Waves and Sound - LANDOSlandos.stri.kmutnb.ac.th/.../uploads/2014/11/L9-Waves-and-Sound.pdf · Waves and Sound . Outline ... provided the end of the slinky is moved in simple harmonic

Lecture 10

Waves and Sound

Outline • What is wave?

• Wave equation

– Linear wave equation

– Wave function (a type of field) as a solution to the wave equation

• Transverse waves: wave in string

• Longitudinal waves: sound wave, pressure wave – Dispersion relation (wave speed, wave number k, angular frequency w)

• Traveling Pulse

• Sinusoidal wave

• Superposition/interference and Standing waves

• Wave in musical instruments: normal modes, harmonics

• Beats

• Sound: sound level (dB scale), intensity, pitch

• Doppler effect and shock wave

16.1 The Nature of Waves

1. A wave is a traveling disturbance.

2. A wave carries energy from place to place.


Longitudinal Wave


Transverse Wave


Water waves are partially transverse and partially longitudinal.

Traveling Pulse

What could be waves?

Any functions y(x,t) that satisfies the linear wave equation

We call the solutions to this differential equation wave functions.

Typically, any function 𝑓, 𝑔 that can be written in the form 𝒇(𝒙 − 𝒗𝒕) or 𝒈(𝒙 + 𝒗𝒕)

(they are traveling waves) are solutions. Superposition of these

functions are also solutions (could be an interference or standing

waves)

16.2 Periodic Waves

Periodic waves consist of cycles or patterns that are produced over and

over again by the source.

In the figures, every segment of the slinky vibrates in simple harmonic

motion, provided the end of the slinky is moved in simple harmonic

motion.

16.2 Periodic Waves

In the drawing, one cycle is shaded in color.

The amplitude (A) is the maximum excursion/displacement of a particle of the medium from

the undisturbed (equilibrium) position.

The wavelength (𝝀) is the horizontal length of one cycle of the wave.(ระยะทางการซ ้ าในสเปซ)

The period (𝑻) is the time required for one complete cycle. (ระยะซ ้ าในเวลา)

The frequency (f) is related to the period and has units of Hz, or s-1.

Tf

1

fT

v

The wave moves a distance of one cycle (one wavelength) in a period T wave speed

16.2 Periodic Waves

Example 1 The Wavelengths of Radio Waves

AM and FM radio waves are transverse waves consisting of electric and

magnetic field disturbances traveling at a speed of 3.00x108m/s. A station

broadcasts AM radio waves whose frequency is 1230x103Hz and an FM

radio wave whose frequency is 91.9x106Hz. Find the distance between

adjacent crests in each wave.

fT

v

f

v

16.2 Periodic Waves

AM m 244Hz101230

sm1000.33

8

f

v

FM m 26.3Hz1091.9

sm1000.36

8

f

v

16.3 The Speed of a Wave on a String

The speed at which the wave moves to the right depends on how quickly

one particle of the string is accelerated upward in response to the net

pulling force.

Tension in the taut string

Linear mass density of the string

𝑣𝑤𝑎𝑣𝑒 =𝑇

(𝑚𝐿

)=

𝑇

𝜇

Elastic/restoring property

Inertial property

Answer: B

Answer: B is not always true.

Not all waves are sinusoidal. Sinusoidal waves are wave with single frequency11

16.3 The Speed of a Wave on a String

Conceptual Example 3 Wave Speed Versus Particle Speed

Is the speed of a transverse wave on a string the same as the speed at

which a particle on the string moves?

=

=𝜔

𝑘

16.4 The Mathematical Description of a Wave: sinusoidal waves

Consider the displacement y at time t of a

particle located at a point P

เลขคล่ืน(เชิงมุม) ความถ่ีเชิงมุม

16.5 The Nature of Sound Waves

LONGITUDINAL SOUND WAVES


The distance between adjacent condensations is equal to the

wavelength of the sound wave.

Speed of an element of the medium is NOT necessarily equal to the traveling speed of the wave


Individual air molecules are not carried along with the wave.


THE FREQUENCY OF A SOUND WAVE

The frequency is the number of cycles

per second.

A sound with a single frequency is called

a pure tone.

The brain interprets the frequency in terms

of the subjective quality called pitch.


