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Start Ch 20Traveling Waves

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WavesMechanical Waves: Waves that travel within a material medium, such as air or water.Ex: Sound waves and water waves.Note: These waves NEED a medium to propagate. You can’t have a water wave if you don’t have water.

FYI:Electromagnetic Waves: Self propagating waves of electric and magnetic fields. These waves do NOT require a medium, they can travel through vacuum.Ex: Light waves, Radio waves.Matter Waves: Particles such as electrons and atoms can behave like waves. Quantum Physics.

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Two main types of traveling waves: Transverse Wave

Transverse Wave: wave where displacements are perpendicular to the direction the wave travels. Ex: A wave travels along a string in a horizontal direction while the particles that make up the string oscillate vertically (displaced vertically).

Demo: String

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Two main types of traveling waves: Longitudinal Wave

Longitudinal Wave: particles in the medium move parallel to the direction of wave travel. Ex: Sound waves in gases and liquids are longitudinal waves. An oscillating loudspeaker cone compresses and expands the air just like the springs in this figure.

Demo: Slinky

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• Transverse and Longitudinal waves are examples of mechanical waves. They require a medium to travel through.

For example, the medium of a water wave is water, the medium of sound waves are air, the medium of a wave on a stretched string is the string.

• A medium must be elastic. It needs a restoring force of some sort to bring the medium back to equilibrium after it is displaced.

The tension in a stretched string pulls the string back straight. Gravity restores the level surface of a lake after a water wave passes by.

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As a wave passes through a medium, the atoms of the medium are displaced from equilibrium. This is a disturbance of the medium.

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What causes a disturbance? A source.

The source of a wave might be a rock thrown into the water, your hand plucking a guitar string, an oscillating loudspeaker cone pushing the air, etc.

Once created, the disturbance travels outward through the medium at the wave speed v.

Note: A disturbance propagates through the medium, but the medium as a whole does not move. The ripples on a pond move outward from the splash of a rock, but there is no outward flow of water from the splash. The particles of a string oscillate up and down but do not move in the direction of a pulse traveling along the string.

A wave transfers energy, but it does not transfer any material or substance outward from the source.

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The wave travels along the string, but the material itself does not travel along the string.

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1-D transverse waves

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1-D Longitudinal Waves

Note: Top graph is ∆x vs. x

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Sine Waves

Recall:

fT 1=

Period: Time for one cycle of motion.

Wavelength: Distance spanned by one cycle of motion.

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In time T the wave moves one wavelength λ.

Velocity is displacement over time.

πωλλλ2

=== fT

v (periodic waves)

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πωλλλ2

=== fT

v (periodic waves)

Can define: λπ2

=k

Giving: vk=ω

k is called the wave number. Units are rad/m.

Note: the wave number k is NOT the spring constant k. These are two different physical quantities with the same variable letter.

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With waves the displacement D of a particle in the medium depends on position x and time t. D(x,t)

For periodic sine waves we have the general equation which describes how the wave propagates:

( )0sin),( φω +−= tkxAtxD(sinusoidal wave traveling in the positive x-direction)

Note: for a sine wave traveling in the negative x-direction we have: ( )0sin),( φω ++= tkxAtxD

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( )0sin),( φω +−= tkxAtxDThis is a sine wave with amplitude A like we have seen.

The difference is now we have a few terms inside the sine term.

1. The kx term describes how the sine wave depends on position.

2. The ωt term describes how the sine wave depends on time.

3. The Φ0 term is a phase offset that is used to ‘fit’ the sine wave with initial conditions (usually at t=0).

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ExampleA sine wave with amplitude 1cm and frequency 100Hz travels at 200m/s in the positive x-direction. At t = 0 s, the point x = 1m is on a crest of the wave.

Determine A, v, λ, k, f, ω, T, Φ0, Write the equation for the wave.

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Phase

The quantity

is called the phase of the wave, denoted by ( )0φω +− tkx

φ

( )φsin),( AtxD =The displacement can be written as:

( )0sin),( φω +−= tkxAtxD

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Phase Difference

( ) ( )010212 φωφωφφφ +−−+−=−=Δ tkxtkx

( )λ

πφ xxkxxk Δ=Δ=−=Δ 212

If ∆x = λ, the phase difference is 2π rad (phase difference between two adjacent wave fronts).

