Ch. 14 Waves and Ch. 14 Waves and SoundSound

An Introduction to Waves and Wave PropertiesAn Introduction to Waves and Wave Properties

Mechanical WaveMechanical WaveA mechanical wave is a disturbance which A mechanical wave is a disturbance which propagates through a medium with little or no propagates through a medium with little or no net displacement of the particles of the medium.net displacement of the particles of the medium.

Wave “Pulse”

Water Waves

Animation courtesy of Dr. Dan Russell, Kettering University

People Wave

•Wavelength ( Wavelength ( λλ ): distance ): distance over which wave repeatsover which wave repeats

•Period (T): time for one Period (T): time for one wavelength to pass a given wavelength to pass a given pointpoint

•Frequency Frequency ff: How often the : How often the wave repeats itself.wave repeats itself.

•Note: T = 1/Note: T = 1/ f f oror f = 1/T f = 1/T

Parts of a WaveParts of a Wave

3

-3

2 4 6 x(m)

y(m)

A: amplitude

: wavelength crest

trough

equilibrium

Speed of a waveSpeed of a wave The speedThe speed of a wave is the distance traveled of a wave is the distance traveled

by a given point on the wave (such as a by a given point on the wave (such as a crest) in a given interval of time.crest) in a given interval of time.

v = d/tv = d/t d: distance (m)d: distance (m) t: time (s)t: time (s)

v = v = ƒƒ v : speed (m /s)v : speed (m /s) : wavelength (m): wavelength (m) ƒ : frequency (sƒ : frequency (s–1–1, Hz), Hz)

Period of a wavePeriod of a wave

T = 1/ƒT = 1/ƒT : period (s)T : period (s)ƒ : frequency (sƒ : frequency (s-1-1, Hz), Hz)

Problem:Problem: Sound travels at approximately 340 Sound travels at approximately 340 m/s, and light travels at 3.0 x 10m/s, and light travels at 3.0 x 1088 m/s. How far m/s. How far away is a lightning strike if the sound of the away is a lightning strike if the sound of the thunder arrives at a location 2.0 seconds after thunder arrives at a location 2.0 seconds after the lightning is seen?the lightning is seen?

Problem:Problem: Sound travels at approximately 340 m/s, Sound travels at approximately 340 m/s, and light travels at 3.0 x 10and light travels at 3.0 x 1088 m/s. How far away is a m/s. How far away is a lightning strike if the sound of the thunder arrives at a lightning strike if the sound of the thunder arrives at a location 2.0 seconds after the lightning is seen?location 2.0 seconds after the lightning is seen?

Problem:Problem: The frequency of an oboe’s A is 440 The frequency of an oboe’s A is 440 Hz. What is the period of this note? What is the Hz. What is the period of this note? What is the wavelength? Assume a speed of sound in air of wavelength? Assume a speed of sound in air of 340 m/s.340 m/s.

Problem:Problem: The frequency of an oboe’s A is 440 The frequency of an oboe’s A is 440 Hz. What is the period of this note? What is the Hz. What is the period of this note? What is the wavelength? Assume a speed of sound in air of wavelength? Assume a speed of sound in air of 340 m/s.340 m/s.

Wave TypesWave Types

A A transverse wavetransverse wave is a wave in which particles of is a wave in which particles of the medium move in a direction the medium move in a direction perpendicularperpendicular to to the direction which the wave moves. the direction which the wave moves.

Example: Waves on a StringExample: Waves on a String

Wave types: transverseWave types: transverse

A A longitudinal wavelongitudinal wave is a wave in which particles is a wave in which particles of the medium move in a direction of the medium move in a direction parallelparallel to the to the direction which the wave moves. These are also direction which the wave moves. These are also called called compression wavescompression waves..

Example: sound Example: sound

Wave types: longitudinalWave types: longitudinal

Longitudinal vs TransverseLongitudinal vs Transverse

Other Wave TypesOther Wave Types

Earthquakes: combinationEarthquakes: combination Ocean waves: surfaceOcean waves: surface Light: electromagneticLight: electromagnetic

Water waves are a combination of transverse and longitudinal waves.

Reflection of wavesReflection of waves

• Occurs when a wave strikes a medium Occurs when a wave strikes a medium boundary and “bounces back” into original boundary and “bounces back” into original medium.medium.

