U.S. Department of the Interior

U.S. Geological Survey

Scientific Investigations Report 2009–5142

High-Frequency Normal Mode Propagation 

in Aluminum Cylinders

TIME, IN MICROSECONDS

100-mm-long 

cylinder

200-mm-long 

cylinder

RELATIVE AMPLITUDE

–1,00015 30 45 60 75

0

1,000

2,000

P-wave crystal

S-wave crystal

Shear wave

P100–1 P100–2

P200–1 P200–2

P200–3

P100–3

High-Frequency Normal Mode Propagation 

in Aluminum Cylinders

By Myung W. Lee and William F. Waite

Scientific Investigations Report 2009–5142

U.S. Department of the Interior

U.S. Geological Survey
U.S. Department of the Interior

KEN SALAZAR, Secretary

U.S. Geological Survey

Suzette M. Kimball, Acting Director

U.S. Geological Survey, Reston, Virginia: 2009

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Suggested citation:

Lee, M.W., and Waite, W.F., 2009, High-frequency normal mode propagation in aluminum cylinders: U.S. 

Geological Survey Scientific Investigations Report 2009–5142, 16 p.
iii

Contents

Abstract...........................................................................................................................................................1

Introduction.....................................................................................................................................................1

Measuring Waveforms in Aluminum Cylinders ........................................................................................2

Theory..............................................................................................................................................................4

Wave Equation Solution.......................................................................................................................4

Phase Velocity.......................................................................................................................................4

Group Velocity .......................................................................................................................................5

Normal Mode Amplitude .....................................................................................................................5

Amplitude of P-Body Wave .................................................................................................................7

Normal Mode and Body-Wave Amplitudes for P-Waves ..............................................................7

S-Wave Amplitude................................................................................................................................8

Discussion.......................................................................................................................................................8

P-Wave Group Velocity........................................................................................................................9

S-Wave Group Velocity........................................................................................................................9

P-Wave Amplitudes............................................................................................................................10

S-Wave Amplitudes............................................................................................................................11

P-Wave Group Velocity: Dependence on Cylinder Radius ..........................................................12

Conclusions...................................................................................................................................................12

Acknowledgments.......................................................................................................................................14

References Cited..........................................................................................................................................14

Appendix A. Low-Frequency and Large Diameter Approximation ..................................................16

Figures

1. Measurement equipment............................................................................................................2

2. Waveforms for 35-mm-diameter and 100-mm and 200-mm nominal-length 

aluminum cylinders surrounded by air at zero effective pressure.......................................3

3. Numerical derivation of theoretical phase velocities for a 35-mm-diameter 

aluminum rod.................................................................................................................................6

4. Relation between wave number and frequency for fundamental and first-order 

normal modes for a 35-mm-diameter aluminum cylinder......................................................7

5. Calculated amplitude of the dispersive P-wave fundamental order normal mode 

for a 35-mm-diameter, 100-mm-long aluminum cylinder........................................................8

6. Group velocities for 35-mm-diameter aluminum cylinders....................................................9

7. Dispersive shear-wave normal mode analysis......................................................................10

8. Predicted relative axial amplitudes for the P-body wave and fundamental and 

first-order P-wave normal modes in a 35-mm-diameter, 200-mm-long 

aluminum cylinder ......................................................................................................................11

9. Predicted relative radial amplitudes of S-wave arrivals for a 35-mm-diameter, 

200-mm-long aluminum cylinder..............................................................................................11

10. Predicted relative radial amplitudes of the fundamental and first-order P-wave

normal modes for a 35-mm-diameter, 100-mm-long aluminum cylinder...........................12
iv

11. Normal mode dependence on cylinder radii..........................................................................13

12. Predicted relative amplitudes for the P-body wave and fundamental and 

first-order normal modes in a 25.4-mm-diameter, 200-mm-long 

aluminum cylinder ......................................................................................................................14

Table

1. Velocities and dominant frequencies for normal mode arrivals in 

aluminum cylinders ......................................................................................................................3
High-Frequency Normal Mode Propagation in Aluminum 