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Example

What is the phase difference between the crest of a wave and an adjacent trough?

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Waves in 2-D and 3-D

The 2-D analog of a 1-D sine wave is called a circular wave. Each circle is spaced one wavelength apart and represents the same phase of the wave (eg, crests).

The 3-D analog is called a spherical wave. Spheres are spaced one wavelength apart and also represent the same phase of the wave.

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2-D and 3-D waves

( ) ( )0sin),( φω +−= tkrrAtrD

Circular and Spherical waves can be described with a sine function. This time instead of the linear distance x, we now work with the radial distance r (where r is centered at the wave source).

Note: In 2 and 3 dimensions the amplitude depends on radius. As waves spread out the amplitude decreases to conserve energy.

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Plane waves

If you are far away from a spherical wave source the waves look like plane waves (straight waves traveling in a plane).

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Sound Waves

Sound waves are a sequence of compressions and rarefactions.

Sound waves can travel through gases, liquids, and solids.

Note: Sound waves need a medium to travel through.“In space, no one can hear you scream.” --Alien

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Speed of Sound

The speed of sound depends on the medium.

In air: vsound = 343 m/s(sound speed in air at 20°C)

It takes sound approximately 3s to travel 1km, or about 5s to travel 1 mile.

A useful technique to estimate how far away lightning strikes are. Also useful to estimate the distance of a sniper (crack/bang).

Video: Sniper Crack/Bang, Rocket Bang

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Finding distance to the rocket

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Sounds we can hearTypical human hearing extends from 20Hz to 20,000Hz.

The range of sounds you can hear usually decreases with age. Also, hearing damage can reduce your range of hearing (among other problems). Hearing damage is caused by loud sounds, physical damage, diseases, etc.

Sounds above human hearing are called Ultrasonic.

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Example

Assume your range of hearing is 20Hz – 20,000Hz. What is the range of wavelengths for sound waves you can hear?

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Intensity

The Intensity of a wave is:

where P is the power of the wave and A is the area.

API =

Units: W/m2

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For a source emitting spherical waves the intensity is:

24 rPI source

π=

If intensity at distance r1 is

and the intensity at r2 is

211 4/ rPI source π=

222 4/ rPI source π=

Then: 21

22

2

1

rr

II=

(Psource not needed)

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What if you could produce a tightly focused beam of waves that traveled in 3-D but acted like a 1-D wave?

Video: LRAD, MRAD

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Sound Intensity LevelOften sound levels are expressed in decibels (dB) as,

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛=

010logdB10

IIβ

where I0 = 1 x 10-12 W/m2

(threshold of hearing)

Note: 0dB doesn’t mean no sound; it means for most people no sound is heard.

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( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛=

010logdB10

IIβ where I0 = 1 x 10-12 W/m2

(threshold of hearing)

Note: Sound Intensity Level increased by 10 dB each time the actual intensity increases by a factor of 10.

Sound is perceived as ‘twice as loud’ when the intensity increases by a factor of 10.

We say the loudness of a sound doubles for each 10 dB increase.

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( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛=

010logdB10

IIβ where I0 = 1 x 10-12 W/m2

(threshold of hearing)

Although the ‘Threshold of pain’ is 130dB, exposure to quieter sounds still causes hearing loss.

Ex: 120 dB can cause damage to hair cells in the ear. Lengthy exposure to 85 dB can produce damage as well.

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Doppler EffectWhen there is relative motion between the wave source and observer it appears like the frequency of oscillation changes.

Demo: Bell ball on string

Ex: A train whistle sounds higher pitched when it is coming towards you than when it is going away. Same thing for a car horn, ambulance, fire truck, etc.

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vvff

vvff

s

s

/1

/1

0

0

+=

−=

−

+ (Doppler effect for an approaching source)

(Doppler effect for a receding source)

vs is speed of the source, v is speed of wave in medium, f0 is frequency of the source.

Doppler Effect: Moving source

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Doppler Effect: Stationary source, Moving observer

( )( ) 00

00

/1/1

fvvffvvf

−=+=

−

+ (observer approaching a source)

(observer receding from a source)

v0 is observer speed relative to source, v is speed of wave in medium.

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End Ch.20