• Completely reflected waves have the Completely reflected waves have the same energy and speed as original wave.same energy and speed as original wave.

Reflection TypesReflection Types

Fixed-end reflection:Fixed-end reflection: The The wave reflects with inverted wave reflects with inverted phase. phase.

Open-end reflection:Open-end reflection: The The wave reflects with the wave reflects with the same phasesame phase

Animation courtesy of Dr. Dan Russell, Kettering University

Refraction of wavesRefraction of waves

• Transmission of wave Transmission of wave from one medium to from one medium to another.another.

• Refracted waves may Refracted waves may change speed and change speed and wavelength.wavelength.

• Refraction is almost Refraction is almost always accompanied by always accompanied by some reflection.some reflection.

• Refracted waves do Refracted waves do not change frequency.not change frequency.

Animation courtesy of Dr. Dan Russell, Kettering University

Wave transfer from Wave transfer from a low density to a a low density to a

high density high density material.material.

Transmitted WaveTransmitted WaveWave speed decreases.Wave speed decreases.Amplitude decreases.Amplitude decreases.Wavelength decreases.Wavelength decreases.Polarity is the same.Polarity is the same.

Reflected WaveReflected WaveWave speed & length Wave speed & length stays the same.stays the same.

Amplitude decreasesAmplitude decreasesPolarity is reversed. Polarity is reversed.

Wave transfer from Wave transfer from a high density to a a high density to a

low density low density material.material.

Transmitted WaveTransmitted WaveAmplitude increases.Amplitude increases.

Wave speed increasesWave speed increases

Wave length increasesWave length increases

Polarity remains the Polarity remains the same.same.

Reflected WaveReflected WaveWave length and speed Wave length and speed stays the same.stays the same.

Polarity remains the Polarity remains the same.same.

Amplitude decreases.Amplitude decreases.

Sound is a longitudinal waveSound is a longitudinal wave Sound travels through the air at approximately 340 Sound travels through the air at approximately 340

m/s.m/s. It travels through other media as well, often much It travels through other media as well, often much

faster than that!faster than that! Sound waves are started by vibrations of some other Sound waves are started by vibrations of some other

material, which starts the air moving.material, which starts the air moving.

Animation courtesy of Dr. Dan Russell, Kettering University

Hearing SoundsHearing Sounds We hear a sound as “high” or “low” depending on its We hear a sound as “high” or “low” depending on its

frequency or wavelength. Sounds with short wavelengths frequency or wavelength. Sounds with short wavelengths and high frequencies sound high-pitched to our ears, and and high frequencies sound high-pitched to our ears, and sounds with long wavelengths and low frequencies sound sounds with long wavelengths and low frequencies sound low-pitched. The range of human hearing is from about 20 low-pitched. The range of human hearing is from about 20 Hz to about 20,000 Hz.Hz to about 20,000 Hz.

The amplitude of a sound’s vibration is interpreted as its The amplitude of a sound’s vibration is interpreted as its loudness. We measure the loudness (also called sound loudness. We measure the loudness (also called sound intensity) on the decibel scale, which is logarithmic.intensity) on the decibel scale, which is logarithmic.

© Tom Henderson, 1996-2004

Sound WavesSound waves are longitudinal waves, similar to the waves on a Slinky:

Here, the wave is a series of compressions and stretches.

Sound Waves

In a sound wave, the density and pressure of the air (or other medium carrying the sound) are the quantities that oscillate.

Sound Waves

The speed of sound is different in different materials; in general, the denser the material, the faster sound travels through it.

Sounds with frequencies greater than Sounds with frequencies greater than 20,000 Hz are called ultrasonic; sounds 20,000 Hz are called ultrasonic; sounds with frequencies less than 20 Hz are with frequencies less than 20 Hz are called infrasonic.called infrasonic.

Ultrasonic waves are familiar from medical Ultrasonic waves are familiar from medical applications; elephants and whales applications; elephants and whales communicate, in part, by infrasonic waves.communicate, in part, by infrasonic waves.

Pure SoundsPure Sounds

Sounds are longitudinal waves, but if we graph Sounds are longitudinal waves, but if we graph them right, we can make them look like them right, we can make them look like transverse waves.transverse waves.