Cylinders

By Myung W. Lee and William F. Waite

Abstract

Acoustic measurements made using compressional-wave 

(P-wave) and shear-wave (S-wave) transducers in aluminum 

cylinders reveal waveform features with high amplitudes 

and with velocities that depend on the feature’s dominant 

frequency. In a given waveform, high-frequency features 

generally arrive earlier than low-frequency features, typical for 

normal mode propagation. To analyze these waveforms, the 

elastic equation is solved in a cylindrical coordinate system for 

the high-frequency case in which the acoustic wavelength is 

small compared to the cylinder geometry, and the surrounding 

medium is air. Dispersive P- and S-wave normal mode propagations are predicted to exist, but owing to complex interference patterns inside a cylinder, the phase and group velocities 

are not smooth functions of frequency. To assess the normal 

mode group velocities and relative amplitudes, approximate 

dispersion relations are derived using Bessel functions. The 

utility of the normal mode theory and approximations from a 

theoretical and experimental standpoint are demonstrated by 

showing how the sequence of P- and S-wave normal mode 

arrivals can vary between samples of different size, and how 

fundamental normal modes can be mistaken for the faster, but 

significantly smaller amplitude, P- and S-body waves from 

which P- and S-wave speeds are calculated.

Introduction

Acoustic transmission through soil is strongly influenced 

by the nature of the intergranular contacts (Santamarina and 

others, 2001). Crystalline solids such as gas hydrate forming in the pore space of unconsolidated sediment can stiffen 

intergranular contacts or become load-bearing components of 

the sediment (Sloan and Koh, 2007), increasing the acoustic velocity through the sediment (Guerin and others, 1999; 

Helgerud and others, 1999; Lee and Waite, 2008; Petersen 

and others, 2007; Yuan and others, 1999). The dependence of 

acoustic velocity on pore-space gas hydrate content depends 

on the transmitted wave’s propagation mode (Yun and others, 

2006) and frequency (Bauer and others, 2005; Priest and others, 2005), the hydrate morphology and degree of saturation 

within the pore space (Dvorkin and others, 2000; Kingston and 

others, 2008; Yun and others, 2005), as well as the porosity 

and lithology of the hydrate-bearing sediment itself (Guerin 

and Goldberg, 2005).

The gas hydrate and sediment test laboratory instrument 

(GHASTLI) (Winters and others, 2000) is one example of a 

laboratory system designed to investigate gas hydrate-bearing 

sediment, which can be found in marine and subpermafrost 

environments (Waite and others, 2004; Winters and others, 

2004; Winters and others, 2007). GHASTLI is instrumented 

with compressional- and shear-wave transducers for pulse 

transmission through gas hydrate-bearing soil samples. 

Because it is designed for triaxial measurements of soil 

strength, GHASTLI requires cylindrical samples with lengths 

between 2 and 2.5 times their diameter (American Society for 

Testing and Materials, 2003). Correctly identifying observed 

compressional- and shear-waveform arrivals in these relatively 

long, narrow cylinders is complicated by the appearance of 

multiple high-amplitude features in the measured waveform 

(Waite and others, 2004, 2008).

To identify the observed arrivals and isolate geometric 

controls on the waveform shape from porous media effects, we 

carried out a series of measurements on aluminum cylinders of 

varying length and diameter. Based on the relation between the 

velocity, frequency, and amplitude of the waveform features, 

we can identify the high-amplitude features using a normal 

mode approach.

Normal mode solutions for layered Earth models have 

been investigated extensively, and many theories are available (for example, Ewing and others, 1957; Brekhovskikh, 

1960; Aki and Richards, 1980). Many wave propagation 

theories in thin cylinders at low frequencies, such as for bar 

waves, have also been developed and extensively analyzed by 

Kolsky (1963) and Graff (1975). However, these theories are 

not applicable to high-frequency measurements on relatively 

long, thin, cylindrical samples. Here we present a theory to 

calculate normal mode velocities and amplitudes and compare 

these results with values measured on aluminum cylinders. We 

derive the amplitude and dispersion relation for normal mode 

velocities using a cylindrical coordinate system, assuming a 

solid cylindrical body of infinite length surrounded by air, with 

a radius exceeding the acoustic wavelength. 
2 High-Frequency Normal Mode Propagation in Aluminum Cylinders

Measuring Waveforms in Aluminum 

Cylinders

In GHASTLI, compressional- and shear-wave acoustic 

waveforms are generated and measured using piezo-electric 

transducer crystals, hereafter referred to as “crystals,” 

mounted in cylindrical titanium endcaps (fig. 1A). Each endcap contains an axially positioned, disk-shaped shear-wave 

(S-wave) crystal. The compressional-wave (P-wave) crystal, 

which measures the waveform’s axial displacement, is an 

annular ring surrounding the S-wave crystal (fig. 1B), which 

measures radial displacements. The crystals are epoxied to 

the back of a 6.35-mm-thick titanium wear plate, the front of 

which is coupled to the aluminum cylinder with Panametrics 

shear-wave couplant.