THE PRESSURE AMPLITUDE OF A SOUND WAVE

Loudness is an attribute of

a sound that depends primarily

on the pressure amplitude

of the wave.

16.6 The Speed of Sound

Sound travels through gases,

liquids, and solids at considerably

different speeds.


In a gas, it is only when molecules collide that the condensations and

rerefactions of a sound wave can move from place to place.

Ideal Gas

m

kTv

m

kTvrms

3

KJ1038.1 23k

5

7or

3

5

Or we just use simple relation 𝑣 ∝ 𝑇 where T is in Kelvin

together with a reference sound speed at 0°𝐶 , 𝑣 𝑇 = 273 𝐾 = 331 𝑚/𝑠

𝑣 𝑇

331 𝑚/𝑠=

𝑇

273 𝐾

(use T in Kelvin)!!!

Or in Celsius,

v(T in℃) = (331 𝑚/𝑠) 1 +𝑇(𝑖𝑛 ℃ )

273


LIQUIDS

Bv

Bulk modulus

mass density

SOLID BARS

Yv

Young’s modulus

mass density

ความบีบอดัไดใ้นของเหลว

แรงคงตวัท่ีเป็นผลของความเคน้บิด

16.7 Sound Intensity

Sound waves carry energy that can be used to do work.

The amount of energy transported per second is called the

power of the wave.

The sound intensity is defined as the power that passes perpendicularly

through a surface divided by the area of that surface.

A

PI


Example 6 Sound Intensities

12x10-5W of sound power passed through the surfaces labeled 1 and 2. The

areas of these surfaces are 4.0m2 and 12m2. Determine the sound intensity

at each surface.


25

2

5

1

1 mW100.34.0m

W1012

A

PI

25

2

5

2

2 mW100.112m

W1012

A

PI


For a 1000 Hz tone, the smallest sound intensity that the human ear

can detect is about 1x10-12W/m2. This intensity is called the threshold

of hearing.

On the other extreme, continuous exposure to intensities greater than

1W/m2 can be painful.

If the source emits sound uniformly in all directions, the intensity depends

on the distance from the source in a simple way.

A Point Source

•A point source will emit sound

waves equally in all directions.

– This can result in a spherical

wave in 3D

•This can be represented as a series

of circular arcs concentric with the

source.

•Each surface of constant phase is a

wave front.

•The radial distance between

adjacent wave fronts that have the

same phase is the wavelength λ of

the wave.

•Radial lines pointing outward from

the source, representing the direction

of propagation, are called rays. Section 17.3


24 r

PI

power of sound source

area of sphere


Conceptual Example 8 Reflected Sound and Sound Intensity

Suppose the person singing in the shower produces a sound power P.

Sound reflects from the surrounding shower stall. At a distance r in front

of the person, does the equation for the intensity of sound emitted uniformly

in all directions underestimate, overestimate, or give the correct sound

intensity?

24 r

PI

16.8 Decibels

The decibel (dB) is a measurement unit used when comparing two sound

intensities.

Because of the way in which the human hearing mechanism responds to

intensity, it is appropriate to use a logarithmic scale called the intensity

level:

oI

IlogdB 10

212 mW1000.1 oI

Note that log(1)=0, so when the intensity

of the sound is equal to the threshold of

hearing, the intensity level is zero.

16.8 Decibels

oI

IlogdB 10

212 mW1000.1 oI

16.8 Decibels

Example 9 Comparing Sound Intensities

Audio system 1 produces a sound intensity level of 90.0 dB, and system

2 produces an intensity level of 93.0 dB. Determine the ratio of intensities.

oI

IlogdB 10

16.8 Decibels

oI

IlogdB 10

oI

I11 logdB 10

oI

I22 logdB 10

1

2

1

21212 logdB 10logdB 10logdB 10logdB 10

I

I

II

II

I

I

I

I

o

o

oo

1

2logdB 10dB 0.3I

I

0.210 30.0

1

2 I

I

1

2log0.30I

I

16.9 The Doppler Effect

The Doppler effect is the

change in frequency or pitch

of the sound detected by

an observer because the sound

source and the observer have

different velocities with respect

to the medium of sound

propagation.

