Chapter 14
Wave Motion
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
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’)
The linear wave equation
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
x  x'vt
f ( x' )  f ( x  vt)
x'  x  vt
y ( x, t )  f ( x  vt)
The linear wave equation
• We considered a wave propagating with velocity v
y1 ( x, t )  f ( x  vt)
• For a medium with isotropic (symmetric) properties,
the wave equation should have a symmetric solution
for a wave propagating with velocity –v
y2 ( x, t )  f ( x  (v)t )
 f ( x  vt)
The linear wave equation
• Therefore, solutions of the wave equation should
have a form
y ( x, t )  f ( x  vt)
• Considering partial derivatives
y ( x, t ) f ( x  vt) f ( x  vt) ( x  vt)


 f ' ( x  vt)
x
x
( x  vt)
x
y ( x, t ) f ( x  vt) f ( x  vt) ( x  vt)


 f ' ( x  vt)  (v)
t
t
( x  vt)
t
The linear wave equation
• Therefore, solutions of the wave equation should
have a form
y ( x, t )  f ( x  vt)
• Considering partial derivatives
 y( x, t )   f ( x  vt)  
 
   f ' ( x  vt)   f ' ' ( x  vt)
2
x
x 
x
 x
2
 y( x, t )   f ( x  vt)  
 
   f ' ( x  vt)  (v) 
2
t
t 
t
 t
2
 f ' ' ( x  vt)  v 2
The linear wave equation
• Therefore, solutions of the wave equation should
have a form
y ( x, t )  f ( x  vt)
• Considering partial derivatives
 2 y ( x, t )
 f ' ' ( x  vt)
2
x
 y ( x, t )
2
2  y ( x, t )
 f ' ' ( x  vt)  v  v
2
t
x 2
2
2
 y ( x, t )
2  y ( x, t )
v
2
t
x 2
2
2
The linear wave equation
• The linear wave equation (not the only one having
solutions of the form y(x,t) = f(x ± vt)):
 y ( x, t )
2  y ( x, t )
v
2
t
x 2
2
2
• 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
v
T

Superposition of waves
• Let us consider two different solutions of the linear
wave equation
 y1
2  y1
v
2
t
x 2
2
2
 2 y1  2 y2

y

y2
2
2
1
 2 v
v
2
2
2
t
t
x
x
 y2
2  y2
v
2
t
x 2
 ( y1  y2 )
2  ( y1  y2 )
v
2
t
x 2
2
2
2
+
2
2
2
• Superposition principle – a sum of two solutions of
the linear wave equation is a solution of the linear
wave equation
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
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:
y ( x  vt)  A sin( k ( x  vt)   )
 A cos( k ( x  vt)     / 2)
• A – amplitude, φ – phase constant
Wavelength
y ( x, t )  A cos( k ( x  vt)   )
• “Freezing” the solution at t = 0 we obtain a
sinusoidal function of x:
y ( x,0)  A cos( kx   )
• Wavelength λ – smallest distance (parallel to the
direction of wave’s travel) between repetitions of the
wave shape
Wave number
y ( x,0)  A cos( kx   )  A cos( k ( x   )   )
 A cos( kx  k   )
• On the other hand:
cos( kx   )  cos( kx  2   )
• Angular wave number: k = 2π / λ
k  2 / 
Angular frequency
y ( x, t )  A cos( k ( x  vt)   )
• Considering motion of the point at x = 0
we observe a simple harmonic motion (oscillation) :
y (0, t )  A cos(  kvt   )  A cos( kvt   )
• For simple harmonic motion (Chapter 13):
y (t )  A cos(t   )
  kv  2v / 
• Angular frequency ω
Frequency, period
• Definitions of frequency and period are the same as
for the case of rotational motion or simple harmonic
motion:
f  1 / T   / 2
T  2 / 
• Therefore, for the wave velocity
v   / k   / T  f
y ( x, t )  A cos( kx  t   )
Chapter 14
Problem 24
Ultrasound used in a medical imager has frequency 4.8 MHz and wavelength
0.31 mm. Find (a) the angular frequency, (b) the wave number, and (c) the wave
speed.
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
y1 ( x, t )  A cos(kx  t )
• The resultant wave:
y2 ( x, t )  A cos(kx  t   )
y( x, t )  y1 ( x, t )  y2 ( x, t )
 A cos( kx  t )  A cos( kx  t   )
 2 A cos( / 2) cos( kx  t   / 2)
  
cos   cos   2 cos
 2
    
 cos

2
 

Interference of waves
y ( x, t )  2 A cos( / 2) cos( kx  t   / 2)
• If φ = 0 (Fully constructive)
y ( x, t )  2 A cos( kx  t )
• If φ = π (Fully destructive)
y ( x, t )  0
• If φ = 2π/3 (Intermediate)
y ( x, t )  2 A cos( / 3) 
 cos( kx  t   / 3)
 A cos( kx  t   / 3)
Interference of waves
• Considering two sinusoidal waves of the same
amplitude, wavelength, but running in opposite
directions
y1 ( x, t )  A cos(kx  t )
y2 ( x, t )  A cos(kx  t   )
• The resultant wave:
y( x, t )  y1 ( x, t )  y2 ( x, t )
 A cos( kx  t )  A cos( kx  t   )
 2 A cos( kx   / 2) cos(t   / 2)
  
cos   cos   2 cos
 2
    
 cos

2
 

Interference of waves
    y ( x, t )  2 A sin( t ) sin( kx)
• If two sinusoidal waves of the same amplitude and
wavelength travel in opposite directions, their
interference with each other produces a standing
wave
kx  ( n  1 )
2
n  0,1,2...
| sin kx | 1
kx  n
n  0,1,2...
sin kx  0
y  2 A sin t
y0
xn