A Panametrics Model 5800 Pulsar-Receiver generates 

the 50-volt pulse used to excite either the P- or S-crystal in 

one endcap. The pulse traverses the sample, is received by the 

corresponding P- or S-crystal in the second endcap, displayed 

on a Tektronix TDS-340 oscilloscope, and stored to disk. For 

each experimental setup, the signal generation and measurement electronics are connected first to the P-crystals; then 

once the P-waveform has been recorded, the electronics are 

connected to the S-crystals to obtain the S-waveform. For both 

the P-crystal pairing and the S-crystal pairing, the waveform 

is essentially independent of which crystal in the pair is the 

signal transmitter and which is the receiver.

Measured waveform arrival times contain a system delay 

time that depends on the wiring configuration, endcap, and 

crystal pair. This delay, which is the time required for a signal 

to propagate through the electronics, crystals, and endcap wear 

plates, must be subtracted from the measured arrival times 

before the velocity of a given waveform feature through the 

sample alone can be calculated. These delay-corrected velocities are given in table 1.

Figure 2 shows measured waveforms for 100-mm and 

200-mm nominal-length aluminum cylinders (actual lengths 

are 101.30 mm and 202.55 mm). The cylinders are 35 mm in 

diameter and surrounded by air. There are three distinct axial 

displacement features measured by the P-wave transducer for 

each aluminum cylinder 

To distinguish between features, the features are designated such that P100-3 indicates the third feature for the 100-

mm aluminum cylinder measured by the P-wave transducer. 

The measured group velocity and dominant frequency for each 

feature is listed in table 1.

The shear waveforms for the 100- and 200-mm-long 

cylinders (fig. 2) indicate the first, second, and third axial displacement P-wave features are also transmitted and recorded 

by the S-wave crystals, though less efficiently. This generation 

and detection of axial displacements by S-wave transducer 

crystals results from their imperfect polarization in the radial 

direction (Ayling and others, 1995). This imperfect polarization can also cause P-wave crystals to generate and detect 

radial displacements.

Figure 1. Measurement equipment. Within 

each endcap (A), the shear-wave (S-wave) and 

compressional-wave (P-wave) transducer crystals, 

shown schematically in (B), are epoxied to the inner 

surface of a wear plate that contacts the sample. The 

100-mm-long, 35-mm-diameter aluminum cylinder is 

shown here, with shear-wave couplant between the 

aluminum and the endcap wear plates. A C-clamp holds 

the system together.

For the purpose of deriving a tractable mathematical 

model of this system, the P- and S-wave signals are assumed 

to be generated from a point source on the cylindrical axis. For 

the annular P-wave crystals (fig. 1B), the validity of assuming P-waves are generated and received on-axis is supported 

by the correspondence between the relative timing of P-wave 

features transmitted and received using the annular P-wave 

crystals, and those transmitted and received by the on-axis 

S-wave crystals (fig. 2). When modeling the S-wave signal, 

however, we must assume the S-wave’s radial motion is measured across the full diameter of the crystal.

A

B
Measuring Waveforms in Aluminum Cylinders 3

Table 1. Velocities and dominant frequencies for normal mode arrivals in aluminum cylinders.

[DPNM and DSNM, dispersive P- and S-wave normal modes, respectively; * implies the two separate arrivals (body 

wave and fundamental normal mode) cannot be distinguished because of their similar velocities and relatively short 

sample lengths; PT, thin aluminum rod. Two measurement suites were run for this study: the 2006 measurements 

utilized transducer crystals with slightly higher frequency content than the crystal set used in the 2007 measurements]

Diameter/

length (mm)

Year 

measured

Waveform

feature 

Velocity

(m/s)

Dominant

frequency

(MHz)