Observed frequency increases Observed frequency decreases

Frequency unchanged


MOVING SOURCE

Tvs

ssss

ofvfv

v

Tv

vvf

vvff

s

so1

1

vvff

s

so1

1source moving

toward a stationary

observer

source moving

away from a stationary

observer

vvff

s

so1

1


MOVING OBSERVER

v

vf

f

vf

vff

os

s

os

oso

1

1

v

vff o

so 1

v

vff o

so 1

Observer moving

towards stationary

source

Observer moving

away from

stationary source


v

vv

v

ffs

o

so

1

1

GENERAL CASE

Numerator: plus sign applies

when observer moves towards

the source

Denominator: minus sign applies

when source moves towards

the observer


Example 10 The Sound of a Passing Train

A high-speed train is traveling at a speed of 44.7 m/s when the engineer

sounds the 415-Hz warning horn. The speed of sound is 343 m/s. What

are the frequency and wavelength of the sound, as perceived by a person

standing at the crossing, when the train is (a) approaching and (b) leaving

the crossing?

vvff

s

so1

1

vvff

s

so1

1

Hz 4771

1Hz 415

sm343

sm7.44

of

approaching leaving

Hz 3671

1Hz 415

sm343

sm7.44

of

Shock wave

Mach angle is the half-angle of the apex of the cone

16.10 Applications of Sound in Medicine

By scanning ultrasonic waves across the body and detecting the echoes

from various locations, it is possible to obtain an image.


Ultrasonic sound waves cause

the tip of the probe to vibrate at

23 kHz and shatter sections of

the tumor that it touches.


When the sound is reflected

from the red blood cells, its

frequency is changed in a

kind of Doppler effect because

the cells are moving.

16.11 The Sensitivity of the Human Ear

Properties of wave

• Mathematical description of waves

• Waves vs particles

• Propagation through a boundary: transmission and

reflection

• Superposition and interference

– Standing waves

• Boundary condition, quantization, normal modes, harmonics (in

musical instrument)

– Beats

• 4 properties: Interference, refraction, diffraction

Reflection of a Wave, Fixed End

•When the pulse reaches the support,

the pulse moves back along the string

in the opposite direction.

•This is the reflection of the pulse.

•The pulse is inverted.

– Due to Newton’s third law

• When the pulse reaches the fixed end of the string, the string produces an upward force on the support.

• The support must exert an equal-magnitude and oppositely directed reaction force on the string.

Section 16.4

Propagation of wave: Transmission and Reflection

Reflection of a Wave, Free End

•With a free end, the string is free to

move vertically.

•The pulse is reflected.

•The pulse is not inverted.

•The reflected pulse has the same

amplitude as the initial pulse.

Section 16.4

Transmission of a Wave

•When the boundary is intermediate between the last two

extremes.

– Part of the energy in the incident pulse is reflected and part

undergoes transmission. • Some energy passes through the boundary.

Section 16.4

Transmission of a Wave, 2

•Assume a light string is attached to a heavier string.

•The pulse travels through the light string and reaches the boundary.

•The part of the pulse that is reflected is inverted.

•The reflected pulse has a smaller amplitude.

•Assume a heavier string is attached

to a light string.

•Part of the pulse is reflected and part

is transmitted.

•The reflected part is not inverted.

Transmission of a Wave, 4

•Conservation of energy governs the pulse

– When a pulse is broken up into reflected and transmitted parts at a boundary, the sum of the energies of the two pulses must equal the energy of the original pulse.

•When a wave or pulse travels from medium A to medium B and vA > vB, it is inverted upon reflection

– B is denser than A.

•When a wave or pulse travels from medium A to medium B and vA < vB, it is not inverted upon reflection.

– A is denser than B.

Section 16.4

Energy in Waves in a String

•Waves transport energy when they propagate through a medium.

•We can model each element of a string as a simple harmonic oscillator with oscillation in

the y-direction. Every element has the same total energy and have a mass of

dm= mdx.

•The kinetic energy associated with the up and down motion of the element is

dK = ½ (dm) vy2.

so that kinetic energy of an element of the string is dK = ½ (m dx) vy2.

