leacture dr.khitam y. elwasife - الصفحات الشخصية | الجامعة...
TRANSCRIPT
Acoustics
is the science that deals with the study of all mechanical waves in gases, liquids, and
solids including vibration, sound, ultrasound and infrasound .
A scientist who works in the field of acoustics is an acoustician while someone
working in the field of acoustics technology may be called an acoustical engineer.
The application of acoustics is present in almost all aspects of modern society with
the most obvious being the audio and noise control industries
When waves are combined in systems with boundary conditions, only certain allowed frequencies can exist.
We say the frequencies are quantized.
Quantization is at the heart of quantum mechanics.
The analysis of waves under boundary conditions explains many quantum phenomena.
Quantization can be used to understand the behavior of the wide array of musical instruments that are based on strings and air columns.
Waves can also combine when they have different frequencies.
Quantization
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
Superposition Principle
Waves can be combined in the same location in space.
To analyze these wave combinations, use the 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 that obey the superposition principle are linear waves.
For mechanical waves, linear waves have amplitudes much smaller than
their wavelengths.
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 means waves pass through each other.
Superposition Example
Two pulses are traveling in opposite
directions (a).
The wave function of the pulse
moving to the right is y1 and for the
one moving to the left is y2.
The pulses have the same speed but
different shapes.
The displacement of the elements is
positive for both.
When the waves start to overlap (b),
the resultant wave function is y1 + y2.
Superposition Example,
When crest meets crest (c) the
resultant wave has a larger amplitude
than either of the original waves.
The two pulses separate (d).
They continue moving in their original directions.
The shapes of the pulses remain unchanged.
This type of superposition is called constructive interference.
Destructive Interference Example
Two pulses traveling in opposite directions.
Their displacements are inverted with respect to each other.
When these pulses overlap, the resultant pulse is y1 + y2.
This type of superposition is called destructive
interference.
Constructive interference occurs when the displacements caused by the two pulses are in the same direction.
The amplitude of the resultant pulse is greater than either individual pulse.
Destructive interference occurs when the displacements caused by the two pulses are in opposite directions.
The amplitude of the resultant pulse is less than either individual pulse.
Types of Interference
Radiation from a monopole source
A monopole is a source which radiates sound equally well in all directions. The simplest example of a monopole source would be a sphere whose radius alternately expands and contracts sinusoidally. The monopole source creates a sound wave by alternately introducing and removing fluid into the surrounding area. A boxed loudspeaker at low frequencies acts as a monopole. The pattern for a monopole source is shown in the figure at right.
.
A dipole source consists of two monopole
sources of equal strength but opposite
phase and separated by a small distance
compared with the wavelength of sound.
While one source expands the other source
contracts. The result is that the fluid (air)
near the two sources sloshes back and
forth to produce the sound. A sphere which
oscillates back and forth acts like a dipole
source, as does an unboxed loudspeaker .
A dipole source does not radiate sound in
all directions equally. The pattern shown at
right; there are two regions where sound is
radiated very well, and two regions where
sound cancels.
.
Radiation from a dipole source
Superposition of Sinusoidal Waves
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:
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, is also sinusoida.l
The resultant wave has the same frequency and wavelength as the original waves.
The amplitude of the resultant wave is 2A cos (f / 2) .
The phase of the resultant wave is f / 2.
Sinusoidal Waves with Constructive-destructive Interference
When f = 0, then cos (f/2) = 1
The amplitude of the resultant wave is 2A.
The crests of the two waves are at the same location in space.
The waves are everywhere in phase.
The waves interfere constructively.
In general, constructive interference occurs when cos (Φ/2) = ± 1.
That is, when Φ = 0, 2π, 4π, … rad
When Φ is an even multiple of π
When f = p, then cos (f/2) = 0
Also any odd multiple of p
The amplitude of the resultant wave is 0.
See the straight red-brown line in the figure.
The waves interfere destructively
Sinusoidal Waves, General Interference
When f is other than 0 or an even multiple of p, the amplitude of the resultant is between 0 and 2A. The wave functions still add
The interference is neither constructive nor destructive.
Constructive interference occurs when f = np where n is an even integer (including 0).
Amplitude of the resultant is 2A
Destructive interference occurs when f = np where n is an odd integer. Amplitude is 0
General interference occurs when 0 < f < np
Amplitude is 0 < Aresultant < 2A
Interference in Sound Waves
Sound from S can reach R by two different
paths.