Waveform feature

identification

35/101.3 2006 P100-1* 6,360 0.78 Fundamental order DPNM

35/101.3 2006 P100-2 3,890 0.49 First-order DPNM

35/101.3 2006 P100-3 2,800 0.44 First-order DSNM 

35/202.55 2006 P200-1* 6,370 0.88 Fundamental order DPNM

35/202.55 2006 P200-2 4,840 0.59 First-order DPNM

35/202.55 2006 P200-3 3,910 0.79 Second-order DPNM

35/101.3 2006 SW100 3,190 0.76 Body S-wave 

35/101.3 2007 S100-1 3,150 0.46 Fundamental order DSNM

35/101.3 2007 S100-2 2,690 0.42 First-order DSNM

35/202.55 2007 S200-1 3,090 0.42 Fundamental order DSNM

35/202.55 2007 S200-2 2,740 0.41 First-order DSNM

25.4/202.55 2007 PT200-1 6,380 0.95 Fundamental order DPNM

25.4/202.55 2007 PT200-2 5,160 0.81 First-order DPNM

Figure 2. Waveforms for 35-mm-diameter and 100-mm and 200-mm nominal-length aluminum cylinders 

surrounded by air at zero effective pressure. To account for the different system delay times when using 

the S-crystals rather than the P-crystals, the shear-transducer waveforms (green curves) were shifted 

left by 2.68 microseconds relative to the P-transducer signals (black curves). This shift aligns the first 

detectable transmission of energy in each waveform for a given sample, highlighting the correspondence 

between the relative arrival times of P-wave features in both signals. Because of the shift, however, the 

plotted arrival times for the shear-waveform features are not used to determine wave speeds.

TIME, IN MICROSECONDS

100-mm-long 

cylinder

200-mm-long 

cylinder

RELATIVE AMPLITUDE

–1,00015 30 45 60 75

0

1,000

2,000

P-wave crystal

S-wave crystal

Shear wave

P100–1 P100–2

P200–1 P200–2

P200–3

P100–3
4 High-Frequency Normal Mode Propagation in Aluminum Cylinders

Theory

Identifying the waveform features listed in table 1 

requires a theoretical relation between each normal mode’s 

dominant frequency and group velocity. We derive that relation in three steps:

1. Derive a solution of the wave propagation equation for 

cylinders.

2. Derive the dispersion relation to obtain the normal mode 

phase velocity dependence on frequency.

3. Use the approximate phase velocity at high frequency to 

calculate the group velocity for each normal mode.

As an independent check on the normal mode approach, 

we also derive the relative amplitudes of the normal modes as 

well as of the P- and S-body waves.

Wave Equation Solution

Under the assumption of axial symmetry, differential 

equations for wave propagation within an infinitely long cylinder can be written using a cylindrical coordinate system (r, θ, 

z) as (for example, Biot, 1952; Lee and Balch, 1982):

∂

∂

+

∂

∂

+

∂

∂ = ∂∂

2

2

2

2 2

2

2

Φ 1 Φ Φ 1 Φ

r r r z t 

∂

∂

+

∂

∂

− +

∂

∂ = ∂∂

2

2 2

2

2 2

2

2

Ψ Ψ Ψ 1 Ψ 1 Ψ

r r r r z t 

 

 (1) , (1)

 and U

z r r z = ∂

∂

+

∂

∂

+ U Φ Ψ Ψ , r z r = ∂

∂ − ∂∂

Φ Ψ

where Φ and Ψ are scalar potentials for P- and S-waves, 

respectively; Ur and Uz are radial and axial displacement, 

respectively; and α, β, and t are P-wave velocity, S-wave 

velocity, and time, respectively. The formal solution of equation 1 in the frequency domain can be written as follows when 

the P-wave source is located at the origin with the strength Vo,

and the S-wave source is located at the origin with the strength 

So:

Φ = + ∫ ∫ − − AJ lr e dk V H lr e dk ikz

o

ikz

0 0

2 ( ) ( ) ( )

Ψ = + ∫ ∫ − − BJ mr e dk S H mr e dk ikz

o

ikz

1 1

2 ( ) ( ) ( ) , (2)

with 

Φ Φ ( ) ( )   = −∞

∞

∫ t e dt i t

l k = − k i k k ⎛

⎝

⎜

⎞

⎠

⎟ > = − − ⎛

⎝

⎜

⎞

⎠

⎟ <





  





 

2

2

2

1 2

2

2

2

/

for / sgn for /

m k = − k i k k ⎛

⎝

⎜

⎞

⎠

⎟ > = − − ⎛

⎝

⎜

⎞

⎠

⎟ <



   



  

2

2

2

1 2

2

2

2

/

for / sgn for /

 k c =  /

where w is an angular frequency, c is the phase velocity, and 

Jn and Hn

( ) 2 are the order n Bessel function and order n Hankel 

function of the second kind, respectively. Note that uniformly 

expanding P- and S-wave source waves are assumed for the 

wave equation solution. 