When we graph the air motion involved in a pure When we graph the air motion involved in a pure sound tone versus position, we get what looks sound tone versus position, we get what looks like a sine or cosine function. like a sine or cosine function.

A tuning fork produces a relatively pure tone. So A tuning fork produces a relatively pure tone. So does a human whistle. does a human whistle.

Later in the period, we will sample various pure Later in the period, we will sample various pure sounds and see what they “look” like.sounds and see what they “look” like.

Graphing a Sound WaveGraphing a Sound Wave

Complex SoundsComplex Sounds

Because of the phenomena of “superposition” Because of the phenomena of “superposition” and “interference” real world waveforms may not and “interference” real world waveforms may not appear to be pure sine or cosine functions.appear to be pure sine or cosine functions.

That is because most real world sounds are That is because most real world sounds are composed of multiple frequencies.composed of multiple frequencies.

The human voice and most musical instruments The human voice and most musical instruments produce complex sounds.produce complex sounds.

Later in the period, we will sample complex Later in the period, we will sample complex sounds.sounds.

The OscilloscopeThe Oscilloscope

With the Oscilloscope we can view waveforms in the “time With the Oscilloscope we can view waveforms in the “time domain”. Pure tones will resemble sine or cosine functions, and domain”. Pure tones will resemble sine or cosine functions, and complex tones will show other repeating patterns that are formed complex tones will show other repeating patterns that are formed from multiple sine and cosine functions added together.from multiple sine and cosine functions added together.

The Fourier TransformThe Fourier TransformWe will also view waveforms in the “frequency We will also view waveforms in the “frequency domain”. A mathematical technique called the Fourier domain”. A mathematical technique called the Fourier Transform will separate a complex waveform into its Transform will separate a complex waveform into its component frequencies.component frequencies.

Doppler EffectDoppler EffectThe Doppler Effect is the The Doppler Effect is the raising or lowering of the raising or lowering of the perceived pitch of a sound perceived pitch of a sound based on the relative based on the relative motion of observer and motion of observer and source of the sound. source of the sound. When a car blowing its When a car blowing its horn races toward you, horn races toward you, the sound of its horn the sound of its horn appears higher in pitch, appears higher in pitch, since the wavelength has since the wavelength has been effectively shortened been effectively shortened by the motion of the car by the motion of the car relative to you. The relative to you. The opposite happens when opposite happens when the car races away.the car races away.

The Doppler effect is the change in pitch of a sound when the source and observer are moving with respect to each other.

When an observer moves toward a source, the wave speed appears to be higher, and the frequency appears to be higher as well.

The new frequency is:

If the observer were moving away from the source, only the sign of the observer’s speed would change:

To summarize:

The Doppler effect from a moving source can be analyzed similarly; now it is the wavelength that appears to change:

We find:

Here is a comparison of the Doppler shifts for a moving source and a moving observer. The two are similar for low speeds but then diverge. If the source moves faster then the speed of sound, a sonic boom is created.

Combining results gives us the case where both observer and source are moving:

Sample Problem: A bus approaching a bus Sample Problem: A bus approaching a bus stop at 24 m/s blows its horn. What the stop at 24 m/s blows its horn. What the perceived frequency that you hear, if the perceived frequency that you hear, if the horn’s true frequency is 150 Hz?horn’s true frequency is 150 Hz?

Doppler EffectDoppler Effect

Stationary source

Moving source

Supersonic sourceAnimations courtesy of Dr. Dan Russell, Kettering University

http://www.kettering.edu/~drussell/Demos/doppler/mach1.mpg

http://www.lon-capa.org/~mmp/applist/doppler/d.htm

The Doppler effect has many practical applications: weather radar, speed radar, medical diagnostics, astronomical measurements.

At left, a Doppler radar shows the hook echo characteristic of tornado formation. At right, a medical technician is using a Doppler blood flow meter.

Principle of SuperpositionPrinciple of Superposition

When two or more waves pass a particular When two or more waves pass a particular point in a medium simultaneously, the point in a medium simultaneously, the resulting displacement at that point in the resulting displacement at that point in the medium is the sum of the displacements medium is the sum of the displacements due to each individual wave.due to each individual wave.

The wavesThe waves interfereinterfere with each other. with each other.

Types of interference.Types of interference.