The distance along any path from speaker
to receiver is called the path length, r.
The lower path length, r1, is fixed.
The upper path length, r2, can be varied.
Whenever Dr = |r2 – r1| = n l, constructive
interference occurs.a maximum in sound
intensity is detected at the receiver.
Whenever Dr = |r2 – r1| = (nl)/2 (n is odd),
destructive interference occurs. No sound is
detected at the receiver.
A phase difference may arise between two
waves generated by the same source when
they travel along paths of unequal lengths.
Standing wave
When a sound wave hits a wall, it is partially absorbed and partially reflected. A person
far enough from the wall will hear the sound twice. This is an echo. In a small room
the sound is also heard more than once, but the time differences are so small. This
is known as reverberation.
Music is the Sound that is produced by instruments or voices. To play most musical
instruments you have to create standing waves on a string or in a tube or pipe.
The pitch of the sound is equal to the frequency of the wave. The higher the frequency,
the higher is the pitch.
Wind instruments produce sounds by means of vibrating air columns. To play a wind
instrument you push the air in a tube with your mouth. The air in the tube starts to
vibrate with the same frequency as your lips. Resonance increases the amplitude of
the vibrations, which can form standing waves in the tube. The length of the air
column determines the resonant frequencies. The mouth or the reed produces a
mixture if different frequencies, but the resonating air column amplifies only the
natural frequencies. The shorter the tube the higher is the pitch. Many instruments
have holes, whose opening and closing controls the effective pitch.
Standing Waves--Free/Open Ends This shows longitudinal standing waves, such as in sound waves in a pipe open at both ends. The top wave is the fundamental or first harmonic. It has displacement antinodes at each end, and a displacement node in the middle. When the displacement is large and to the right at one end, it is large and to the left at the other, and they're separated by one node. In other words, it is one-half wavelength from one end to the other. The next two waves show the next two harmonics, the second and the third. The pipe holds, respectively, two and three half-wavelengths, and shorter wavelengths correspond to higher frequencies--the frequencies are two and three times the fundamental.
L
Type of air column
Closed at both ends
Closed at one end
open at the other
Open at both ends
The longest standing wave in a tube of length L with two open ends has displacement antinodes (pressure nodes) at both ends. It is called the fundamental.
The longest standing wave in a tube of length L with
two open ends has displacement antinodes
(pressure nodes) at both ends. It is called the
fundamental.
The next longest standing wave in a tube of length L with two open ends is the second harmonic.
It also has displacement antinodes at each end.An integer number of half wavelength have to fit into the tube of length L. L = nλ/2, λ = 2L/n, f = v/λ = nv/(2L). For a tube with two open ends all frequencies fn = nv/(2L) = nf1, with n equal to an integer, are natural frequencies.
The quality of a sound depends on the relative
intensities of the waves with the natural
frequencies. It depends on the spectrum of the
sound. The sound quality of a musical
instrument is called its timbre.
A sinusoidal sound wave of frequency f is a
pure tone. A note played by a musical
instrument is not a pure tone. Its wave
function is not sinusoidal, i.e. it is not of the
form ∆P(x,t) = ∆Pmaxsin(kx - ωt + φ). The wave
function is a sum of sinusoidal wave functions
with frequencies nf, (n = 1, 2, 3, ...,) with
different amplitudes, which decrease as n
increases. The harmonic waves with different
frequencies which sum to the final wave are
called a Fourier series. Breaking up the
original sound wave into its sinusoidal
components is called Fourier analysis.
21
-1
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
equilibrium position of particles
instantaneous position of
particles
sine curve showing
instantaneous displacement of
particles from equilibrium
instantaneous pressure
distribution
time averaged pressure
fluctuations
Enter t/T
Normal modes in a pipe with an open and a closed end (stopped pipe)
,...)5,3,1(4
4 n
n
LornL n
n ll
,...)5,3,1(4
nL
vnfn
Standing waves in air column
25
Problem
What are the natural frequencies of vibration for a human ear?
Why do sounds ~ (3000 – 4000) Hz appear loudest?
Solution
Assume the ear acts as pipe open at the atmosphere and closed at the eardrum.
The length of the auditory canal is about 25 mm.
