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Transcript of Mechanical Waves
Mechanical waves
Dr. Eng. Hesham SalamahDr. Eng. Hesham Salamah
[email protected][email protected]
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Waves Waves
Wave is everywhere in nature!
Why can you hear me?
Why can you receive a TV signal emitted
from a TV station?
• A wave is the motion of a disturbanceA wave is the motion of a disturbance
Types of WavesTypes of Waves
Waves are classified into different
types according to their natures :
Transverse waves Longitudinal waves
Mechanical waves
Transverse waves
Electromagnetic waves
WAVES
Types of Waves; continueTypes of Waves; continue
Classification According to natures of matter:
1. 1. Mechanical Waves Mechanical Waves
Examples: water waves, sound waves, and seismic waves.
They are governed by Newton’s laws, and they can exist
only within a material medium, such as water, air, and
rock.
These waves have certain central features:
2. 2. Electromagnetic WavesElectromagnetic Waves
Examples: Visible and ultraviolet light, radio and
television waves, microwaves, X rays, and radar
waves.
They don’t require material medium to exist. These
waves have a limited wave speed in vacuum,. m/s 458 792299c
3. 3. Matter WavesMatter Waves
Examples: Electrons, protons, and other fundamental
particles; ultracold atoms.
The most remarkable property of the matter waves is
that wave functions of matter waves are referred to
as probability amplitudes of waves.
Classification According to Oscillation Types:
1. 1. Transverse WavesTransverse Waves
The direction of oscillation of medium elements is
perpendicular to the direction of travel of the wave.
2. 2. Longitudinal WavesLongitudinal Waves
The direction of oscillation of medium elements is
parallel to the direction of the wave’s travel, the motion
is said to be longitudinal, and the wave is said to be a
longitudinal wave.
Both a transverse wave and a longitudinal wave are said tobe traveling waves because they both travel from one pointto another, as from one end of the string to the other end.
Creation of Waves and PropagationCreation of Waves and Propagation
For a source of simple harmonic oscillation with angular frequency , the traveling in the positive direction of x axis and oscillating parallel to the y axis, the displacementy of the element located at position x at time t is given by
Conditions of a Mechanical waveConditions of a Mechanical wave
1. A Wave Source.
2. Medium in which a mechanical wave propagates.
A wave means that the oscillation state propagates with time. It does not mean the particles of medium move forward with wave. Particles of the medium oscillate only around their corresponding equilibrium positions.
).sin(),( tkxytxy m
Displacement Phase
Angularfrequency
Angular wave number
Amplitude ym is the amplitude ofa wave.
The phase of the waveis the argument
kx – t)
λ
2π k
f π2 T
2π ω Move to right
Space Periodicity and Time PeriodicitySpace Periodicity and Time Periodicity
Wavelength and Angular Wave Number
The wavelength of a wave is the distance (parallel to
the direction of the wave’s travel) between repetitions of
the shape of the wave (or wave shape).
According to the property of sine function, it begins to
repeat itself when its angle is increased by 2 rad,
That is
2
k (Angular wave number)
Period, Frequency, and Angular Frequency
You can choose any time t1, through a period, the
oscillation is repeated.
The period of oscillation T of a wave is defined to be the
time any mass element takes to moves through one full
oscillation.
).sin(sin 11 Ttyty mm
This can be true only ifT =2or
.2T
Angular frequency
Frequency ƒ is defined by
.2
1
Tf
Wavefronts
Wavefronts are surfaces over which the oscillations of wave have
the same value; usually such surfaces are represented by whole
or partial circles in a two-dimensional drawing for a point
source.
If point A retains its displacement as it moves, its phase must remain a constant, that is:
constant. a kxt
Taking a derivative, we have
,0dtdx
k
or.
kv
dtdx
By using definitions of and k, we can rewrite v as
fT
v
Phase speed
The Speed of a Traveling WaveThe Speed of a Traveling Wave
Question:
How to describe a wave traveling in negative direction of x?
Similarly to the discussion of positive direction of x, the
wave speed of a wave traveling in the negative x direction
should be have a form:
kdtdx
v
This corresponding to the condition:
constant a kxt
This means that the wave function should have a form:
)sin(),( tkxytxy m
It seems that k may be have a property of vector, its
magnitude is
vk
and its direction is in that of wave traveling, or wave
velocity.
For a traveling wave to right, this means the oscillation at x is:
For a traveling wave to left, this means the oscillation at x is:
)sin(),( tkxytxy m
)sin(),( tkxytxy m
(a) (b)
A BF F
v
Wave Speed on a Stretched StringWave Speed on a Stretched String
v
The speed of a wave along a stretched ideal string depends only on the tension and linear density of the string and not on the frequency of the wave.
DensityMassLinear
ForceAppliedv
L
m
Length
mass
Energy and Power Energy and Power of a Traveling String Waveof a Traveling String Wave
In general, the average energy flow through an unit area of
perpendicular to the propagating direction, or density of energy
flow, or intensity of the wave, is given by:
22 2
1 maverage yvP
Where is the linear mass density of wave.
v
Example (1): Example (1): The equation of a wave is
m
4
)4010(2
sin05.0),( txtxy
Find: (a) the wavelength, the frequency, and the wave velocity;
(b) the particle velocity and acceleration at x = 0.5 m and t = 0.05 s.
k = 2/ = 5 rad/m and so = 0.4 m.
The angular frequency is = 2 f = 20 rad/s; thus f = 10 Hz.
The wav velocity is υ = f = /k = 4 m/s in the x direction.
