Waves and Sound

Remember Periodic Motion?

• Motion which repeats in a regular cycle

• Pendulum, vibrating spring, vibrating guitar string

Simple Harmonic Motion

• Motion around a point of equilibrium

• Force proportional to displacement of object from equilibrium

What is a wave?

• Wave=disturbance that carries energy through matter or space

• Note that the actual matter does not travel far but the energy can- the energy in this wave could have traveled from Alaska!

Classification of Waves

Waves Are:• Mechanical or Non-Mechanical

• One (or More) Pulses or Periodic

• Longitudinal or Transverse or Combined

Mechanical Waves• Require A Medium For Transmission

– Medium = Mass / Atoms / Material

• Transmitted Via Vibration Of Particles In The Medium Around A “Rest” Position

• Examples– Sound– Water Wave

Non-Mechanical Waves• No Medium Is Required For Transmission• Can Be Transmitted Through Empty Space• Examples:

– Visible Light– Infrared Or Ultraviolet Light– Radio/TV Waves Microwaves

Any Electromagnetic Radiation

Pulse vs. Periodic• Pulse

– A Single Vibratory Disturbance

• Periodic Wave– A Series Of

Regular Disturbances

– Regular: Identical & Evenly Timed

Transverse waves

• Disturbance is perpendicular to the motion of the wave

• http://www.youtube.com/watch?v=cPKGa2DsIs0

Longitudinal Waves

• Disturbance is parallel to motion of wave

• Ex- sound waves• Fluids usually only

transmit longitudinal waves

Surface Waves/Elliptical Waves

• Underwater, waves are longitudinal but at the surface they have elements of both longitudinal and transverse

• Motion of a particle on the surface is an ellipse

Torsional Waves

• Twist around a central axis

• Like Tacoma Narrows Bridge

Wave properties

• Equilibrium• Crest• Trough• Amplitude

• Phase• Wavelength

Amplitude

• Maximum displacement of a particle in a wave from the equilibrium

• Examples: brightness of a light, loudness of a sound

Wavelength

• Distance between 2 corresponding locations

• Usually measured from crest to crest or trough to trough

• Symbol is

Amplitude and Wavelength

• These waves have the same wavelength but different amplitudes

• These waves have the same amplitude but different wavelengths

Phase• Points On A Periodic

Wave Are In Phase If They Have:– Same Displacement

From Rest Position

AND– Same Direction Of

Motion

• C and F are “In Phase”

Phase

• Points that are “in phase” act the same- they are a whole multiple of a wave apart

• Since wavelength is one complete cycle, we usually refer to it as 360

• So in phase= n360

• Points that are “out of phase” are not a whole multiple of 360 apart- they can be any # of degrees apart

• We usually look at 90, 180, and 270 apart

Phase Problems-

• Using A as a reference, which point(s) are:– 360 in phase– 90out of phase– 180 out of phase– 270out of phase

Frequency

• Number of vibrations per second

• Symbol is f• Unit is Hz (1/s)

Period• Time to complete one cycle• Symbol is T• Unit is s• T=1/f

Speed

• Speed of a wave= wavelength x frequency

• v= f• Examples- we see

the baseball hit the bat before we hear it b/c light wave travels faster than sound wave

Comparing Wave Speeds• Light:3.00 x 108 m/s• Sound: 3.31 x 102 m/s• We See The Lightning Flash Before We Hear

The Thunder.• We See The Bat Hit The Ball Before The Crack

Is Heard

Speed of a Wave on a String

• For faster waves: tighter string (more tension) or lighter string (less mass per length)

• Mass/length is known as the linear mass density

Speed of wave problems

• A ball of string is purchased at a local hardware store. According to the manufacturer, the package contains 100 yards (91.5 meters) of string and has a mass of 12 oz (341 grams)

• What is the string's linear mass density?• If the string's tensile strength is 90 N, what is

the maximum speed a pulse could travel along the string?

solutions

• Mass/length= 3.73 x 10-3 kg/meter

• Speed=155.3 m/sec

Wave Graphs- same shape but different info

• Vibration graph- shows behavior at one spot

• Waveform graph shows wave behavior in many spots at one time

ProblemsProblems• A periodic wave goes through twenty complete A periodic wave goes through twenty complete

cycles of its motion in 4.0 secondscycles of its motion in 4.0 seconds

• What is the frequency of the wave?What is the frequency of the wave?