If the waves are “in phase”, that is crests If the waves are “in phase”, that is crests and troughs are aligned, the amplitude is and troughs are aligned, the amplitude is increased. This is called increased. This is called constructive constructive interferenceinterference..

If the waves are “out of phase”, that is If the waves are “out of phase”, that is crests and troughs are completely crests and troughs are completely misaligned, the amplitude is decreased misaligned, the amplitude is decreased and can even be zero. This is called and can even be zero. This is called destructive interferencedestructive interference..

Constructive InterferenceConstructive Interferencecrests aligned with crest

waves are “in phase”

Constructive InterferenceConstructive Interference

Destructive InterferenceDestructive Interferencecrests aligned with troughs

waves are “out of phase”

Destructive InterferenceDestructive Interference

Sample Problem: Draw the waveform from Sample Problem: Draw the waveform from its two components.its two components.

Sample Problem: Draw the waveform from Sample Problem: Draw the waveform from its two components.its two components.

Two-dimensional waves exhibit interference as well. This is an example of an interference pattern.b

Here is another example of an interference pattern, this one from two sources. If the sources are in phase, points where the distance to the sources differs by an equal number of wavelengths will interfere constructively; in between the interference will be destructive.

Standing WaveStanding Wave

A standing wave is a wave which is A standing wave is a wave which is reflected back and forth between fixed reflected back and forth between fixed ends (of a string or pipe, for example).ends (of a string or pipe, for example).

Reflection may be fixed or open-ended.Reflection may be fixed or open-ended. Superposition of the wave upon itself Superposition of the wave upon itself

results in a pattern of constructive and results in a pattern of constructive and destructive interference and an enhanced destructive interference and an enhanced wavewave

A standing wave is fixed in location, but oscillates with time. These waves are found on strings with both ends fixed, such as in a musical instrument, and also in vibrating columns of air.

The fundamental, or lowest, frequency on a fixed string has a wavelength twice the length of the string. Higher frequencies are called harmonics.

There must be an integral number of half-wavelengths on the string; this means that only certain frequencies are possible.

Points on the string which never move are called nodes; those which have the maximum movement are called antinodes.

In a piano, the strings vary in both length and density. This gives the sound box of a grand piano its characteristic shape.

Once the length and material of the string is decided, individual strings may be tuned to the exact desired frequencies by changing the tension.

Musical instruments are usually designed so that the variation in tension between the different strings is small; this helps prevent warping and other damage.

Standing waves can also be excited in columns of air, such as soda bottles, woodwind instruments, or organ pipes.

As indicated in the drawing, one end is a node (N), and the other is an antinode (A).

Fixed-end standing wavesFixed-end standing waves(violin string)(violin string)

1st harmonic

2nd harmonic

3rd harmonic

http://id.mind.net/~zona/mstm/physics/waves/standingWaves/standingWaves1/StandingWaves1.html

Animation available at:

Fixed-end standing wavesFixed-end standing waves(violin string)(violin string)

FundamentalFirst harmonic = 2L

First OvertoneSecond harmonic = L

Second OvertoneThird harmonic = 2L/3

L

If the tube is open at both ends, both ends are antinodes, and the sequence of harmonics is the same as that on a string.

Open-end standing wavesOpen-end standing waves(organ pipes)(organ pipes)

FundamentalFirst harmonic = 2L

First OvertoneSecond harmonic = L

2nd OvertoneThird harmonic = 2L/3

L

In this case, the fundamental wavelength is four times the length of the pipe, and only odd-numbered harmonics appear.

Mixed standing wavesMixed standing waves(some organ pipes)(some organ pipes)

First harmonic = 4L

Second harmonic = (4/3)L

Third harmonic = (4/5)L

L

Sample ProblemSample Problem How long do you need to make an organ pipe that produces a How long do you need to make an organ pipe that produces a

fundamental frequency of middle C (256 Hz)? The speed of the sound fundamental frequency of middle C (256 Hz)? The speed of the sound in air is 340 m/s.in air is 340 m/s.

A) Draw the standing wave for the first harmonicA) Draw the standing wave for the first harmonic

B) Calculate the pipe length.B) Calculate the pipe length.

