chapters 16 – 18 waves. types of waves mechanical – governed by newton’s laws and exist in a...

Chapters 16 – 18

Waves

Types of waves

• Mechanical – governed by Newton’s laws and exist in a material medium (water, air, rock, ect.)

• Electromagnetic – governed by electricity and magnetism equations, may exist without any medium

• Matter – governed by quantum mechanical equations

Types of waves

Depending on the direction of the displacement relative to the direction of propagation, we can define wave motion as:

• Transverse – if the direction of displacement is perpendicular to the direction of propagation

• Longitudinal – if the direction of displacement is parallel to the direction of propagation

The linear wave equation

• Let us consider transverse waves propagating without change in shape and with a constant wave

velocity v

• We will describe waves via vertical displacement

y(x,t)

• For an observer moving with the wave

the wave shape doesn’t depend on time y(x’) = f(x’)


For an observer at rest:

• the wave shape depends on time y(x,t)

• the reference frame linked to the wave is moving

with the velocity of the wave v

vtxx ' vtxx '

)()'( vtxfxf )(),( vtxftxy


• We considered a wave propagating with velocity v

• For a medium with isotropic (symmetric) properties, the wave equation should have a symmetric solution

for a wave propagating with velocity –v

)(),(1 vtxftxy

))((),(2 tvxftxy

)( vtxf


• Therefore, solutions of the wave equation should have a form

• Considering partial derivatives

)(),( vtxftxy

x

vtxf

x

txy

)(),(

x

vtx

vtx

vtxf

)(

)(

)()(' vtxf

t

vtxf

t

txy

)(),(

t

vtx

vtx

vtxf

)(

)(

)()()(' vvtxf




)(),( vtxftxy

x

vtxf

xx

txy )(),(2

2

)(' vtxfx

)('' vtxf

t

vtxf

tt

txy )(),(2

2

)()(' vvtxft

2)('' vvtxf




)(),( vtxftxy

)(''),(

2

2

vtxfx

txy

22

2

)(''),(

vvtxft

txy

2

22 ),(

x

txyv

2

22

2

2 ),(),(

x

txyv

t

txy


• The linear wave equation (not the only one having solutions of the form y(x,t) = f(x ± vt)):

• It works for longitudinal waves as well

• v is a constant and is determined by the properties of the medium. E.g., for a stretched string with linear

density μ = m/l under tension T

T

v

2

22

2

2 ),(),(

x

txyv

t

txy

Superposition of waves

• Let us consider two different solutions of the linear wave equation

• Superposition principle – a sum of two solutions of the linear wave equation is a solution of the linear wave equation

21

22

21

2

x

yv

t

y

22

22

22

2

x

yv

t

y

22

22

21

22

22

2

21

2

x

yv

x

yv

t

y

t

y

221

22

221

2 )()(

x

yyv

t

yy

+

Superposition of waves

• Overlapping solutions of the linear wave equation algebraically add to produce a resultant (net) wave

• Overlapping solutions of the linear wave equation do not in any way alter the travel of each other

Chapter 16Problem 44

(a) Show that the function y(x, t) = x2 + v2t2 is a solution to the wave equation. (b) Show that the function in part (a) can be written as f(x + vt) + g(x – vt) and determine the functional forms for f and g. (c) Repeat parts (a) and (b) for the function y(x, t) = sin (x) cos (vt).

Reflection of waves at boundaries

• Within media with boundaries, solutions to the wave equation should satisfy boundary conditions. As a results, waves may be reflected from boundaries

• Hard reflection – a fixed zero value of deformation at the boundary – a reflected wave is inverted

• Soft reflection – a free value of deformation at the boundary – a reflected wave is not inverted

Sinusoidal waves

• One of the most characteristic solutions of the linear wave equation is a sinusoidal wave:

• A – amplitude, φ – phase constant

)2/)(cos(

))(sin()(

vtxkA

vtxkAvtxy

Wavelength

• “Freezing” the solution at t = 0 we obtain a

sinusoidal function of x:

• Wavelength λ – smallest distance (parallel to the direction of wave’s travel) between repetitions of the wave shape

))(cos(),( vtxkAtxy

)cos()0,( kxAxy

Wave number

• On the other hand:

• Angular wave number: k = 2π / λ

)cos()0,( kxAxy ))(cos( xkA

)cos( kkxA

)2cos()cos( kxkx /2k

Angular frequency

• Considering motion of the point at x = 0 we observe a simple harmonic motion (oscillation) :

• For simple harmonic motion (Chapter 15):

• Angular frequency ω

))(cos(),( vtxkAtxy

)cos(),0( kvtAty )cos( kvtA

)cos()( tAty

/2 vkv

Frequency, period

• Definitions of frequency and period are the same as for the case of rotational motion or simple harmonic motion:

• Therefore, for the wave velocity

2//1 Tf /2T

fTkv //

)cos(),( tkxAtxy


A transverse sinusoidal wave on a string has a period T = 25.0 ms and travels in the negative x direction with a speed of 30.0 m/s. At t = 0, an element of the string at x = 0 has a transverse position of 2.00 cm and is traveling downward with a speed of 2.00 m/s. (a) What is the amplitude of the wave? (b) What is the initial phase angle? (c) What is the maximum transverse speed of an element of the string? (d) Write the wave function for the wave.