• What is its period?What is its period?

• Determine the frequency of a wave whose period is Determine the frequency of a wave whose period is 5.0 seconds5.0 seconds

Wavefront

• The Locus Of Adjacent Points Which Are In Phase – Such As The Crest Of A Water Wave

Spherical Wavefront

Spherical Wavefront

Periodic Wave Phenomena

• Superposition/Interference• Resonance • Doppler Effect• Diffraction• Reflection • Refraction

Waves at An InterfaceWaves at An Interface

• InterfaceInterface– A Boundary With A Different MediumA Boundary With A Different Medium

• Part Of The Wave Is ReflectedPart Of The Wave Is Reflected• Part Is Transmitted Through The Part Is Transmitted Through The

Second MediumSecond Medium• Part Is Absorbed (Turns Into Heat)Part Is Absorbed (Turns Into Heat)• Speed can changeSpeed can change

Reflection

At a rigid boundary, when wave hits with an upward force, the boundary medium will react with a downward force so reflected wave is

INVERTED

• If boundary is nonrigid (it can move) wave will reflect in same orientation

Refraction

• When a wave enters a new medium velocity can change causing wave to bend

Doppler Effect• A Variation In Observed Frequency When There Is Relative Motion

Between A Source And An Observer

• Approaching:– Higher Frequency

Observed

• Receding– Lower Frequency

Observed

Sound Pitch Changes

Light Color Changes

Doppler Effect ExamplesSiren Passing

Doppler Effect

Calculations involving Doppler Effect

• Let fs be the source frequency and fd be the detected frequency

• If source moving towards you, frequency will increase so choose + or - accordingly– fd=(v+vd)/(v+vs) *fs

• Thus, if moving away, frequency will be lower • If moving towards, frequency will be higher

Example: Doppler

• A car is traveling 20 m/s away from a stationary observer. If the car’s horn emits a frequency of 600Hz, what frequency will the observer hear?

• Use v=340m/s for the speed of sound

Solution

• Since car is traveling away from observer, frequency will be lower

• fd=(340+0)/(340+20) * 600Hz= 567Hz

Breaking the sound barrier

• Speed of sound varies in different mediums

• When something travels faster than the local speed of sound it “breaks the sound barrier”

Breaking the sound barrierRegions of constructive interference=SHOCK WAVES

Superposition of waves

• When 2 waves meet, the displacement in the medium is the sum of the individual displacements

• They then continue, unchanged by their meeting

Constructive InterferenceConstructive Interference• Maximum Maximum ConstructiveConstructive Interference Interference

Occurs When The Phase Difference Is Occurs When The Phase Difference Is 0° “In Phase”0° “In Phase”

Destructive InterferenceDestructive Interference

• Maximum Maximum Destructive Destructive Interference Interference Occurs When The Phase Occurs When The Phase Difference Is 180 “Out of Phase”Difference Is 180 “Out of Phase”

Interference PatternsInterference Patterns

• Symmetrical Patterns Produced By Symmetrical Patterns Produced By Sources In Phase In The Same MediumSources In Phase In The Same Medium

Interference Can Produce Interference Can Produce ColorsColors

Interference PatternsInterference Patterns

Interference PatternsInterference Patterns• Maximum Maximum

destructivedestructive interference interference produces produces nodesnodes

• Maximum Maximum constructiveconstructive interference interference produces produces anti-nodesanti-nodes

