Waves & Energy Transfer. Waves Three types of waves:1. Mechanical 2. EM 3. Matter Waves -transfer of energy (by particles) or by waves. Transfer energy.

Waves & Energy Transfer

Waves

Three types of waves:1. Mechanical

2. EM

3. Matter

Waves -transfer of energy (by particles) or by waves.

• Transfer energy without changing position

Mechanical Waves

Three type of mechanical waves:

1. Transverse – particles of the medium vibrate perpendicular to the direction of motion.

Transverse WavesTransverse Waves-1

• Seismic waves ~ secondary waves

Transverse Waves

Mechanical Waves

2. Longitudinal (Compressional) – particles of the medium move parallel to the direction of motion.

3. Surface – mixture of transverse & longitudinal waves. Waves

• Also known as compressional waves• Longitudinal wave - Wikipedia, the free encyclopedia

• Seismic waves ~ p –waves – primary waves

Longitudinal Waves

Surface Waves

Wave Pulse vs. Traveling Wave

Wave pulse – single disturbance that travels through a medium.

Traveling wave – a series of pulses at regular intervals. (a source vibrating with SHM will produce traveling waves.

The Measures of a Wave

• Period (T) – the time needed for the motion of the wave to repeat itself. (complete one wavelength).

• Frequency (f) – the number of complete vibrations per second measured at a fixed location.

- measured in Hertz (Hz)- 1 Hz = 1 vibration per second- Frequency & Period relate:

f = 1/T T = 1/f

The Measures of a Wave (cont.)

• Wavelength (λ) – shortest distance between two points where the wave pattern repeats itself.

crests – high points of a wave

troughs – low points of a wave

The Measures of a Wave (cont.)

• Velocity of a wave (v) – for any object, the velocity is the distance it moved divided by the time interval. For a wave, the distance equals the wavelength, and the time interval is the period, T (or multiple thereof)v = λ/T or v = λ f

***frequency always remains constant, so, a higher velocity ~ longer λ, lower velocity ~ shorter λ

The Measures of a Wave (cont.)

• Amplitude – maximum displacement from the rest position or equilibrium position.

- in order to achieve a higher amplitude, more work has to be done

- a wave with a larger amplitude is transferring more energy

- the energy transferred by a wave depends on the square of its amplitude. If you double the amplitude, you increase the energy by 4x.

Wave Interference at Boundaries

Waves at a boundary between media:• Speed (Velocity) only depends on the properties

of a medium.examples: water-depth

rope-mass and length sound- temperature of the air

As long as the medium is the same, thespeed of a higher frequency and a lower frequencywave are the same.

Wave Interference at Boundaries

Reflection and Transmission- When a wave approaches a boundary, it can be

reflected (bounced back) or transmitted (passed through)

Incident wave – incoming waveReflected wave – outgoing wave (bounced)Transmitted wave – wave allowed to pass through

Wave Interference at Boundaries

The Effect of the Medium

If the difference between the media is small, then the amplitude of the transmitted wave will be almost as large as the incident wave, and the amplitude of the reflected wave will be small.

If the difference is large, then most of the energy is reflected.

Wave Interference at Boundaries

• Waves passing from one medium to another have the same frequency

• Velocity changes, and wavelength depends on the velocity change so that

f=v/λ remains constant.

• As velocity increases, so does wavelength

TwoMediums

Wave Interference at Boundaries

• Reflection of Waves from Boundaries

Wave Interference: Superposition

Principle of Superposition – the displacement of a medium caused by two or more waves is the algebraic sum of the displacements caused by the individual waves.

Interference – the result of the superposition of two or more waves.

1) Constructive2) Destructive Waves

Wave Interference: Superposition

Constructive – occurs when the wave displacements are the same direction (above or below equilibrium) and are in step.

Result: A larger amplitude than any individual wave (sum of the waves).

The two pulses retain their original size and shape after they pass.

Wave Interference: Superposition

Destructive – occurs when the displacements are opposite and in step.

Result: A smaller or no displacement

The two pulses retain their original size and shape after they pass.

Standing Waves

• Occur when there are nodes at the end (of a rope) and an antinode in the middle.

• They are stationary and appear to be standing still.

• Standing waves are the result of pulses with equal but opposite amplitudes meeting. These are incident and reflected waves that interfere.

Standing Waves

sta1fix Sta2fix

When an incident and reflected wave meet as in the picture, max constructive & destructive interference occurs.

Result:

Standing Waves

• Node – one point that is completely undisturbed (produced by DESTRUCTIVE INTERFERENCE)

• Antinode – point that undergoes the greatest displacement (produced by CONSTRUCTIVE INTERFERENCE)

Standing Waves

Resonance

• Resonance occurs when a period (T) of a medium’s vibration equals the time required for a wave to travel to a fixed point and back (T).

• A large oscillation occurs (STANDING WAVE)

• The natural frequency of a medium equals that of the wave’s frequency.

Diffraction

• When waves meet a barrier with holes in it, they bend around the edges of the barrier forming circular waves that spread out.