Interference of waves

• Interference – a phenomenon of combining waves, which follows from the superposition principle

• Considering two sinusoidal waves of the same amplitude, wavelength, and direction of propagation

• The resultant wave:

)cos(),(2 tkxAtxy)cos(),(1 tkxAtxy

),(),(),( 21 txytxytxy

)cos()cos( tkxAtkxA

2

cos2

cos2coscos

)2/cos()2/cos(2 tkxA


• If φ = 0 (Fully constructive)

• If φ = π (Fully destructive)

• If φ = 2π/3 (Intermediate)

)2/cos()2/cos(2),( tkxAtxy

)cos(2),( tkxAtxy

0),( txy

)3/cos(

)3/cos(2),(

tkx

Atxy

)3/cos( tkxA


• Considering two sinusoidal waves of the same amplitude, wavelength, but running in opposite directions

• The resultant wave:

)cos(),(2 tkxAtxy)cos(),(1 tkxAtxy

),(),(),( 21 txytxytxy

)cos()cos( tkxAtkxA

2

cos2

cos2coscos

)2/cos()2/cos(2 tkxA


• If two sinusoidal waves of the same amplitude and wavelength travel in opposite directions, their interference with each other produces a standing wave

)sin()sin(2),( kxtAtxy

...2,1,0

)21(

n

nkx

22

1

nx

Antinodes

1|sin| kx

tAy sin2

...2,1,0

n

nkx

0sin kx

0y

2

nx

Nodes


A standing wave pattern is observed in a thin wire with a length of 3.00 m. The wave function is y = (0.002 m) sin (πx) cos (100πt), where x is in meters and t is in seconds. (a) How many loops does this pattern exhibit? (b) What is the fundamental frequency of vibration of the wire? (c) If the original frequency is held constant and the tension in the wire is increased by a factor of 9, how many loops are present in the new pattern?

Standing waves and resonance

• For a medium with fixed boundaries (hard reflection) standing waves can be generated because of the reflection from both boundaries: resonance

• Depending on the number of antinodes, different resonances can occur

Standing waves and resonance

• Resonance wavelengths

• Resonance frequencies

L2

2

2L

3

2L

...3,2,1,2

nn

L

v

f ...3,2,1,2

nL

nv

Harmonic series

• Harmonic series – collection of all possible modes - resonant oscillations (n – harmonic number)

• First harmonic (fundamental mode):

...3,2,1,2

nL

vnfn

L

vf

21

More about standing waves

• Longitudinal standing waves can also be produced

• Standing waves can be produced in 2 and 3 dimensions as well

Rate of energy transmission

• As the wave travels it transports energy, even though the particles of the medium don’t propagate with the wave

• The average power of energy transmission for the sinusoidal solution of the wave equation

• Exact expression depends on the medium or the system through which the wave is propagating

vAPavg22

Sound waves

• Sound – longitudinal waves in a substance (air, water, metal, etc.) with frequencies detectable by human ears (between ~ 20 Hz and ~ 20 KHz)

• Ultrasound – longitudinal waves in a substance (air, water, metal, etc.) with frequencies higher than detectable by human ears (> 20 KHz)

• Infrasound – longitudinal waves in a substance (air, water, metal, etc.) with frequencies lower than detectable by human ears (< 20 Hz)

Speed of sound

• Speed of sound:

ρ – density of a medium, B – bulk modulus of a medium

• Traveling sound waves

B

v

V

VBP

)cos(

))(cos(),(

tkxs

vtxkstxs

m

m


As a certain sound wave travels through the air, it produces pressure variations (above and below atmospheric pressure) given by ΔP = 1.27 sin (πx – 340πt) in SI units. Find (a) the amplitude of the pressure variations, (b) the frequency, (c) the wavelength in air, and (d) the speed of the sound wave.

