ULTRASONIC REFLECTIONS OF PRESSURE WAVES BASED UPON ACOUSTIC IMPEDANCE BETWEEN A BUFFER AND A SPECIMEN
Materials that are available as thin rods or wires are uniquely suited to magnetostrictive
delay line applications. Some anisotropic or non-homogeneous materials available as thin
rods, such as solid rocket propellant or samples of wood or bone, may be evaluated using
the magnetostrictive delay line technique. The magnetostrictive induced ultrasound will
permit comparisons between similar materials. In addition, this technique may be used in
the most hostile of environments provided that the specimen remains intact during the
measurement of the ultrasonic propagation velocities.
The magnetostrictive delay line technique is used for the measurement of both extensional
(compressional) and torsional (shear) wave velocities. The propagation of these waves is
non-dispersive if the wavelength is greater than five times the diameter of the thin rod. The
two velocities are then used, together with the mass density, to derive the various moduli.
In order to receive the two interface reflections (delay line to specimen, and specimen to air)
with nearly equal amplitudes for both wave velocities, a solution to a binomial equation is
required. Let the delay line to specimen interface reflection be "A" amplitude, and let the
specimen to air reflection be "B" amplitude and "C" the second echo from the specimen to
air interface. The assumptions for the boundary conditions may be found in "Pressure
Reflectivity and Transmissivity.
As a guide, the following is generally true for most metal specimens that are bonded to the
magnetostrictive delay line. The following summary will be developed later in this paper.
For extensional waves, the echo amplitudes from the specimen/delay line interface and
from the end of the specimen, A = B when the specimen diameter is approximately half the
line diameter. Then, A/B = 4 for torsional waves. Try for (A/B)ext . = (B/A)tors ., and the
specimen diameter will be about 0.5 times the delay line diameter for extensional waves and
about 0.7 times the delay line diameter for torsional waves. The optimum diameter ratio
for extensional waves is 2:1, and the optimum diameter ratio for torsional waves is 2 1/2 :1,
or 1.4:1. The average for the two calculations will be about 0.6 times the delay line
diameter. Non-metal specimens may have diameters approximating the diameter of the
delay line. The optimum impedance ratio is the governing factor for establishing the
optimum echo amplitudes. The length of a specimen should approximate an integer
multiple of a half wavelength.
This analysis is based upon an ultrasonic transducer that is attached to one end of a buffer
rod and the opposite end of the buffer rod is attached to a specimen. The acoustic
impedance of the buffer is Z1 ; the specimen is Z2 . The Reflection Coefficient (R) and the
Transmission Coefficient (T) corresponds to the pressure wave distribution on the outgoing
wave. The returning pressure waves, upon encountering the acoustic interfaces, have a
Reflection Coefficient of (R ' ) and a Transmission Coefficient of (T ' ).
Let: The acoustic impedance ratio for pressure waves: r = Z2/Z1
Then: R = ( r - 1 ) ( r + 1 ); T = 2 r/( r + 1 ); T = 1 + R
R ' = ( 1 - r ) ( 1 + r ) ; T ' = 2/( 1 + r );
T ' = 1 + R ' = 1 - R
The first interface echo, A1 , returning from the buffer-specimen interface, and the
subsequent echoes A2 ...AN (where N is the echo number) returning from the free end of
the specimen are related in amplitude by the corresponding acoustic impedances, (Z1 , Z2)
so long as the echoes are non-interfering with each other. The echo relative amplitudes,
assuming that the attenuation is negligibly small, are equal to:
A1
A2 = ( 1-R² ) = ( 1-R )( 1+R ) = TT'
A3 = R'( 1-R² )
AN = R' exp ( N-2 ) TT' Where N is >= 3
The end echo of a series of repeating acoustic interfaces will be proportional to
± R' exp N TT ' , depending upon whether the termination is "open" or "short" circuited,
respectively. If the attenuation is considered, the correction for the attenuation coefficient
in nepers per unit length, a, is exp ( -2 aL ) A2 ; exp ( -4 aL ) A3 . Also, see Appendix C.
Where: a = ln( |R A2 A3| )/2L.
The ability to match the reflected amplitudes of the echoes returning from the buffer-specimen
interface with the echo returning from the free end of the specimen would be
desirable. Small changes in the relative diameters of the buffer or the specimen will result
in large changes in the acoustic impedance, resulting in large and unconstrained variations
in the echo amplitudes. Such amplitude variations can cause one echo to saturate and at the
same time put the other echo below the noise threshold. Thin rod, or wire, technology in
the application of ultrasound easily avails itself to adjustment of the acoustic impedance.
As long as the compressional wavelength is greater than five times the diameter of the
buffer and the specimen, the waves will propagate without dispersion. The relationship of
the diameter (D) to wavelength (
) may be shown as follows:
D/ <= 0.2
D/ <= 1.2
Where l is the acoustic velocity divided by the acoustic frequency and the subscript
denotes the extensional (o), or the torsional (t) wave mode.
