Traveling Waves
Wave motion.
Periodic waves: on a string, Sound and
electromagnetic waves
Waves in Three Dimensions. Intensity
Waves encountering barriers: Reflection,
Refraction and Difraction,
The Doppler Effect
Superposition, Interference Standing waves
INTRODUCTI0N. TRAVELING WAVES
a wave is a disturbance that travels through space and time, usually
accompanied by the transfer of energy.
Waves travel and the wave motion transfers energy from one point
to another, often with no permanent displacement of the particles of
the medium—that is, with little or no associated mass transport.
They consist instead of oscillations or vibrations around almost fixed
locations. For example, a cork on rippling water will bob up and
down, staying in about the same place while the wave itself moves
onwards
INTRODUCTION.TYPE OF WAVE
One type of wave is a mechanical wave, which propagates through a medium
in which the substance of this medium is deformed. The deformation reverses
itself owing to restoring forces resulting from its deformation. For example,
sound waves propagate via air molecules bumping into their neighbors. This
transfers some energy to these neighbors, which will cause a cascade of
collisions between neighbouring molecules. When air molecules collide with
their neighbors, they also bounce away from them (restoring force). This keeps
the molecules from continuing to travel in the direction of the wave.
Another type of wave can travel through a vacuum, e.g. electromagnetic
radiation (including visible light, ultraviolet radiation, infrared radiation, gamma
rays, X-rays, microwaves and radio waves). This type of wave consists of
periodic oscillations in electrical and magnetic fields.
Transverse waves: The oscillations
occur perpendicularly to the direction
of energy transfer. Exemple: a wave in
a tense string. Here the varying
magnitude is the distance from the
equilibrium horizontal position
Longitudinal waves: Those in
which the direction of vibration is
the same as their direction of
propagation. So the movement of
the particles of the medium is
either in the same or in the
opposite direction to the motion of
the wave. Exemple: sound waves,
what changes in this case is the
pressure of the medium (air, water
or whatever it be).
Pulses
Speed of wave
The shape of pulse is
described by the function
f(x)
y  f ( x  vt)
y  f ( x  vt)
The wave function
provides the
mathematical description
of the traveling pulse
y: disturbance of medium from the equilibrium position
v: speed of propagation of wave
The wave function are particular solutions of the
differential equation called wave equation, which
can derived from Newton´s Law
2 y 1 2 y
 2 2
2
x
v t
Traveling pulses. An example
sen 2 x  t 
y
2
1  2 x  t 
Wave function
where x, y are in
meter, t in
seconds, v = 0.50
m/s
t

sen 2 x  
 2
y
2
t

1  4 x  
 2
Let us to write the wave
equation in such a way that the
group x+v·t appears explicitly.
This pulse moves to the right
(positive direction of X axis)
with a velocity of 0.50 m/s
0,5
y (m)
Plotting for
differents values
of time
0,4
t=0
0,3
t=2
0,2
t=4
0,1
0,0
-0,1
-0,2
-0,3
-0,4
-0,5
-4
-3
-2
-1
0
1
2
3
4
x (m)
Speed of waves
A general property of waves is that their speed relative to medium depends on the
properties of medium but is independent of the motion of the source of waves.
If the observer is in motion with respect to the medium, the velocity of wave
propagation relative to the observer wil be different. A remarkable exception is
encountered in the case of light
Speed of a wave
on a String
The 25-m-long string has a mass of 0.25 kg
and is kept taut by a hanging object of mass
10 kg. What is the speed of the pulse?. If the
10-kg mass is replaced with 20-kg mass, what
is the speed on the string?
Transverse waves travel at 150 m/s on a wire
of length that is under a tension of 550 N.
What is the mass of the wire?
A steel piano wire is 0,7 m long and has a
mass of 5 g. It is stretched with a tension of
500N. What is the speed of transverse waves
on the wire?
v
FT

FT tension on the string
 linear mass density ( kg / m )
Speed of waves (2)
Sound
(in a elastic
material)

v

β bulk modulus
ρ density
 
P
V
V
For sound waves in a gas such air, the pressure changes occur too
rapidly for appreciable heat transfer, and so the process is adiabatic.
Sound
(in air)
Solids
v
v
 RT
M
γ adiabatic coefficient, for air 1,4
R universal gas constant 8.314 J/(mol.K)
M: molar mass of gas, for air 28.96x10-3 Kg/mol
T: absolute temperature
Y
  density of the solid (kg/m3)