•Integrating over all the elements, the total kinetic energy in one

wavelength is K = ¼m w2A 2

•The total potential energy in one wavelength is U = ¼m w2A 2

•This gives a total energy of

E = K + U = ½m w2A 2

Section 16.5

Power Associated with a Wave

•The power is the rate at which the energy is being transferred:

•The power transfer by a sinusoidal wave on a string is proportional to the

– Square of the frequency

– Square of the amplitude

– Wave speed

The rate of energy transfer in any sinusoidal wave is proportional to the square of the angular frequency and to the square of the amplitude.

AE

P A vT T

2 2

2 2

1122

mw mw

Section 16.5

Waves vs. Particles

•Waves are very different from particles.

Particles have zero size. Waves have a characteristic size – their

wavelength.

Multiple particles must exist at

different locations.

Multiple waves can combine at one point in

the same medium – they can be present at

the same location.

Introduction

Quantization

•When waves are combined in systems with boundary conditions, only certain allowed frequencies can exist = (eigen)mode or standing wave

– We say the frequencies are quantized.

– Quantization is at the heart of quantum mechanics

•Quantization can be used to understand the behavior of the wide array of musical instruments that are based on strings and air columns.

Superposition principle:

•If two or more traveling waves are moving through a medium, the

resultant value of the wave function at any point is the algebraic sum of

the values of the wave functions of the individual waves.

•Waves can also combine when they have different frequencies. (same

location in space and time)

Introduction

Superposition and Interference

•Two traveling waves can pass through each other without

being destroyed or altered.

– A consequence of the superposition principle.

•The combination of separate waves in the same region of

space to produce a resultant wave is called interference. – The term interference has a very specific usage in physics.

– It means waves pass through each other.

Section 18.1

Constructive

interference

Destructive interference

17.2 Constructive and Destructive Interference of Sound Waves

Adding 2 sinusoidal waves (case I: same direction)

•Assume two waves are traveling in the same direction in a linear medium, with the same frequency, wavelength and amplitude.

•The waves differ only in phase : f = the phase difference

y1 = A sin (kx-wt)

y2 = A sin (kx-wt +f)

y = y1+y2 = 2A cos(f /2)sin (kx-wt +f /2)

The resultant wave function y(x,t) is also sinusoidal with the same frequency and wavelength as the original waves.

•The amplitude of the resultant wave is 2Acos(f / 2) .

•The phase of the resultant wave is f/2.

Adding 2 sinusoidal waves (case II: opposite direction)

Standing Waves

•Assume two waves with the same amplitude, frequency and wavelength, traveling in opposite directions in a medium.

•The waves combine in accordance with the waves in interference model.

• y1 = A sin (kx – wt) and

• y2 = A sin (kx + wt)

•They interfere according to the superposition principle.

Section 18.2

Standing Waves

•The resultant wave will be y = (2A sin kx) cos wt.

•This is the wave function of a standing wave. – There is no kx–wt term, and therefore it is not a traveling wave.

•In observing a standing wave, there is no sense of motion in the direction of propagation of either of the original waves.

Section 18.2

Standing Wave Example

•Note the stationary outline that results from the superposition of two identical waves traveling in opposite directions.

•The amplitude of the simple harmonic motion of a given element is 2A sin kx.

– This depends on the location x of the element in the medium.

•Each individual element vibrates at w

Section 18.2

17.5 Transverse Standing Waves

Transverse standing wave patters.

17.5 Transverse Standing Waves

,4,3,2,1 2

n

L

vnfnString fixed at both ends

Note on Amplitudes

•There are three types of amplitudes used in describing

waves.

– The amplitude of the individual waves, A

– The amplitude of the simple harmonic motion of the elements in

the medium,

• 2A sin kx

• A given element in the standing wave vibrates within the constraints

of the envelope function 2 A sin k x.

– The amplitude of the standing wave, 2A

Section 18.2

•A node occurs at a point of zero amplitude. – These correspond to positions of x where

•An antinode occurs at a point of maximum displacement, 2A.

– These correspond to positions of x where

0,1, 2, 3,2

nx n

1, 3, 5,4

nx n

Section 18.2

Features of Nodes and

Antinodes •The distance between adjacent antinodes is /2.

•The distance between adjacent nodes is /2.

•The distance between a node and an adjacent antinode is /4.

Section 18.2

Nodes and Antinodes, cont

•The diagrams above show standing-wave patterns produced at various times by two waves of equal amplitude traveling in opposite directions.

