sees” its prey.

Interference - Superposition Property that distinguishes

waves from particles: waves can superpose (when overlapping) and as the result a lot of possible craziness can happen. After two waves overlap they carry on with exactly the same properties as before, as if nothing had happened.

Principle of superposition

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

Standing wavesare the result of the interference of two identical waves with the same frequency and the same amplitude traveling in opposite direction. A antinode is a point where the standing wave has maximal amplitudeA node is a point where the standing wave has minimal amplitudeDistance between two nodes is λ/2

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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only standing wave that has wavelength

λn=2Ln

can be formed on the string of length L.

Wave with 1 = 2L has freq. f 1=vλ1

= v2 L

Wave with 2 = L has freq. f 2=vλ2

= vL …

Wave with n = 2L/n has freq. f n=vλn

=n v2L

=n f 1

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.

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. The beat frequency is equal to the absolute value of the difference in frequencies of the two waves. The beat frequency is equal to the absolute value of the difference in frequencies of the two waves. f=|f 1−f 2|

Example: A hiker shouts toward a vertical cliff 465 m away. The echo is heard 2.75 s later. (a) What is the speed of sound in air there? (b) The wavelength of the sound is 0.75 m. What is the frequency of the wave?

(a) v = distance/time = 2d/t = 2·465/2.75 = 338 m/s(c) What is its period? (b) v = f → f = v/ = 338/0.75 = 451 Hz (c) T = 1/f = 2.22x10-3 s

If you wanted to increase the wavelength of waves in a rope should you shake it at a higher or lower frequency? v = fv depends only on the medium. Therefore for given medium it is constant. So if wavelength increases the frequency decreases. You should shake it at lower frequencies. CHECK IT, PLEASE

Example: 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 = λf so λ = v/f λ = 0.0068 m = 0.7 cm

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

Example: (a) Calculate the wavelengths of i) FM radio waves of frequency 96 MHz.

λ = c/f = 3x108/96x106 = 3.1 m ii) AM long wave radio waves of frequency 200 kHz.

λ = c/f = 3x108/200x103 = 1500 m

(b) Use your answers to (i) and (ii) to explain why if your car is tuned to FM, it cuts out when you enter a tunnel but doesn’t if you are tuned to long wave reception.

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 (~1 MHz, 300 m wavelength) signals can diffract around a building, 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.

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

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

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