Take the speed of sound in air as 340 m.s-1.
L = 25 mm = 0.025 m v = 340 m.s-1
For air column closed at one end and open at the other
L = l1 / 4 l1 = 4 L f1 = v / l1 = (340)/{(4)(0.025)} = 3400 Hz
When the ear is excited at a natural frequency of vibration large amplitude
oscillations (resonance) sounds will appear loudest ~ (3000 – 4000) Hz.
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.
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.
yright Asin 2px
l ft
A sin 2px
l
cos2p ft cos 2p
x
l
sin2p ft
yleft Asin 2px
l ft
A sin 2px
l
cos2p ft cos 2p
x
l
sin2p ft
yright yleft 2Asin 2px
l
cos2p ft
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
Standing Waves, Definitions
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
l
1, 3, 5,4
nx n
l
Nodes and Antinodes
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).
Standing Waves in a String
Consider a string fixed at both ends, The string has length L.
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.
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 λ/4.
We identify an analysis model called waves under boundary conditions.
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
Standing Wave Example
Standing Waves in a String
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 (b) corresponds to to l = L.
The third mode (c) corresponds to l = 2L/3.
This is the first normal mode that is consistent with
the boundary conditions.
There are nodes at both ends.
There is one antinode in the middle.
This is the longest wavelength mode:
½l1 = L so l1 = 2L
The section of the standing wave between nodes is
called a loop.
In the first normal mode, the string vibrates in one
loop.
Standing Waves on a String, Summary
The wavelengths of the normal modes for a string of length L fixed at both ends are ln = 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
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.
Waves on a String, Harmonic Series
Resonance,
Because an oscillating system exhibits a large amplitude when driven at any of its
natural frequencies, these frequencies are referred to as resonance frequencies.
If the system is driven at a frequency that is not one of the natural frequencies, the
oscillations are of low amplitude and exhibit no stable pattern.
Resonance occurs when a system is able to store and easily transfer energy between
two or more different storage modes .However, there are some losses from cycle to
cycle, called damping. When damping is small, the resonant frequency is
approximately equal to the natural frequency of the system, which is a frequency of
unforced vibrations. Some systems have multiple, distinct, resonant frequencies.
Resonance phenomena occur with all types of vibrations or waves there is mechanic
cal resonance, acoustic resonance, electromagnetic resonance, nuclear magnetic
resonance(NMR), electron spin resonance (ESR) and resonance of quantum wave
functions. Resonant systems can be used to generate vibrations of a specific
frequency (e.g., musical instruments),
Resonance
• When we apply a periodically varying force to a
system that can oscillate, the system is forced to
oscillate with a frequency equal to the frequency of
the applied force (driving frequency): forced
oscillation. When the applied frequency is close to a
characteristic frequency of the system, a
phenomenon called resonance occurs.
• Resonance also occurs when a periodically varying force
is applied to a system with normal modes.
•of the applied
force is close to one of normal modes of the system,
resonance
occurs.
35
Why does a tree howl? The branches of trees vibrate because of the wind.
The vibrations produce the howling sound.
N A
Length of limb L = 2.0 m
Wave speed in wood v = 4.0103 m.s-1
Fundamental L = l / 4 l = 4 L
v = f l
f = v / l = (4.0 103) / {(4)(2)} Hz
f = 500 Hz
Fundamental mode of vibration
Problem
The sound waves generated by the
fork are reinforced when the length
of the air column corresponds to one
of the resonant frequencies of the
tube. Suppose the smallest value of
L for which a peak occurs in the
sound intensity is 9.00 cm.
Problem
Resonance
Lsmalles t= 9.00 cm
(a) Find the frequency of the
tuning fork.
-1 2
11 345 m.s 9.00 10 mn v L
1 2
1
345 Hz 985 Hz
4 4(9.00 10
vf
L
(b) Find the wavelength and the next two water levels giving resonance.
