Acoustic waves - VGTU Elektronikos fakultetas

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ELEKTRONIKOS ĮTAISAI
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Acoustic Waves
Acoustic wave: A longitudinal wave that (a) consists of a sequence of
pressure pulses or elastic displacements of the material, whether gas, liquid, or
solid, in which the wave propagates. In solids, the wave consists of a sequence
of elastic compression and expansion waves that travel though the solid.
In acoustic wave devices acoustic waves are transmitted on a miniature solid
substrate.
In a crystalline solid a sound wave is transmitted as a result of the
displacement of the lattice points about their mean position. The wave is
transmitted as an elastic wave. (Elastic substance is able to return to its
original shape or size after being pulled or pressed out).
The term sound wave is sometimes confined to waves with the frequency
falling within the audible range of the human ear, i. e. from about 20 Hz to
20 kHz. Waves of frequency greater than 20 kHz are ultrasonic waves. Waves
of frequency 109–1013 Hz are called hypersonic waves.
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Acoustic Waves
There are various kinds of acoustic waves. Bulk acoustic waves are acoustic
waves propagated through the bulk substrate material.
If the motions of the matter particles conveying the wave are perpendicular to
the direction of propagation of the wave itself, we have a transverse wave.
If the motion of particles is back and forth along the direction of propagation, we
have a longitudinal wave.
Surface acoustic waves propagate along the surface of a substrate. There are
some types of the surface acoustic waves. In the case of the Rayleigh waves
particles in the surface layer move both up and down and back and forth tracing
out elliptical paths.
Acoustic waves
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Acoustic Waves
Let us consider that longitudinal vibrations are excited in a rod.
d m  S d x
2
a 
d s
dt
Hooke’s law:
2
 s
t
2
2

E  s
 x
2
ds
dx

F
F
ES
x

2
dF
dF 

dm
x
dx
1 F
 S x
2
 ES
 s
x
2
s ( x , t )  A exp[ j  ( t  x / v l )]  B exp[ j  ( t  x / v l )]
vl 
E/
  E
Here s is displacement; σ is stress.
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F
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s
x
.
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Acoustic Waves
Mechanical waves are described using terms such as wavelength, period,
frequency, amplitude and speed of propagation.
Mechanical vibrations propagate along the rod as an elastic wave. The
longitudinal displacement is given by the sum of waves that propagate in
opposite directions:
s ( x , t )  A exp[ j  ( t  x / v l )]  B exp[ j  ( t  x / v l )]
The displacement causes the longitudinal mechanical stress given by
s

 ( x, t )  E
 j
E  A exp[ j  ( t  x / v l )]  B exp[ j  ( t  x / v l )] .
x
vl
The speed of the longitudinal wave is given by
vl 
E/
The velocity is about 105 times less than the velocity of the
electromagnetic wave. So using mechanical waves we can obtain
sufficient delay time at short distances.
 
vl
f
 v lT
– the length of the acoustic wave at radio frequency is
short.
At v = 5000 m/s and f = 1 MHz,  = 5 mm.
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Acoustic Waves
Let us consider that the length of the rod is l and its left end is fixed. Then
B  A
s (0, t )  0
s ( x , t )  C sin
x
e
j t
vl
 ( x, t )  C

E cos
vl
 (l , t )  0
cos
l
0
 ln
4
2n  1 vl
4

j t
π
Natural frequencies:
2
E
4l

 ln 
vl
f

4
2n  1
l
At natural frequency odd number of wavelength quarters
fit in the length of the rod.
Vibrations of a rod
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 ( 2 n  1)
( 2 n  1)
l
e
vl
vl
vl
f ln 
l  ( 2 n  1)
l
x
Vibrations of a rod 1
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Acoustic Waves
l  ( 2 n  1)
 ln
l  n
4
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λ ln
2
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Flexural Bending Mode Shapes and Boundary Conditions
Dan Russell, Ph.D., Applied Physics, Kettering University
All text and images on this page are ©2006 by Daniel A. Russell and may not used in other web pages or reports without permission
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Dan Russell, Ph.D., Applied Physics, Kettering University
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Acoustic Waves and Mechanical Resonators
1. Mechanical resonators are small and have high quality (to 107).
2. Because resonant frequency is dependent on the resonator dimensions
and elasticity modulus, materials having small thermal expansion
coefficient and stable elasticity modulus must be used for mechanical
resonators.
3. The velocity of acoustic wave is dependent on the type of the wave.
4. The natural frequencies depend on the types of acoustic wave.
5. A mechanical resonator have many resonant frequencies dependent on
the number n, type of the acoustic wave and other factors. This property
is not desirable in electronic applications.
6. The desirable type of vibrations may be selected by the selection of the
form of a resonator and the method used for excitation of vibrations.
Rectangular
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Circular
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Dan Russell, Ph.D., Applied Physics, Kettering University
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2009
Dan Russell, Ph.D., Applied Physics, Kettering University
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ELEKTRONIKOS ĮTAISAI
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2009
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