Chapter 11

Two ways to transport energy and momentum◦ Streaming particles◦ Flowing waves

                                         

                                                                                                                              Image right: Recent Cassini images of Saturn's moon Enceladus backlit by the sun show the fountain-like sources of the fine spray of material that towers over the south polar region. Image credit: NASA/JPL/Space Science Institute+ Full image and caption+ Movie: Enceladus plumes+ Browse version of image

Two ways to study◦ Psychological (mind) and physiological (body)

What we hear◦ Physical

What sound is – compression wave

Moving self-sustained disturbance of a medium

Medium◦ Field ◦ Substance

Mechanical wave – in material media

Atoms◦ Push together – repel◦ Pull apart – attract◦ Objects are made of atoms

When atoms are distorted they act like attached by springs

Displacement causes a wave

Self-sustaining disturbance Examples

◦ String◦ Liquid waves◦ Sound waves◦ Compression waves

The main difference between particle stream and wave is:◦ Medium stays in place as the wave progresses

Longitudinal ◦ Sustaining medium is displaced parallel to the

direction of propagation◦ Ex – Sound waves

Transverse◦ When the sustaining medium is displaced

perpendicular to the direction of propagation◦ Ex – Guitar string

Torsion◦ Variation of transverse waves

Water waves◦ Combination of Longitudinal and Transverse waves

Longitudinal – move back and forth

Transverse – moveup and down

Water – move in circle

One cycle of a wave Profile – outline or shape of the wavepulse

◦ Determined by the driver of the wave Speed – Determined by the medium Examples

◦ Gunshot◦ Grunt◦ Tsunami

Disturbance of waves with a beginning and end

Amplitude varies Carrier wavelength – Steady sinusoidal

oscillation

Ideal disturbance composed of endless repeats of the same profile wave

Period – how long it takes one profile to pass a point in space

Frequency – number of profile waves passing per second

Wavelength – λ (lambda) - distance in space over which the wavetrain executes one cycle

Amplitude – Height of the waves

v = fλ◦ V – velocity (m/s)◦ f – frequency (cycles/sec or Hz)◦ λ – wavelength (m)

Waves pass the length of a 4.5 m boat. It takes 1.5 seconds for the wave to go from end to end. If the waves are 0.5 seconds apart, what is the period, frequency and wavelength?

T = 0.5 seconds f = 1/T = 1/0.5 = 2.0 Hz v = L/t = 4.5 m/1.5 sec = 3.0 m/s λ = v/f = 3.0 m/s / 2.0 Hz = 1.5 m

Speed of the waves is determined by the properties of the medium, not in any way the motion of the source

Velocity of wave in string◦v = √FT/m/L◦v - velocity (m/s)◦FT = Tension (N)◦m /L – mass/unit length

What is the speed of a wave pulse in a 20 cm, 40 g guitar string with the tension of 19.6 N?◦ v = √FT/m/L◦ v - ?◦ FT = 19.6 N◦ m /L – .040 kg / 0.20 m = 0.020 kg/m

◦v = √FT/m/L = √19.6 N / 0.020 kg/m = 31 m/s

Reflected – carries all the original energy

Absorbed – Friction stops wave

Transmission – moving from one media to another◦ Velocity may change when

moving between medias

Solids – longitudinal elastic wave◦ Ex – Earth quake

Fluids – acoustic waves◦ Ex – sound waves

Parts◦ Rarefaction – distance between atoms is elongated◦ Compression – distance between atoms is squeezed◦ Direction of movement – in the direction of

oscillation Each atom is in SHM

Can be determined by the restoring force and its density◦ Use

Bulk Modulus Bernoulli’s equation Young’s Modulus

Dolphins use chirps to locate items underwater◦ Size of wave – 1.4 cm◦ Can “see” fish and other small items◦ Above our hearing range - 10⁵ Hz

Autofocus cameras Bats Medicine

◦ Tumor and Kidney stone destruction◦ Probe body

Joints Baby

Wave lengths below our hearing range (less than 20 Hz)

Examples◦ Elephants◦ Submarines◦ Subwoofers in Rock Bands

Vibrate our internal organs

http://www.pbs.org/wnet/nature/animalspredict/video2.html

Human hearing range – 20 Hz to 20 khz◦ Usually can not hear through entire range◦ Diminishes with age (above 20 years) and loud

noises

First considered in Rome◦ Marco Vitruvius Pollio – designed amphitheaters◦ Though sound travel through air like water waves

Sound needs a media to travel through◦ No sound in a vacuum◦ No sound in explosions in space

Speaker vibrates Creates pressure variations

◦ Quiet – less than 0.002 Pa◦ Loud – about 10 Pa

Loudness – depends on how far the air molecules move

Period and Frequency – depends on time for speaker to move through a cycle

Wavelength – distancebetween rarefactions

What is the wavelength of a tuning note (A440) which is 440 Hz. The speed of sound at room temperature is 343.9 m/s?