2
14 4(9.00 10 ) m 0.360 mLl
m. 450.02/ m, 270.02/ 2312 ll LLLL
Beats and interference
Wave interference is the phenomenon that occurs when two waves meet while
traveling along the same medium. The interference of waves causes the medium
to take on a shape that results from the net effect of the two individual waves
upon the particles of the medium.
if two upward displaced pulses having the same shape meet up with one another
while traveling in opposite directions along a medium, the medium will take on
the shape of an upward displaced pulse with twice the amplitude of the two
interfering pulses. This type of interference is known as constructive
interference
If an upward displaced pulse and a downward displaced pulse having the same
shape meet up with one another while traveling in opposite directions along a
medium, the two pulses will cancel each other's effect upon the displacement of
the medium and the medium will assume the equilibrium position. This type of
interference is known as destructive interference
The diagrams below show two waves - one is blue and the other is red -
interfering in such a way to produce a resultant shape in a medium; the
resultant is shown in green. In two cases (on the left and in the middle),
constructive interference occurs and in the third case (on the far right,
destructive interference occurs.
An animation below shows two sound waves interfering constructively in order to produce very large oscillations in pressure at a variety of anti-nodal locations. Note that compressions are labeled with a C and rarefactions are labeled with an R.
Beats
Two sound waves with different but close frequencies give rise
to BEATS s1 x,t sm cosw1tConsider
s2 x,t sm cosw2tw1 w2
s s1 s2 2sm cos w t coswt
cos cos 2cos1
2 cos
1
2
w 1
2w1 w2 Very
small w
1
2w1 w2
≈w1≈w2
s 2sm cos w t coswt
On top of the almost same frequency, the amplitude takes
maximum twice in a cycle: cosw’t = 1 and -1: Beats Beat frequency fbeat: fbeat f1 f2
Sum and Difference Frequencies
When you superimpose two sine waves of different frequencies, you get components at
the sum and difference of the two frequencies. This can be shown by using a sum
rule from trigonometry. For equal amplitude sine waves
The first term gives the phenomenon of beats with a beat frequency equal to the
difference between the frequencies mixed. The beat frequency is given by
since the first term above drives the output to zero (or a minimum for unequal
amplitudes) at this beat frequency. Both the sum and difference frequencies are
exploited in radio communication, forming the upper and lower sidebands and
determining the transmitted bandwidth.
When you say that the beat frequency is f1-f2 rather than (f1-f2)/2, that requires some
explanation. For the difference frequency you can just say that you get a minumum
when the modulating term reaches zero, which it does twice per cycle, so that the
number of minima per second is f1-f2.
The beat frequency
refers to the rate at which the volume is heard to be oscillating from high to low volume.
For example, if two complete cycles of high and low volumes are heard every second,
the beat frequency is 2 Hz. The beat frequency is always equal to the difference in
frequency of the two notes that interfere to produce the beats. So if two sound waves
with frequencies of 256 Hz and 254 Hz are played simultaneously, a beat frequency of 2
Hz will be detected. A common physics demonstration involves producing beats using
two tuning forks with very similar frequencies. If a tine on one of two identical tuning
forks is wrapped with a rubber band, then that tuning forks frequency will be lowered. If
both tuning forks are vibrated together, then they produce sounds with slightly different
frequencies. These sounds will interfere to produce detectable beats. The human ear is
capable of detecting beats with frequencies of 7 Hz and below.
The diagram below shows several two-point source interference patterns.
The crests of each wave is denoted by a thick line while the troughs are
denoted by a thin line. Subsequently, the antinodes are the points where
either the thick lines are meeting or the thin lines are meeting. The nodes
are the points where a thick line meets a thin line
How is Sound Converted Into Electrical Signals?
All types of terminal devices used in electronic communications operate by
converting data into electrical signals. Telephones convert sound into electricity and
electricity into sound as they send signals over long distances.
Let's think about the microphone. Sound (vibrations in the air) causes vibration in
an acoustic membrane which applies distortion to a piezoelectric element which
generates electricity. The greater the vibration in the membrane, the greater the
distortion in the element, and the higher the voltage produced
The microphone uses audio vibrations to create distortion in a piezoelectric
element, where they are converted into electrical signals.
Crystals which acquire a charge when compressed, twisted or distorted are said to
be piezoelectric. This provides a convenient transducer effect between electrical
and mechanical oscillations
Radio frequencies occupy the range from a few hertz to 300 GHz, although
commercially important uses of radio use only a small part of this
spectrum. Other types of electromagnetic radiation, with frequencies above
the RF range, are infrared, visible light, ultraviolet,X-rays and gamma rays.
Since the energy of an individual photon of radio frequency is too low to
remove an electron from an atom, radio waves are classified as non-ionizin
radiation.