• http://www.physicsclassroom.com/mmedia/waves/ipd.cfm

Nodes and Antinodes

• Nodes- – net displacement=0

• Antinodes- – Net displacement=

max

Nodes & Anti-NodesNodes & Anti-Nodes

• NodesNodes– Points of NO DISPLACEMENTPoints of NO DISPLACEMENT

• Anti-NodesAnti-Nodes– Points of MAXIMUM DISPLACEMENTPoints of MAXIMUM DISPLACEMENT

• Line of NodesLine of Nodes– ““Smooth Area”Smooth Area”

Standing WavesStanding Waves• Each object has a “natural frequency” at which it is Each object has a “natural frequency” at which it is

“willing” to vibrate“willing” to vibrate• If you force a vibration at this frequency, object will If you force a vibration at this frequency, object will

resonate resonate or vibrate at increasing amplitudeor vibrate at increasing amplitude• Next shower, try to test this- sing different notes until Next shower, try to test this- sing different notes until

you reach one that is significantly louder (increased you reach one that is significantly louder (increased amplitude)amplitude)

• We can make this using a wave reflecting off a We can make this using a wave reflecting off a boundary at the same frequency, amplitude, and boundary at the same frequency, amplitude, and wavelengthwavelength

Resonance

• If small, regular forces applied at just the right time it can increase the amplitude of vibration

• Ex- trampoline, maybe Tacoma Narrows Bridge?

• In a string, this depends on its length- always draw!

• http://www.ngsir.netfirms.com/englishhtm/StatWave.htm

Nodes, Antinodes in Standing Waves

• Nodes and antinodes alternate• Each node is 1/2 from the last• We use this to determine how standing waves

form

Standing WavesStanding Waves

Notice there are distinct wavelengths that can produce these standing waves. Note: 1st overtone=2nd harmonic

fundamental

2nd harmonic 3rd harmonic 4th harmonic

Harmonics vs overtones

• Let’s take a guitar string: the fundamental (also known as 1st harmonic) is when there is λ/2 wavelengths

2nd harmonic (1st overtone)

• The second harmonic occurs when you have twice the fundamental so 2(λ/2)=1 wavelength

3rd harmonic

• Now we have 3x the fundamental or 3(λ/2)=3λ/2 or 1.5 wavelengths

• Are we seeing the pattern here?

• NOTE:use the term “harmonic” since it matches the math

Determining harmonic frequencies

• Consider an 80-cm long guitar string that has a fundamental frequency (1st harmonic) of 400 Hz.

• What is the wavelength of the 2nd harmonic?

• wavelength is 160 cm or 1.60 m.

• Now what is the speed of the wave?

• speed = frequency • wavelength

• speed = 400 Hz • 1.6 mspeed = 640 m/s

• Now, the speed of the other harmonics is the same- you can use their wavelengths to determine the frequency of each harmonic

• What is the frequency of the 2nd harmonic?

solution

• Wavelength of 2nd harmonic would be 0.8m

• f=v/λ

• f=640/0.8=800Hz

• Now calculate frequency of 3rd harmonic

Standing waves- strings

• Strings are fixed on both ends

• How does the fundamental frequency compare to the length of the string?

• Draw the fundamental and 2 harmonics

Resonance in Strings- Probs• Two identical vibrating strings are 2.0 meters

long and uniform in density. One is fixed on both ends, the other is free on one end.

• What is the wavelength of the fundamental frequency along the fixed-end string?

• What is the wavelength of the fundamental frequency along the free-end string?

• What is the wavelength of the first overtone along the free-end string?

• What is the wavelength of the second overtone along the fixed-end string?

Resonance and Wave Speed

• Now put it together with wave speed in a spring:

• Suppose a string of length 100 meters has a mass of 300 grams: determine the fundamental frequency in this string when a 5-kg mass is suspended from a 1-meter vibrating section.