C) What is the wavelength and frequency of the 2C) What is the wavelength and frequency of the 2ndnd harmonic? harmonic? Draw the standing waveDraw the standing wave

Sample ProblemSample Problem How long do you need to make an organ pipe whose fundamental How long do you need to make an organ pipe whose fundamental

frequency is a middle C (256 Hz)? The pipe is closed on one end, frequency is a middle C (256 Hz)? The pipe is closed on one end, and the speed of sound in air is 340 m/s.and the speed of sound in air is 340 m/s.

A) Draw the situation.A) Draw the situation.

B) Calculate the pipe length.B) Calculate the pipe length.

C) What is the wavelength and frequency of the 2C) What is the wavelength and frequency of the 2ndnd harmonic? harmonic?

ResonanceResonance

Resonance occurs when a vibration from Resonance occurs when a vibration from one oscillator occurs at a natural one oscillator occurs at a natural frequency for another oscillator.frequency for another oscillator.

The first oscillator will cause the second to The first oscillator will cause the second to vibrate.vibrate.

Demonstration.Demonstration.

BeatsBeats

““Beats is the word physicists use to Beats is the word physicists use to describe the characteristic loud-soft describe the characteristic loud-soft pattern that characterizes two nearly (but pattern that characterizes two nearly (but not exactly) matched frequencies.not exactly) matched frequencies.

Musicians call this “being out of tune”.Musicians call this “being out of tune”. Let’s hear (and see) a demo of this Let’s hear (and see) a demo of this

phenomenon.phenomenon.

Beats are an interference pattern in time, rather than in space. If two sounds are very close in frequency, their sum also has a periodic time dependence, although with a much lower frequency.

What word best describes this to What word best describes this to physicists?physicists?

Amplitude

Answer: beats

What word best describes this to What word best describes this to musicians?musicians?

Amplitude

Answer: bad intonation(being out of tune)

DiffractionDiffraction

Diffraction is defined as the bending of a Diffraction is defined as the bending of a wave around a barrier.wave around a barrier.

Diffraction of waves combined with Diffraction of waves combined with interference of the diffracted waves interference of the diffracted waves causes “diffraction patterns”.causes “diffraction patterns”.

Let’s look at the diffraction phenomenon Let’s look at the diffraction phenomenon using a “ripple tank”.using a “ripple tank”. http://www.falstad.com/ripple/

Double-slit or multi-slit diffractionDouble-slit or multi-slit diffraction

n=0

n=1

n=2

n=1

Double slit diffractionDouble slit diffraction

nn = d sin = d sin n: bright band number (n = 0 for central)n: bright band number (n = 0 for central) : wavelength (m): wavelength (m) d: space between slits (m)d: space between slits (m) : angle defined by central band, slit, and : angle defined by central band, slit, and

band nband n This also works for diffraction gratings This also works for diffraction gratings

consisting of many, many slits that allow consisting of many, many slits that allow the light to pass through. Each slit acts as the light to pass through. Each slit acts as a separate light source.a separate light source.

Single slit diffractionSingle slit diffraction

nn = s sin = s sin n: dark band number n: dark band number : wavelength (m): wavelength (m) s: slit width (m)s: slit width (m) : angle defined by central band, slit, and : angle defined by central band, slit, and

dark band dark band

Sample ProblemSample Problem Light of wavelength 360 nm is passed through a diffraction Light of wavelength 360 nm is passed through a diffraction

grating that has 10,000 slits per cm. If the screen is 2.0 m from grating that has 10,000 slits per cm. If the screen is 2.0 m from the grating, how far from the central bright band is the first order the grating, how far from the central bright band is the first order bright band?bright band?

Sample ProblemSample Problem Light of wavelength 560 nm is passed through two slits. It is Light of wavelength 560 nm is passed through two slits. It is

found that, on a screen 1.0 m from the slits, a bright spot is found that, on a screen 1.0 m from the slits, a bright spot is formed at x = 0, and another is formed at x = 0.03 m? What is formed at x = 0, and another is formed at x = 0.03 m? What is the spacing between the slits?the spacing between the slits?

Sample ProblemSample Problem Light is passed through a single slit of width 2.1 x 10Light is passed through a single slit of width 2.1 x 10-6-6 m. How far m. How far

from the central bright band do the first and second order dark from the central bright band do the first and second order dark bands appear if the screen is 3.0 meters away from the slit?bands appear if the screen is 3.0 meters away from the slit?