(b) The particle velocity and acceleration are
m/s22.2)42
5(cos)05.0()20(
t
y
222
2
m/s140)42
5(sin)05.0()20(
t
ya
Example (2): Example (2): One end of a string is fixed. It hangs over a
pulley and has a block of mass 2.00 kg attached to the other
end, as in Fig. 2.4. The horizontal part has a length of 1.6 m
and a mass of 20.0 g. What is the speed of a transverse pulse
on the string?
m
L = 1.6 m
Solution:Solution:
The tension is simple the weight of the block; that is, F = 19.6 N.
The linear mass density is (20 103)/(1.6) = 1.25 102 kg/m.
m/skg/m
N9.66.39
1025.1
12
F
v
Example (3): Example (3): A rod vibrating at 12 Hz generates harmonic waves with
an amplitude of 1.5 mm in a string of linear mass density 2 g/m. If the
tension in the string is 15 N, what is he average power supplied by the
source?
m/skg10(2
N3-
6.8615
F
v
The angular frequency is rad/s.4.752 f
mW1.1)6.86()105.1()4.75()102(2
1 2323
2
2
1av
vyP m
Interference of
waves
Dr. Eng. Hesham SalamahDr. Eng. Hesham Salamah
[email protected][email protected]
Please, Switch off the Please, Switch off the MobileMobile
Suppose that two waves travel simultaneously along the same stretched string. Let y1(x, t) and y2(x, t) be the displacements that the string would experience if each wave traveled alone. The displacement of the string when the waves overlap is then the algebraic sum:
).,(),(),( 21 txytxytxY
It means
Overlapping waves algebraically add to produce a resultant wave (or net wave).
Superposition for WavesSuperposition for Waves
The Principle of Superposition for Waves
Overlapping waves do not in any way alter the travel of each other.
Interference of Waves
Suppose that there are two sinusoidal
waves of the same wavelength and
amplitude in the same direction along
a stretched string.
)sin(),(1 tkxytxy m
)sin(),(2 tkxytxy m
Where, is the phase shift
)2
1sin(]
2
1cos2[
)sin()sin(
),(),(),( 21
tkxy
tkxytkxy
txytxytxY
m
mm
The new resultant amplitude resultant amplitude depends on phase :
1. If = 0 rad, the two interfering waves are exactly in phase.
2
1cos2 mm yY
)sin(2),( tkxytxY m
In this case, the interference produces the greatestpossible amplitude which is called fully constructiveinterference.
From the principle of superposition, the resultant wave is:
2. If = 0),( txY
Although we send two waves along the string, we see no
motion of the string. This type of interference is called
fully destructive interference (see fig. (e)).
3. When interference is neither fully constructive nor
fully destructive, it is called intermediate interference.
Figures (c) and (f) for =2/3.
Two waves with the same wavelength are in phase if
their phase difference is zero or any integer number of
wavelengths. Thus, the integer part of any phase
difference expressed in wavelengths may be discarded.
According to the previous section, the condition for
fully constructive interference is
.210for ),2( ,,,nn
Similarly, fully destructive interference occurs when
.,2,1,0for ,)12( nn
Standing WavesStanding Waves
Let’s consider two waves with the same oscillating
frequencies, but traveling in opposite directions.
To analyze a standing wave, we represent the two
waves with the equations:
The principle of superposition gives, for the combinedwaves,
tkxy
tkxytkxytxY
m
mm
cos]sin2[
)sin()sin(),(
)sin(),( tkxytxy m
)sin(),( tkxytxy m
The wave like this — two sinusoidal waves of the same amplitude and wavelength travel in opposite directions, their interference with each other produces a standing wave.
Standing Waves and Resonance Standing Waves and Resonance
• For For certain frequenciescertain frequencies, , the interference produces a the interference produces a standing wave pattern (or standing wave pattern (or an an oscillation mode)oscillation mode) Such a standing wave is Such a standing wave is said to be produced at said to be produced at resonanceresonance, and the string , and the string is said to is said to resonateresonate at these at these certain frequencies, called certain frequencies, called resonant frequenciesresonant frequencies. .
For n=1, the frequency f =v/2L. This Lowest frequency
is called the fundamental mode or the first harmonic .
For n=2, we have the second harmonic oscillation mode.
For n=3, we have the third harmonic oscillation mode.
)2
t ω sin(k x 2
cos y 2 t)y(x, m
4
t 40 x 3sin 0.17 4
t 40 x 3sin 4
cos 0.12 * 2 t)y(x,
s 0.05 T T
π2 40
T
π 2 ω
m/s 13.33 3
40
k
ω υ
Example (1):Example (1):
The following two waves interfere :
y1 = 0.12 sin[3x – 40t], y2 = 0.12 sin[3x – 40t – 0.5)
•Find the equation of the resultant wave
•What is the periodic time of the wave
•What is the speed of the resultant wave
)0.20.3(sin0.4(1 txy cm)
2y ),0.20.3(sin0.4( tx cm)
Example (2): Example (2): Two waves traveling in opposite directions produce a standing wave. The individual wave functions are
and
where x and y are measured in cm. Find: (a) the amplitude of the simple harmonic motion
of the element of the medium located at x = 2.3 cm; (c) the maximum displacement of an
element located at an antinode?
Here, ym = 4.0 cm, k = 3.0 rad/cm and = 2.0 rad/s.
) cos(2.0 ) sin(3.0 )0.8()(sin()cos(2),( txkxtytxy m
cm63.4]180
(2.3) [(3.0)sin cm) 0.8( cm3.2 xamplitudeThe
(b) The maximum displacement of an element at an antinode is the amplitude of
the standing wave. From part (a) of the recent example, we have the value of this
maximum being 8.0 cm.
Hz 2 12
24 1 f
Hz 4 6
24 2 f
Hz 6 4
24 3 f
Example (3)Example (3)
The speed of a wave on a particular string is 24 m/s. if the
string is 6 m long, what is the fundamental frequency, first
harmonic and second harmonic?