The cross sectional area or the polar moment of inertia, depending upon whether the wave
is propagating as longitudinal (extensional) or shear (torsional), respectively, influences the
acoustic impedance. For buffers and specimens that can be described as solid right circular
cylinders, the acoustic impedance is:
Zo = 
Co D²/4 Extensional waves
Zt = Ct D´/32 Torsinal waves
Where r is the density, D is the diameter, and C is the respective sound velocity. The
impedance for torsional waves take into account the polar moment of inertia for a solid right
circular cylinder, rather than the cross sectional area used to determine the impedance for
extensional waves. Notice the strong effect the diameter has on the acoustic impedance.
The determination of the optimum reflection coefficient may proceed from the reflected
intensity relationship and the solution of the quadratic equation as follows:
Let: r = Z2 / Z1
Then: R = [ ( r - 1 ) ( r + 1 )]²
R + T = 1
A = Io R (buffer/specimen interface echo amplitude)
B = Io T ² (specimen free end echo amplitude)
Let: A = B (equal amplitude reflections)
2
T2 = ( 1 - R )²
Io R = Io ( 1 - R ) 2 = Io ( R² - 2 R + 1 )
R = R² - 2 R + 1
Subtract R: 0 = R² - 3 R + 1
QUADRATIC SOLUTION:
R = ( 3 ± 5 exp½ )/2 = 2.618;0. 382 (wave intensity)
R exp½ = ( r - 1 )( r + 1 ) = ±1.619; ±0.618 (wave pressure)
Solve for r in Terms of R:
r = ( 1 + R exp½ ) (1 - R exp½ )
r = Z
2 /Z1 = ±4.236; r = Z2 /Z 1 = ±0.236
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Only the positive roots are used since the negative roots are imaginary.
Note: Buffer-Specimen relationship where the buffer diameter is equal to or greater than 30
: 2L/C BUFFER:= N(2thk)/C SPECIMEN , for bulk specimens also holds for thin rods with one
exception. Where N is the number of the echo; thk is the thickness of the specimen, and L
is the length of the buffer. In the application of the buffer to thin rod wave technology, the
large diameter buffer is not required. The buffer, or delay line, may be adjusted in diameter
to match the impedance of the magnetostrictive line on the one end and the specimen on the
other end. A tapered delay line is sometimes referred to as a transformer, or impedance
transformer.
PRESSURE REFLECTIVITY AND TRANSMISSIVITY ²
The assumption for the following analysis is that the waves are generated in the first
medium and travel into the second medium. Some of the pressure signal is reflected at the
interface of the two mediums, but once entering the second medium, the pressure signal
propagates without reflection. Assume the boundary at X = 0, so that the exponentials
e (exp) ± jkx = 1. The total pressure is the same on both sides of the boundary; The particle
velocity into the boundary equals the particle velocity out of the boundary on the other side.
The acoustic wave pressure distribution may be developed from transmission line theory.
The subscripts 1 and 2 refer to the first and second media.
Let:2/Z1 (Impedance ratio at boundary)
P2+ (Pressure striking the boundary)
P2- (Pressure reflected from the boundary)
P1 = ( P1+ P1 ); P2 = ( P2+ + P2- ) (Pressure amplitudes)
uX1 = 1/Z1 ( P1+ + P1- ) ; uX2 = 1/Z2 ( P2+ + P2- ) (Particle velocity)
From the assumed boundary conditions,
r P1+ - r P2 - = P2+
........ or ........ Z2 ( P1+ - P1- ) = Z1 P2+
r P1+ - r P1- = P1+ P1-
r P1+ - P1+ = r P1-+ P1-
P( r - 1 ) = P1- ( r + 1 )
P1- /P1+ = ( r - 1 ) ( r + 1 ) (normal incidence pressure reflectivity)
P2+ = P1+ + P1- (pressure partitioning)
P1- = P2+ - P1+
Z 2 ( P1+ - P1-) = Z1P2+
Z 2 ( P1+ - ( P2+ - P1+ ) ) = Z 1 P2+
Z 2 ( P1+ - P2+ + P1+ ) = Z 1 P2+
Z 2 ( 2 P1+ - P2+ ) = Z 1 P2+
2 Z 2 P1+ - Z 2 P2+ = Z 1 P2+
2 Z 2 P1+ = Z 2 P2+ + Z 2P2+ = P2+ ( Z 1 + Z 2 )
P2+ /P1+ = 2 Z 2/ ( Z 1 + Z 2 ) = 2 r / ( r + 1) ; (normal incidence pressure
transmissivity)
Full size picture
( C ²o - 4C ²t )/(C ²o - 3C ²t )
and 
Where k = Radius of gyration
for rectangular bars,
flexural vibration mode
Plus Harmonics