2
Nodes
1

x  n  
2 2

Antinodes
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
2L

, n  1,2,3...
n
  2L
2L

2
2L

• Resonance frequencies
3
v
nv
f  
, n  1,2,3...
 2L
Harmonic series
• Harmonic series – collection of all possible modes resonant oscillations (n – harmonic number)
v
fn  n
, n  1,2,3...
2L
• First harmonic (fundamental mode):
v
f1 
2L
More about standing waves
• Longitudinal standing waves can also be produced
• Standing waves can be produced in 2 and 3
dimensions as well
More about standing waves
• Longitudinal standing waves can also be produced
• Standing waves can be produced in 2 and 3
dimensions as well
Chapter 14
Problem 40
A 2.0-m-long string is clamped at both ends. (a) Find the longest wavelength
standing wave possible on this string. (b) If the wave speed is 56 m/s, what’s
the lowest standing-wave frequency?
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
Pavg  A  v
2
2
• Exact expression depends on the medium or the
system through which the wave is propagating
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:
P
v

ρ – density of a medium, γ – characteristic constant,
P – pressure
F
P
A
• Traveling sound waves
s( x, t )  sm cos( k ( x  vt)) 
 sm cos( kx  t )
Intensity of sound
• Intensity of sound – average rate of sound energy
transmission per unit area
P
I
A
1
2 2
• For a sinusoidal traveling wave: I  vsm
2
I
• Decibel scale
  (10dB) log
I0
β – sound level; I0 = 10-12 W/m2 – lower limit of human
hearing
Chapter 14
Problem 65
Show that a doubling of sound intensity corresponds to approximately a 3-dB
increase in the decibel level.
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:
2L
v

, f n
n
2L
n  1,2,3...
Organ pipes
• Organ pipes are open on one end and closed on the
other
• For such pipes the resonance condition is modified:

L  n ; n  1,3,5...
4
4L
v

, f n
n
4L
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
• Resonances in musical instruments are not
necessarily 1D, and often involve different parts of
the instrument
• Guitar resonances at medium frequencies:
Musical instruments
• Resonances in musical instruments are not
necessarily 1D, and often involve different parts of
the instrument
• Guitar resonances at high frequencies:
Beats
• Beats – interference of two waves with close
frequencies
s1  sm cos 1t
+
s2  sm cos 2t
s  s1  s2  sm cos 1t  sm cos 2t
 1  2   1  2 
 2sm cos
t  cos
t
 2
  2

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 in 2D
• Far from the point source wavefronts can be
approximated as planes – planar waves
• Waves from two sources interfere to produce
regions of low and high amplitude: constructive
interference and destructive interference
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
Ps
P

I
2
A 4r
(inverse square law)
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
v

f

f

Speed of sound waves is v
• In the detector (moving) reference frame:
Speed of detector is 0
Speed of sound waves is v + vD
f '
v'


v  vD

v  vD
 f
v
v

f

v
Doppler effect
• For a moving detector (ear) and a stationary source
v  vD
f ' f
v
• If the detector is moving away from the source:
v  vD
f ' f
v
• For both cases:
v  vD
f ' f
v
Doppler effect
• For a stationary detector (ear) and a moving source
• In the detector (stationary) reference frame:
v
v
f '
 f
*
v  vS
• In the moving (source) frame:
v  vS
v

v
S
f 
* 
*
f
Doppler effect
• For a stationary detector and a moving source
v
f ' f
v  vS
• If the source is moving away from the detector:
v
f ' f
v  vS
• For both cases:
v
f ' f
v  vS
Doppler effect
• For a moving detector and a moving source
v  vD
f ' f
v  vS
• Doppler radar:
Supersonic speeds
• For a source moving faster than the speed of sound
the wavefronts form the Mach cone
• Mach number
v s vs t
1


v
vt sin 
Ernst Mach
(1838-1916)
Supersonic speeds
• The Mach cone produces a sonic boom
Questions?
Answers to the even-numbered problems
Chapter 14
Problem 18
3.38 m
Answers to the even-numbered problems
Chapter 14
Problem 22
(a) 0.582 m-1
(b) 1.5 s-1
Answers to the even-numbered problems
Chapter 14
Problem 38
every 6.0 s
Answers to the even-numbered problems
Chapter 14
Problem 64
(a) 5.1×10-2 N/m2
(b) 1.6×10-5 N/m2