Bulk Waves, Surface Waves, and

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4. ??? ??
Bulk Waves, Surface Waves, and Plate Waves
and Some of their Properties
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Learning Objectives Bulk waves Mode
conversion Critical angles Angle beam shear
wave transducer Rayleigh (surface) waves Lamb
(plate) waves
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Mode Conversion
incident
reflected
fluid
P-wave
P-wave
q
q
r
c
,
p1
p1
p1
1
p
p
v
v
inc
reflt
x
p
v
trans
solid
q
transmitted
r
p2
c
c
,
,
s
f)
p2
s2
2
P-wave (
v
q
trans
s2
y
transmitted
y
)
S-wave (
Generalized Snell's law
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Critical Angles
P
P
P
qp1
qp1
qp1
P
S
S
inhomogeneous P-wave
inhomogeneous P, S waves
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Angle Beam Shear Wave Transducer
weld inspections
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Rayleigh (Surface)Wave Transducer
stress-free surface
In the late 1800's Lord Rayleigh looked for a
wave confined near the stress-free surface of an
elastic solid of the form
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x
y
By satisfying the equations of motion
and the stress-free boundary conditions
on y 0
Rayleigh found that the wave speed, c, must
satisfy
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Rayleigh's equation
There is always one real root of this equation,
where
A good approximation of this root is
Poisson's ratio
The Rayleigh wave travels about 90 of the shear
wave speed
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displacements
stresses
tyy
ux
txx
txx
uy
uy
ux
txy
tyy ,
txy
The depth of penetration is a function of the
frequency
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Lamb (Plate) Waves
w 2pf frequency (rad/sec)
2h
If one looks for solutions of the form
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then solutions of the following two types are
found
extensional waves
flexural waves
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satisfying the boundary conditions
on
gives the Rayleigh-Lamb equations
extensional waves - flexural waves
There are multiple solutions of these equations.
For each solution the wave speed, c, is a
different function of frequency. Each of these
different solutions is called a "mode" of the
plate.
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consider the extensional waves
If we let
(high frequency)
then both tanh functions are
and we find
so we just have Rayleigh waves on both
stress-free surfaces
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In contrast for kh ltlt1 (low frequency)
we find
and the Rayleigh-Lamb equation reduces to
which can be solved for c to give
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fundamental extensional mode
c
phase velocity (m/s)
c cR
c cplate
fundamental flexural mode
frequency-thickness (MHz-mm)