• The same effect occurs when waves meet small obstacles.

• The smaller the λ in comparison to the size of the obstacle, the less diffraction

• Diffraction of water waves (opening)

Diffraction

Reflection

Law of Reflection – for a wave, the angle at which it approaches a barrier is equal to the angle at which it is reflected.

Normal – line drawn perpendicular to the barrier (surface)

Angle of Incidence – θi the incoming angle measured between the ray and the normal

Angle of Reflection- θr the outgoing angle measured between the ray and the normal.

Reflection

θi θr

normal

Regular Reflection Diffuse Reflection

(see above)

Sound

Sound – a longitudinal wave, consisting of compressions and rarefactions.

• In air, the sound source produces regular variations in air pressure. Tunefork Waves

Velocity of sound depends on the medium

In air ~ it’s the temperature of the medium• at sea level, temp = 20°, velocity = 343m/s• Velocity increases 0.6 m/s for each °C increase• Velocity is greater in solids and liquids than in

gases

Sound

Sound waves share the same general properties of other waves:

• Reflection

• Refraction

• Interference

• Diffraction

• v = λf

• Echo – reflected sound wave

Loudness of Sound

Amplitude – the measure of variation in pressure along the wave.

Loudness depends on the amplitude of the pressure wave

• the human ear is extremely sensitive to pressure variations or amplitudes

• It can detect amplitudes as small as 2x 10-5 Pa up to 20 Pa (recall 1 atm = 105 Pa)

• (See packet 22 Decibel scale)

The Doppler Effect (Shift)

The Doppler Shift occurs in all waves, both mechanical and EM.

• When a sound source moves relative to an observer, the detected frequency is altered

• If the sound source moves toward the observer, the detected frequency is higher

v waves are compressed

wavelength is shorter

The Doppler Effect (Shift)

• If the sound source moves away from the observer, the detected frequency is lower

v

waves are rarefacted

wavelengths are longer Velocity DOES NOT CHANGE, so

frequency must!Ex: Radar, Red Shift/ Blue Shift, UltrasoundDopplerWaveFronts DopplerEffect

Pitch & Frequency

Pitch – essentially, the frequency of the sound wave (hat you hear)

Pitch has values on the musical scalemiddle C = 262 Hz

E = 327 HzMost people hear between 20-20,000 Hz

Loudness as perceived by the human ear depends on both frequency & sound level.

Resonance In Air Columns

• Closed Pipe – when a wave hits the closed end, it is reflected back; if a high pressure meets a high pressure, resonance occurs (reflected wave is erect)

• Open Pipe – when the wave meets the open end, it is reflected inverted. When a high meets a high, resonance occurs.

Closed Pipe & Open Pipe Resonators

• Closed Pipe – the shortest column of air that can have a node at the open end and an antinode at the closed end is ¼ λ. Resonance occurs when L = odd # of ¼ λ.

• Open Pipe – the shortest column of air that can have nodes at the open ends is ½ λ. Resonance occurs when L = even # of ¼ λ.

Music & Sound Quality

Timbre – difference between two waves (amplitudes & frequencies) AKA: Tone color or tone quality.

• When two waves superimpose, we experience timbre.

Fundamental – the lowest resonant frequency an instrument will make.

Music & Sound Quality

Harmonic – the higher frequencies that are multiples of the fundamental frequency.

(Depending on whether an open or closed pipe)

When sounds of two separate pitches are played simultaneously (chords), interference occurs.

Consonance – pleasant sound resulting from two waves with a difference of frequency of at least 7 Hz.

YouTube - Musician Tips & Careers : What Makes a Good Song

Music & Sound QualityDissonance – unpleasant sound resulting from

two waves with a difference of frequency of at least 7 Hz.

YouTube - Musician Tips & Careers : What Makes a Good Song

Music Intervals – Ratios of FrequenciesTemperament

Octave (1:2) YouTube - Guitar Octaves

Perfect 5th (2:3) YouTube - Position playing Lesson 7 (perfect fifth)

Perfect 4th (3:4) Major 3rd (4:5) YouTube - Guitar Theory: Major 3rd Intervals : Shaping Major 3rd Guitar Intervals

Beats – 2 frequencies that are nearly identical interfere to produce high and low levels called beats. Beats

Light

Visible Light – electromagnetic wave that the human eye can detect. (All EM waves behave the way visible light does ~ diffraction, refraction, etc.)

Range of Frequencies (of visible light):

4 x 10-7 m 7 x 10-7 m Shortest λ = violet longest λ = red

(see EM Spectrum Chart)

Light

All EM waves travel at the speed of light• Galileo first hypothesized that light had a finite

speed• Roemer calculated the speed of light based on

observations of Jupiter’s moon, Io (It took 22 minutes to cross the diameter of Earth)

• Albert A. Michaelson precisely calculated the speed 2.997996 + .00004 x 108 m/s (Nobel Peace Prize)

• 1960’s, the invention of the laser helped provide an accurate measure

Light

Equation: velocity = c

c = λ f

c = speed of lightC = 3.0 x 108 m/s

Officially: 1 meter = distance light travels in 1/299,792,458 seconds

Light

Color

Visible light can be separated into its component colors or spectrum (by the use of a prism)

White light red

violet

Red bends the least.