Sketch the problem

solution

T = mg T = (5)(9.81) = 49.05 N• mass/length = 0.3 / 100 = 3 x 10-3 kg/m• for the fundamental, L = ½λ = 1 meter λ = 2

meters• vw = √[49.05/(3 x 10-3)]• vw = 127.8 m/sec • vw = fλ• 127.8 = f(2)• f = 63.9 hz

Free and Fixed End Reflectors• Free end reflector: one

end is fixed and one is free– Reflection is in phase

(crest reflects as a crest)– http://www.physicsclassroom.com/mmedia/waves/fix.cfm

• Fixed end reflector: both ends fixed– Reflection is out of phase

(crest reflects as a trough)– Try it :)– http://www.physicsclassroom.com/mmedia/waves/free.cfm

Standing waves in sound- Closed Tube Sound waves are longitudinal but we can draw them as transverse to see them easily

• In a closed tube, the far end is a node (particles can’t compress)

• The possible length of a tube follows a distinct pattern based on the fundamental being 1/4

• Note that only odd harmonics occur

Open-tube

• Standing waves can still be established if the end is open- the waves reflect off the open air

• In this case, the end is air which can vibrate so it is an antinode

• Open tubes can support all harmonics

Harmonics and music

• String instruments often have multiple harmonics vibrating simultaneously- this produces the particular timbre of the instrument

• http://dev.physicslab.org/asp/applets/string/help.asp

• The pentatonic scale-= these overtones and fundamental :

• http://www.youtube.com/watch?v=ne6tB2KiZuk

Sound Waves

• Mechanical• Need a medium• No sound in space

Sound Waves

• Longitudinal• Vibrations create

pressure variations in medium

• Compressions• Rarefactions• Feel your throat as

you hum• http://www.physicsclassroom.com/mmedia/waves/tfl.cfm

Visualizing a longitudinal wave

• If you graph the difference in density/pressure in the medium, you can graph this as the amplitude and visualize it as a sinusoidal graph

Properties of sound waves

• Share properties of waves• Speed v=f

– Speed depends on medium and temperature

• Reflection– Echo, sonar– Reflections of multiple or

rough surfaces called reverberations

• Interference– Dead spots(nodes)– amplification

Interference: BEATS• Sound waves interfere just as other waves• If pitches of 2 waves are close, we hear resulting

interference as “beats”- pulsating changes in amplitude (loudness)- the beats we hear are the areas of constructive interference

Beats Problems

• Suppose you sound two tuning forks simultaneously: one fork has a frequency of 256 hz and the other has a frequency of 260 hz. – How many beats would be heard each

second?– What is the pitch of these beats?

Beats

• The frequency of the beats is the difference between the 2 original frequencies– beat frequency = |f2 - f1|

• The pitch of the beats is the average of the 2 frequencies– beat pitch = ½(f2+f1)

Perception of Sound

• Loudness = amplitude

• Measured in decibel (dB)

• Exposure to loud sounds can cause your ear to lose sensitivity

Perception of Sound

• Our perception of the loudness of a sound is not directly proportional to the pressure- intensity is actually logarithmic so for each 10 decibel increase, the intensity goes up 10X

• Also depends on pitch, pure tone vs. combined tones

Speed of Sound• Depends on medium

– Greater elasticity= faster – Metals conduct sound

quickly

• Depends on temp– Hot=fast

• In same medium, all sound waves travel at same speed– When band plays- all

sounds reach you at same speed regardless of pitch, amplitude, instrument

Speed of Sound in Air

• In dry air, speed of sound is function of temp:

• vw = 331 + 0.6 T

• As air becomes more humid, speed increases

• If air temp not constant, can cause refraction– When this happens with

light, we see a mirage

Dampening

• Sound waves cause vibrations in the medium so energy is lost to heat- thus the wave is damped

• Lower frequency cause less motion so can travel farther- thus use low frequency for fog horns

Physics of Music

• Sound produced by vibrating object which causes pressure oscillations in air

• Vibrating string• Vibrating reed(s)• Vibrating column of air