•In a standing wave, the elements of the medium alternate between the extremes shown in (a) and (c).

Section 18.2

Standing Waves in a String

•Consider a string of length L fixed at both ends

•Waves can travel both ways on the string. Standing waves are set up by a continuous superposition of waves incident on and reflected from the ends.

•There is a boundary condition on the waves.

–The ends of the strings must necessarily be nodes. They are fixed and therefore must have zero displacement.

Boundary conditions:

y(0) = 0, y(L) = 0

Section 18.3

Standing Waves in a String, 2

•The boundary condition results in the string having a set of natural patterns of oscillation, called normal modes.

– Each mode has a characteristic frequency.

• This situation in which only certain frequencies of oscillations are allowed is called quantization.

– The normal modes of oscillation for the string can be described by imposing the requirements that the ends be nodes and that the nodes and antinodes are separated by l/4.

•We identify an analysis model called waves under boundary conditions.

Section 18.3


•This is the first normal mode that is

consistent with the boundary

conditions: nodes at both ends, 1

antinode in the middle.

•longest wavelength mode:

½1 = L so 1 = 2L

•The section of the standing wave

between nodes is called a loop. In the

first normal mode, the string vibrates

in one loop.

Section 18.3


•Consecutive normal modes add a loop at each step.

– The section of the standing wave from one node to the next is called a loop.

•The second mode (c) corresponds to to = L.

•The third mode (d) corresponds to = 2L/3.

Section 18.3

Standing Waves in a String, Summary

•The wavelengths of the normal modes for a string of length L fixed at both ends are n = 2L / n, n = 1, 2, 3, …

– n is the nth normal mode of oscillation

– These are the possible modes for the string:

•The natural frequencies are

– Also called quantized frequencies

ƒ2 2

n

v n Tn

L L m

Section 18.3

Waves on a String, Harmonic Series

•The fundamental frequency corresponds to n = 1. – It is the lowest frequency, ƒ1

•The frequencies of the remaining natural modes are integer multiples of the fundamental frequency.

– ƒn = nƒ1

•Frequencies of normal modes that exhibit this relationship form a harmonic series.

•The normal modes are called harmonics.

Section 18.3

Standing Waves in a tube

Section 18.5

Notes About Musical Instruments

•As the temperature rises:

– Sounds produced by air columns become sharp

• Higher frequency

• Higher speed due to the higher temperature

– Sounds produced by strings become flat

• Lower frequency

• The strings expand due to the higher temperature.

• As the strings expand, their tension decreases.

•Musical instruments based on air columns are generally excited by resonance. –The air column is presented with a sound wave rich in many frequencies.

–The sound is provided by:

– A vibrating reed in woodwinds

– Vibrations of the player’s lips in brasses

– Blowing over the edge of the mouthpiece in a flute

Section 18.5

Example: resonance in Air Columns

•A tuning fork is placed near the

top of the tube.

•When L corresponds to a

resonance frequency of the pipe,

the sound is louder.

•The water acts as a closed end

of a tube.

•The wavelengths can be

calculated from the lengths where

resonance occurs.

Section 18.5

Example: Standing Waves in Rods

Section 18.6

If the rod is clamped at ¼ of the length

: 2nd normal mode is produced instead

Standing Waves in Membranes

•Two-dimensional oscillations may be set up in a flexible membrane stretched over a circular hoop.

•The resulting sound is not harmonic because the standing waves have frequencies that are not related by integer multiples.

– The sound may be more correctly described as noise instead of music.

•The fundamental frequency contains one nodal curve.

Beats and Beat Frequency •Beating is the periodic variation in amplitude at a given point due to the superposition

of two waves having slightly different frequencies.

•The number of amplitude maxima one hears per second is the beat frequency.

•It equals the difference between the frequencies of the two sources.

•The human ear can detect a beat frequency up to about 20 beats/sec.

17.4 Beats

The beat frequency is the difference between the two sound

frequencies.

17.7 Complex Sound Waves

Exercise

Answer : a) t = ¾ sec, b) x = 1 m

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Waves and Sound - LANDOSlandos.stri.kmutnb.ac.th/.../uploads/2014/11/L9-Waves-and-Sound.pdf · Waves and Sound . Outline ... provided the end of the slinky is moved in simple harmonic