NDTnet - March 1996, Vol.1 No.03

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Chapter III

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    A particularly useful technique for measuring ultrasonic velocities in a variety of materials at elevated temperatures was established at the laboratory by Dunegan ll from a compilation of experiments performed by McSkimin,l2 and Reynolds.l3 This technique required special machining of the specimens into two concentric continuous solid cylinders of different diameters and lengths. The larger diameter cylinder was threaded with 32 threads per inch to prevent interfering reflections from the circumference. This specimen allowed simultaneous measurement of the longitudinal and shear velocities from very low temperatures to about 870°C. Special furnaces and cooling collars would have to be assembled before the test could be repeated. The technique worked out by Reynolds is shown on the following page:

    The rod should be machined as a smooth, solid, right-circular cylinder with perpendicular and parallel faces. Solve for the shear velocity through transcendental equations. Nominal dimensions for most metals are 16 mm diameter by 64 mm long.

    Full size picture

    Ultrasonic pulse echo waveform showing the initial pulse with longitudinal echoes and DTs.

    In another technique, the ultrasonic velocities were measured through the various axes of BeO, ZnO, and CdSe, hexagonal single crystals, for the purpose of calculating the elastic constants.14 This technique was adopted from work performed by McSkimin l2 and involved the measurement of velocities through very small single crystals mounted to the end of a special quartz buffer rod. The longitudinal velocity was also measured with the aid of a water collimator-transducer system. These measurements used a tone burst generator with an adjustable pulse duration. The frequency tuning and long pulse duration resulted in ultrasonic echo interferometry. Constructive and distructive interference alternated with an increasing frequency as a result of specimen resonance.

    Acoustic analysis of solid materials catagorized as thin rods, thin plates, or thin bars is often performed somewhat differently from the earlier discription on bulk materials. Sound generated into thin material may have a wavelength greater than five times the smallest material dimension. A typical test material might be wires. An analysis of the wave properties in uniform diameter thin materials such as wire is shown below:

    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

    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:

    A2 = ( 1-R² ) = ( 1-R )( 1+R ) = TT'
    3 = 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
    t = 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

    Let: A = B (equal amplitude reflections)
      Io R = Io T2
      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


    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 > 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.

    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.


      r = Z2/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,


    Rp = ( r - 1 ) ( r + 1 )
    p = 2r/( r + 1 )


    RE ( r - 1 ) ² / ( r + 1 )²
    E = 4 Z 2 Z1 / ( Z2 + Z 1 )² = 4 r / ( r + 1 ) ²

    The acoustic measurements for the extensional and torsional (shear) wave velocities may be used to calculate the various moduli in homogeneous, isotropic solids. The following Table of Material Properties provides a listing of typical formulae for determining the moduli.


    Poisson's Ratio: n = [ C²o (2 C ²t ] - 1
    Young's Modulus: E = C²o
    Shear Modulus: G = C²t
    Bulk Modulus: K = C ²o t /[3(3C²t - C²o ) ]
    Lame Constant: = vE /[(1 + v ) ( 1 - 2 v ) ]
    Bulk Compressional Velocity: C l = Ct( C ²o - 4C ²t )/(C ²o - 3C ²t )



    Acoustic and ultrasonic waves may propagate in relatively thin plates, in terms of the wavelength of the propagating sound. The sound waves will propagate as resonances of the structure and may, under some circumstances, be highly dispersive. There will often be either plane harmonic waves or overtones formed within such structures and these waves become the subject of Modal analysis and Acoustic Emission analysis. The following Table suggests some of the modes of sound propagation found in thin wall structures:


    Mode Displacement fD = frequency X thickness, m = integer
    Symmetric Normal ( 2 m - 1 ) (C s / 2 )
    Asymmetric Normal 2 m ( C s / 2 )
    Symmetric Normal ( 2 m - 1 ) ( C l/ 2 )
    Asymmetric Normal 2 m ( Cl / 2 )
    Symmetric In Plane 2 m ( Cs / 2 )
    Asymmetric In Plane ( 2 m - 1 ) Cs / 2 )
    Symmetric In Plane mCl Cs /( C²l - C²s) exp ½
    Asymmetric In Plane xm Cl Cs / [l - C²s )exp½ ]*

    * Transcendental solution for Xm :

    tan x = bx, where
    Solve for the roots,

    Flexural Waves

    Criteria: Width >> Thickness; Waves are highly dispersive; Cf oo f; the larger the area, the greater the coupling to fluids.

    k = Radius of gyration: Plates: a /12
    a = Thickness
    Y = Young's modulus
    w = 2 phi f
    n = Poisson's Ratio
    r = density

    Rectangular Bar Clamped at One End

    Where k = Radius of gyration
    for rectangular bars,

    where a = Thickness of bar in the direction of vibration.
    k = a/2 for round bars, where a = diameter
    f2 = 6. 267 f1
    f3 = 17. 55 f1 For the second through fourth tone
    f4 = 34.39 f1

    Bar, Both Ends Free

    flexural vibration mode

    Longitudinal Vibration of Bars

    Plus Harmonics

    Circular Plate unsupported (Free)

    Where: t = thickness, R = radius, v = Poisson's Ratio Modes of Vibration

    Circular Plate Lightly Supported at the Outside Edge

    Circular Plate Clamped at Outside Edge

    With Overtones

    f02 = 3.91f01 ; f03 = 8.75f01 ; f11 = 2.09f01 ; f21 = 3.43f01 ; f12 = 5.98f01

    Pipes, Closed One End

    f 1 = C / = C /4L

    Open Pipes

    f 1 = C / = C /2L
    You can continue with Chapter IV, and start with: CONCLUSIONS

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    Rolf Diederichs 13.Febr.1996, info@ndt.net

    /DB:Article /AU:Brown_A_E /CN:US /CT:UT /ED:1996-03