Intensity of sound

• Intensity of sound – average rate of sound energy transmission per unit area

• For a sinusoidal traveling wave:

• Decibel scale

β – sound level; I0 = 10-12 W/m2 – lower limit of human

hearing

AI

P

22

2

1 mvsI

0

log)10(I

IdB

Sources of musical sound

• Music produced by musical instruments is a combination of sound waves with frequencies corresponding to a superposition of harmonics (resonances) of those musical instruments

• In a musical instrument, energy of resonant oscillations is transferred to a resonator of a fixed or adjustable geometry

Open pipe resonance

• In an open pipe soft reflection of the waves at the ends of the pipe (less effective than form the closed ends) produces standing waves

• Fundamental mode (first harmonic): n = 1

• Higher harmonics:

...3,2,12

,2

nL

vnf

n

L

Organ pipes

• Organ pipes are open on one end and closed on the other

• For such pipes the resonance condition is modified:

L

vnf

n

L

nnL

4,

4

...5,3,1;4

Musical instruments

• The size of the musical instrument reflects the range of frequencies over which the instrument is designed to function

• Smaller size implies higher frequencies, larger size implies lower frequencies

Musical instruments

• Resonances in musical instruments are not necessarily 1D, and often involve different parts of the instrument

• Guitar resonances (exaggerated) at low frequencies:

Musical instruments


• Guitar resonances at medium frequencies:

Musical instruments


• Guitar resonances at high frequencies:

Beats

• Beats – interference of two waves with close frequencies

tss m 11 cos

tss m 22 cos+ tstssss mm 2121 coscos

ttsm 2cos

2cos2 2121

Sound from a point source

• Point source – source with size negligible compared to the wavelength

• Point sources produce spherical waves

• Wavefronts – surfaces over which oscillations have the same value

• Rays – lines perpendicular to wavefronts indicating direction of travel of wavefronts

Interference of sound waves

• Far from the point source wavefronts can be approximated as planes – planar waves

• Phase difference and path length difference are related:

• Fully constructive interference

• Fully destructive interference

2212 LLL

,...2,1,0L

,...2

5,

2

3,

2

1

L

Variation of intensity with distance

• A single point emits sound isotropically – with equal intensity in all directions (mechanical energy of the sound wave is conserved)

• All the energy emitted by the source must pass through the surface of imaginary sphere of radius r

• Sound intensity

(inverse square law)

A

PI

24 r

Ps


Two small speakers emit sound waves of different frequencies equally in all directions. Speaker A has an output of 1.00 mW, and speaker B has an output of 1.50 mW. Determine the sound level (in decibels) at point C assuming (a) only speaker A emits sound, (b) only speaker B emits sound, and (c) both speakers emit sound.

Doppler effect

• Doppler effect – change in the frequency due to relative motion of a source and an observer (detector)

Andreas Christian Johann Doppler

(1803 -1853)

Doppler effect

• For a moving detector (ear) and a stationary source

• In the source (stationary) reference frame:Speed of detector is –vD

Speed of sound waves is v

• In the detector (moving) reference frame:Speed of detector is 0

Speed of sound waves is v + vD

fv v

f

'

'v

f

Dvv

f

v

v

vvf D

Doppler effect

• For a moving detector (ear) and a stationary source

• If the detector is moving away from the source:

• For both cases:

v

vvff D

'

v

vvff D

'

v

vvff D

'

Doppler effect

• For a stationary detector (ear) and a moving source

• In the detector (stationary) reference frame:

• In the moving (source) frame:

*'

v

f

*Svv

f

f

vv S*

Svv

vf

Doppler effect

• For a stationary detector and a moving source

• If the source is moving away from the detector:

• For both cases:

Svv

vff

'

Svv

vff

'

Svv

vff

'

Doppler effect

• For a moving detector and a moving source

• Doppler radar:

S

D

vv

vvff

'


A tuning fork vibrating at 512 Hz falls from rest and accelerates at 9.80 m/s2. How far below the point of release is the tuning fork when waves of frequency 485 Hz reach the release point? Take the speed of sound in air to be 340 m/s.

Supersonic speeds

• For a source moving faster than the speed of soundthe wavefronts form the Mach cone

• Mach number

Ernst Mach(1838-1916)

v

vs

vt

tvssin

1

Supersonic speeds

• The Mach cone produces a sonic boom

Questions?

Answers to the even-numbered problems

Chapter 16

Problem 80.800 m/s


Chapter 17

Problem 16(a) 5.00 × 10-17 W(b) 5.00 × 10-5 W


Chapter 17

Problem 4046.4°


Chapter 17

Problem 54The gap between bat and insect is

closing at 1.69 m/s.

chapters 16 – 18 waves. types of waves mechanical – governed by newton’s laws and exist in a...

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