Y
stress
F/A

strain L / L
Young
modulus
Calculate the speed of sound in air at (a) 0ºC and (b) 20ºC
The bulk modulus for water is 2.0x109 N/m2. Use it to find the speed of sound in water
(b) The speed of sound in mercury is 1410 m/s What is the bulk modulus for mercury
(ρ = 13.6 x 103 Kg/m3 )
PERIODIC WAVES
Harmonic waves
Harmonic waves are the most basic type of periodic waves. All waves,
wether they are periodic or not, can be modeled as a superposition of
harmonic waves.
If one end of a string is attached to a vibrating point that is moving up and
down with simple harmonic motion, a sinusoidal wave train propagates
along the string. If a harmonic wave is traveling through a medium , each
point of the medium oscillates in simple harmonic motion.
Harmonic waves: The harmonic function
Harmonic waves are the most basic type of periodic waves. All waves, wether they are periodic or
not, can be modeled as a superposition of harmonic waves.
The sinusoidal shape is
described by the sine function
crest
y  A sin( 2
λ, wavelength: the minimun distance after which the
wave repeat (distance between crests, per example)
v

T
 f
Basic relationship between
wavelength,λ , speed,v,
period, T, and frequency, f
k: wave number
k
2

x

)
For a wave traveling in the direction
of increasing x, with a speed v,
replace x by x –vt, with δ = 0
y  A sin( 2
x  vt

)
x
y  A sin 2 (  ft) 

y  A sin( kx   t )
Harmonic waves: Energy transfer on a string
The energy on one vibrating point,
considering that describes a harmonic
motion, is
1
1
Etotal  U  K  k A2  m A2 2
2
2
Energy transfer
For the string where a harmonic
wave has been generated, the energy
of a particle of mass dm will be
Energy is being transferred from the
initial vibrating point to the whole string,
m
because when the wave reaches new
dm  dx
l
portions of the string, they begin to
1
1m
oscillate gaining energy. The energy
Etotal  dm A2 2 
dxA2 2
transferred by the unit time, that is, the
2
2 l
power, will be
Power 
E passin g
dt
1 m dx 2 2 1 m 2 2

A 
vA 
2 l dt
2 l
Harmonic Waves:
Energy on Sound Waves
The wave function of harmonic sound waves
can be writen considering longitudinal
displacements of air mollecules around the
equilibrium position s(x,t),
s( x, t )  so sin( kx   t )
Energy transfer
The average energy of a harmonic
sound wave in a volume element dV,
will be that corresponding a vibrating
particle with a mas dm = ρ dV, that is
1
1
dm A2 2   dV so2 2
2
2
dEtotal 1 2 2
Energy per unit of
  so 
volume
dV
2
dEtotal 
The vibration of air mollecules lead to
variation of pressure
p( x, t )  po sin( kx   t 
po    v so

2
)
Waves in Three Dimensions.
Intensity
Wave Intensity. Case study: Sound Wave
The Wave Intensity, I, is the average power per unit area that is incident
perpendicular to the direction of the propagation
P
P
I 
A 4 r 2
For the case of point source that
emits waves uniformly in all
directions
The rate of transfer of
energy is the passing
into the shell
dE
dE
dE

dV 
Adr
dt dtdV
dV dt
dE dr dE

A 
Av
dV dt dV
P
P dE
1 2 2
1 po2
I 
v   so  v 
A dV
2
2 v
Wave intensity for a sound
wave
A loudspeaker diafragm 30 cm in diameter is vibrating at 1 kHz, with an amplitude of 0.020
mm. Assuming that the close air mollecules vibrates with the same amplitude, find (a) the
pressure amplitude (b) the sound intensity in front of diaphragm (c) the acoustic power
being radiated (d) if the sound is radiated uniformly in the hemisphere, find the intensity at 5
m from the loudspeaker
Intensity level and loudness. The human ear
Range of human ear response to sound wave intensity:
Threshold of hearing 10-12 W/m2
Pain 1 W/m2
The perception of
loudness is not
proportional to the
intensity but varies
logaritmically. We use
a logaritmic scale to
describe the intensity
level for the human
ear, which is measured
in decibels, (dB)
  10 log 10
I
Io
Estimate the sound pressure
variations for the range of sound
intensity in the case of human ear
Waves encountering barriers:
Reflection, refraction and Difraction
Light beam exhibiting reflection,
refraction, transmission and
dispersion when encountering a
prism
Waves encountering barriers:
refraction
Refraction is the phenomenon of a wave changing its speed. Typically,
refraction occurs when a wave passes from one medium into another. The
amount by which a wave is refracted by a material is given by the refractive
index of the material. The directions of incidence and refraction are related to
the refractive indices of the two materials by Snell's law.
Sinusoidal traveling plane wave entering
a region of lower wave velocity at an
angle, illustrating the decrease in
wavelength and change of direction
(refraction) that results.
Waves encountering barriers: Difraction
A wave exhibits diffraction when it encounters an obstacle that bends the wave or when it
spreads after emerging from an opening. Diffraction effects are more pronounced when
the size of the obstacle or opening is comparable to the wavelength of the wave.
The Doppler effect (a)
The Doppler effect (or Doppler shift), is the change in frequency of a wave for an
observer moving relative to the source of the wave The received frequency is higher
(compared to the emitted frequency) during the approach, it is identical at the instant
of passing by, and it is lower during the recession.
All motions are relative to
medium
Stationary receiver
The number of wave crests
passing the receiver per unit
time
fr 
v
v