Violet bends the most.

Formation of Color in Thin Films

(soap bubbles or oil slicks)Colors result because of constructive and

destructive interference of white light. (see sheet) I 1/4λ

R1

1/4λ

R2

R1 travels 1/4λ in the time it takes I to travel to the back side of the bubble. When R2 reflects, it is erect because of the less dense medium, and the two reflected waves meet in step ~ CONSTRUCTIVE INTERFERENCE

Formation of Color in Thin Films

¼ λ ½ λ round trip

R1 R2

R1 ¾ λ 1 ½ λ round trip

R2

When a transverse wave is reflected from a more optically dense medium, it is inverted.

R1 = inverted R2 = erect

When R1 meets R2 it is constructive interference

Formation of Color in Thin Films

Other wavelengths suffer partial or complete destruction.

• Different colors satisfy the different ¼ λ requirement

• As thickness increases, shorter λ’s will be mostly reflected. R ¼ λ

O ¾ λ Y 5/4 λ G 7/4 λ B 9/4 λ V 11/4 λ

Polarization Of Light• Proof that light is transverse.

Ordinary light contains EM waves vibrating in every direction perpendicular to the direction of travel.

Transverse wave model:

Light travels in all directions perpendicular to the motion.

Polarization Of Light

Polaroid Materials – contain molecules that are long and allow EM waves of one direction to pass through.

Polarizing Axis – one direction of the Polaroid material…only waves vibrating parallel to the axis can pass through.

Polarizer – first polaroid filter – filters ½ of waves that pass through (intensity is ½ )

Analyzer – second filter – placed perpendicular to the polarizer and blocks the remainder of light.

Optics

Refraction of Light

• Light travels at different speeds in different media.Refraction – the bending of light at the boundary between

two media.optically dense – material that slows the speed of light.

Index of Refraction (n)• measure of the amount that light bends when passing

into a medium from a vacuum• Refraction occurs because the speed of light depends on

the medium through which it travels

n = c/v

Refraction

air

glass

air

glass

When light passes into a more optically dense medium, the light bends TOWARDS THE NORMAL

Θi > Θr Smaller velocity, smaller Θ

When light passes into a less optically dense medium, the light bends AWAY FROM THE NORMAL

Θi < Θr Larger velocity, larger Θ

When light passes into another medium ON THE NORMAL, nothing happens.

Snell’s Law

• A ray of light bends in such a way that the ratio of the sine of the angle of incidence to the sine of the angle of refraction is constant.

• For light traveling from a vacuum into another medium:

Index of Refraction (for the medium)

n = sin θi

sin θr Refraction

Snell’s Law

• In general, light traveling from one medium into another:

ni sin θi = nr sin θr

ni = sin θr

nr sin θi

Total Internal Reflection

• This occurs when light passes from a more optically dense medium into a less optically dense medium, at an angle so great that there is no refracted ray.

Critical Angle – Angle of Incidence• Angle of incidence that results in the refracted

ray (angle) to fall on the boundary itself. (90°)

• If going into air (or vacuum): sin θc = 1/n

• Otherwise nr sinθr = ni sinθc

nr sin(90°) = ni sinθc

Reflection and Refraction

Dispersion of Light

• Light in a vacuum ~ 3.0 x 108 m/s • It travels more slowly in other media.Waves of different λ travel at different

speeds in certain media.The index of refraction of a material

depends on the λ of the incident light• Frequency remains the same for all λ’s.Dispersion – separation of white light into

its spectrum

Dispersion (cont.)

Dispersive Material – medium in which velocity depends on wavelength (λ)

• Each color has its own index of refraction so when the light passes into the dispersive medium, the individual colors will have their own velocities, THUS, each BENDS DIFFERENTLY.

• Red – has a small index of refraction; bends less• Violet – has a large index of refraction; bends

more Optics

Dispersion (cont.)

• Polychromatic Light – light that contains waves of different wavelengths (white light)

• Monochromatic Light – light that contains only one wavelength

• When polychromatic light passes into a dispersive medium, colors will result.

• Ex: rainbows, colors from a prism

Dispersion (cont.)

Diffraction and Interference of Light

Thomas Young (1773- 1829) studied wave interference.

Double Slit ExperimentYoung set up two closely spaced narrow slits, and

let light fall upon them. The light passing through them was spread out or

diffracted.When the wavelets overlapped, an Interference

Pattern resulted (interference fringes)Diffraction Optics

Diffraction and Interference of Light

Young used monochromatic light and placed a narrow slit in front of the source.

This resulted in COHERENT LIGHT – waves of the same λ and are in step.

Measuring the wavelength:X = λ λ = x dL d L

When white light passes through a double-slit, a continuous spectrum is formed.

Diffraction and Interference of Light

x = difference between the first bright band and the central band on the screen

d = separation between the slits

L = distance between the screen and the slits

λ = wavelength