λ = v/f = 343.9 m/s / 440 Hz = 0.782 m

Waves can move through the same area of space and have a combined effect

Are not changed or scattered Superposition Principle -When two waves

overlap, the resultant is the algebraic sum of various contributions at each point

Jean Baptiste Joseph, Baron de Fourier

Proved that a periodic wave having a wavelength can be synthesized by a sum of harmonic waves

A wave profile is a result of overlapping sines and cosines

Waves move out in a circle or sphere In-phase at different distances As the wave moves out it becomes diffused

Power – Joules/sec – Watts P = Work/sec

Joules – Newton-meters Work = Force x Distance Measuring –

◦ Depends on area the detector ◦ Depends on the amount of time

The average power divided by the perpendicular area across which it is transported

I = Pav/A (Watt/meter²)

Area of spherical wave = 4ΠR²◦ The farther from the source, the greater the area,

therefore the less the intensity

An underwater explosion is detected 100 m away, where the intensity is 1.00 GW/m². About 1 second later the sound wave is recorded 1.5 km away from the explosion. What will its intensity be?

R1 = 100 m R2 = 1.5 kmI1 = 1.00 GW/m² Δt = 1 sec Power in first square = power in second square I1 4ΠR² = I2 4ΠR² I2 = (1 x 10⁹ W/m²) (100 m)² / (1500 m)² = 4.4 x 10⁶ W/m²

In 1636, Father Mersen used echoes to measure speed of sound

Speed of sound increases with temperature of air

Air temperatures aren’t constant◦ Velocity varies depending on the gas

Speed of sound does not depend on frequency ◦ All waves get there simultaneously

During a thunder storm, you hear thunder 3.50 seconds after you see a bolt of lightening. How far away, in meters and miles, did the lightening strike?

Three parts of ear◦ Outer – From outer ear to ear drum

Sound resonates in canal Amplifies waves from 3 kHz to 4 kHz

◦ Middle – links eardrum to 3 bones to oval window Increases sound pressure

◦ Inner – Transducer that converts pressure to electrical impulses Hairs in the cochlear vibrate at different frequencies

and amplitude

Human response to frequency Pure tone – sine wave Higher the frequency, the

higher the pitch◦ Varies in people◦ Increasing intensity makes you

think you also increased pitch Human voices

◦ Men 80 Hz – 240 Hz (700 Hz in song)

◦ Woman 140 Hz – 500 Hz (1100 Hz in song)

Waveform blend of:◦ Harmonic – fundamental tone (f)◦ Overtones – tones that are over the harmonic

May or may not be harmonics (2f, 3f, etc) Combination of harmonic and overtones makes the timbre

Intensity-level ◦ Number of factors of 10 that is its intensity is

above the threshold of sound◦ measured in bel (In honor of Alexander Graham

Bell)◦ Io(hearing) = 1.0 x 10 ¹² W/m²

Decibel (dB) – 1/10th of a bel◦ Unitless◦ β = 10 log10 I / Io

Condenses the range from 1.0 to a million millionth to 0dB to 120 dB

Log A/B = log A – log B

Log AB = log A + log B

β = 10 log I / Io

Δβ = 10 log I1 / I2 This means that if you have a 12-W system and

want to make it 2X louder, you have to increase the power to 120-W

Noise – Unrelated jumble of disturbances◦ Non-periodic◦ Continuous frequency

White noise – broad bandwidth of sounds out equal intensities◦ Ex – wind, pouring water, radio static

We can distinguish between wavepulses up to about 20 beats per second – then it becomes a hum

Interference caused in sound waves of different frequency

Used to tune guitars and pianos Carrier wave = f1 + f2 / 2 Beat frequency = f1 - f2

◦ f1 = higher f

Waves reflected back and forth in a finite medium

Very common◦ All instruments◦ Our speaking and singing voice◦ Ringing bells◦ Lasers

Nodes – when the resultant is zero Antinodes – midway between nodes Wavelength – twice the node-to-nodes

distance

First harmonic◦ Fundamental

2nd harmonic◦ 1st overtone

3rd harmonic◦ 2nd overtone

4th harmonic◦ 3rd overtone

5th harmonic◦ 4th overtone

Resonance in the system Amplifies the input Guitar

◦ Each string has a different tension and linear mass-density

◦ Fingering – Changes the length of the string – increases the fundamental frequency

L = ½ Nλ (N - whole number of nodes) fN = N/2L FT/m/L

Falsetto voice – increase tension to increase frequency

What must the tension on a 300 mm fiddle string be to be tuned to 660 Hz? The mass-length is 0.38 g/m.

FT = (m/L)(2Lf)² ==0.38 g/m (2 x 0.300 m x 660

Hz)² =72 N (about 16 lbs)

Strings are not loud◦ Sounding board

(piano)◦ Sounding box

(guitar and violin)◦ Pick-ups in

electric guitars

Made in air-filled chambers (wind pipe, trumpets, organ pipes)

Made by – vibrating reeds (saxophone), lips on mouth pieces(trumpet) , fluttering jet of air (flute)

Only frequencies that fit the standing wave mode will be sustained and amplified

◦ Open on both ends◦ Closed on both ends◦ Open on one end and closed on the other

Types of columns

◦ L = ¼ Nλ◦ fN = Nv/4L (v = speed of sound)

◦ L = ½ Nλ◦ fN = Nv/2L (v = speed of sound)

Blowing slow – fundamental Blowing fast – harmonics

Frequency of pitch changes from high to low

Eeeeeoooooo Think of bug in water swimming forward

fo = fsv + vo v + vs◦ The amount of frequency shift depends on who

is moving ◦ Uses – radar guns, weather tracking devices,

satellites, blood flow Doppler effect when bounced off an

approaching target and returned◦ fo = (v + vt) fs/ v - vt

◦ Red-shift of light – galaxies are moving away from us

◦ Based on recession rates and apparent size of universe – the Big Bang happened 15 thousand million years ago