v
fr  
fs
 v  us
During time Ts, -period of the source- the source
moves a distance usTs and the 5th wavefront
travels a distance vTs . The wavelength in front of
source is (v-us)Ts
v  us
  ( v  us )Ts 
fs
In front of the source the minus sign applies.
Behind the source the plus sign applies.
The Doppler effect (b)
The Doppler effect (or Doppler shift), is the change in frequency of a wave
for an observer moving relative to the source of the wave.
All motions are relative to medium
Moving receiver
fr 
Sign plus is used in the case of
receiver moving in the direction
opposite to that of the wave
v  ur

The number of wave
crests passing the
receiver per unit time
Source and receiver are moving relative to medium
  ( v  us )Ts 
v  us
fs
fr 
v  ur
fs
v  us
If the receiver is moving toward the source the plus sign is selected in
the numerator. If the source is moving to the receiver the minus sign is
selected in the denominator. The general rule is that the frequency
tends to increase when the source moves toward the receiver and
when the receiver moves toward the source
The Doppler effect. Summary and exercises
The Doppler effect (or Doppler shift), is the change in frequency of a wave for an observer
moving relative to the source of the wave The received frequency is higher (compared to the
emitted frequency) during the approach, it is identical at the instant of passing by, and it is lower
during the recession.
All motions are relative to
medium
v  ur
fr 
fs
v  us
fr receiver frequency;
fs source frequency
v wave propagation speed
ur receiver speed;
us source speed;
Choose the signs that give an up-shift in
frequency for an approaching source or
receiver, and vice-versa.
If the receiver is moving toward the source the
plus sign is selected in the numerator. If the
source is moving to the receiver the minus sign is
selected in the denominator
The frequency of a train horn is 400 Hz. If the train
speed train is 122 km/h, (a) the wavelength and the
frequency of the sound passing a stationary receiver
placed in front of train; (b) the same if the stationary
receiever were placed behind of train; (c). If the
receiver is approaching to the train with a speed 120
km/h respect to ground, in the same way but in
oppsite direction, what the received frequency will be?
** The trafic stationary radar unit emits waves with a
frequency of 1.5x109 Hz. The receiver unit measures
the reflected waves from the car moving away. The
frequency of this reflected wave differs from the
emiting by 500 Hz . What is the car speed?. **
A ship at rest is equipped with sonar that sends out
pulses of sound at 40 MHz. Reflected pulses are
received from a submarine directly below with a time
delay of 80 ms, at a frequency of 39.958 MHz. Find
(a) the depth of the submarine (b) ts vertical speed.
Speed of sound in seawater is 1.54 KHz
Shock waves (Shock front)
If a source moves with speed greater than the wave speed , then there will be no
waves in front of the source. Instead, the waves pile up behind the source to form a
shock wave.
U.S. Navy F/A-18 breaking
the sound barrier. The white
halo is formed by
condensed water droplets
thought to result from a
drop in air pressure around
the aircraft
SUPERPOSITION OF WAVES
INTERFERENCE
STANDING WAVES
When two or more waves ovelap in space,their individual disturbances
superimpose and add algebraically, creating a resultant wave. This
property of waves is called the principle of superposition
Under certain circumstances the superposition of harmonic waves of
the same frequency produce s sustained wave patterns in the space.
This phenomenon is called interference.
Interference and difraction are what distinguish between wave motion
and particle motion
SUPERPOSITION OF WAVES
y  y1  y2  A sin( kx  t )  A sin( kx  t   ) 
2 A cos( 12  ) sin( kx  t  12  )
Constant
phase 0
Interference
Constant
phase π/2
Interference:
Constructive and
destructive
Phase difference
due to path
difference
STANDING WAVES
A standing wave, also known as a stationary wave, is a wave that remains in a
constant position. This phenomenon can occur because the medium is moving
in the opposite direction to the wave, or it can arise in a stationary medium as
a result of interference between two waves traveling in opposite directions.
The sum of two counter-propagating waves (of equal amplitude and
frequency) creates a standing wave. Standing waves commonly arise when a
boundary blocks further propagation of the wave, thus causing wave reflection,
and therefore introducing a counter-propagating wave. For example when a
violin string is displaced, transverse waves propagate out to where the string is
held in place at the bridge and the nut, where the waves are reflected back. At
the bridge and nut, the two opposed waves are in antiphase and cancel each
other, producing a node. Halfway between two nodes there is an antinode,
where the two counter-propagating waves enhance each other maximally.
There is no net propagation of energy over time
STANDING WAVES
When waves are confined in spaces, multiple reflections cause
superposing waves that interfer according the superposition principle.
For a given string or pipe, there are certain frequencies for which
superpposition results in a stationary vibration pattern: standing wave