We use two boundary conditions in the solution of equation 2: vanishing radial stress, prr= 0, and vanishing tangential 

stress, prz = 0 at the cylinder surface. Stresses are given by:

 p r r z rr = − +

∂

∂ − ∂

∂ ∂









2

2

2

2

2

2 Φ Φ Ψ ( )

 , (3)

 

p

r z z rz = ∂∂ ∂ − − ∂

∂ 



 (2 2 )

2 2

2

2

2

Φ Ψ Ψ

 

where l and are the Lamé parameters.

Substituting boundary conditions into equation 3, the following matrix equation can be derived:

 Q

a a

a a

A

B

S

S [ ] ≡ ⎡

⎣

⎢ ⎤

⎦

⎥

⎡

⎣

⎢ ⎤

⎦

⎥ = ⎡

⎣

⎢ ⎤

⎦

⎥ 11 12

21 22

1

2

 (4)

with

a k J la lJ la a 11

2 2 2

0

2

1 = − +     ( ) ( ) ( ) / 2 2

a ik mJ ma J ma

a 12 o

1 = 2 − ⎛

⎝

⎜ ⎞

⎠  ⎟ ( ) ( )

a i k lJ la 21 1 = 2  ( )

a k J ma 22

2 2 2

1 = −    ( ) ( ) 2

S V k H la V lH la a 1 o o

2 2 2

0

2 2

1

2 = −    ( ) ( ) 2 − 2  ( ) / ( ) ( )

− ⎡ − ⎣ ⎤

⎦ 2 0

2

1

2 ik S mH ma H ma  o ( ) ( ) /  ( )

S V iklH la S k H ma 2 o 1 o

2 2 2 2

1

2 = − 2 − −    2 ( ) ( ) ( / ) ( )

Phase Velocity

The dispersion relation, meaning the normal mode phase 

velocity dependence on frequency, is given by requiring the 

determinant of equation 4 to be zero (Ewing and others, 1957). 

The determinant is given by a a a a 11 22 12 21 − :

Δ = 4 − + 2 2 4

1 0 1 k lJ la mJ ma J ma a   ( )[ ( ) ( ) / ]

 (5)

      2 2 2 2

1

2 2 2

0

2

1 ( ) ( )[( ) ( ) ( ) / ] 2k J ma k J la lJ la a − 2 − + 2

in agreement with the equation shown by Graff (1975).

Using the small argument approximation of the 

Bessel functions in equation 5, we derive the low-frequency 
Theory 5

dispersion relation in Appendix A, recovering the bar-wave 

velocity in a cylinder (Kolsky, 1963) and the Rayleigh surfacewave dispersion relation for large-diameter cylinders (White, 

1965). The low-frequency corroboration with well-known 

wave behavior provides an important check on our theoretical derivation. As described in Appendix A, however, for the 

high-frequency range of interest here, our observed waveform 

features (fig. 2) are neither the bar waves nor the extensional 

mode waveforms described by Kolsky (1963) and Graff 

(1975).

For the frequencies and length scales of interest here, 

the dispersion relation for the dispersive P-wave normal 

mode (DPNM) is derived by requiring the determinant given 

in equation 5 to be zero and the P-wave normal mode phase 

velocity to be greater than α. This restriction forces l and m

to be real numbers. There is no simple analytic solution for 

finding the zeroes of equation 5. The dispersion relation can 

be numerically calculated, however. The phase velocity of a 

P-wave normal mode, Ca, is given by ∆ = 0 and can be written 

as:

( ) ( )[ ( ) ( )]

( )[(

1 * * 2

4 2

2

2

3

2

1 0

1

1

2

−

=

− 















C

C

l J la m J ma J ma

a

J ma

C



  2 0

2

1 1 2

− + ) ( ) ( )]

*

J la l J la

a

, (6)

where l C * = −1 / 2 2   and m C * = −1 / 2 2   . Allowable 

phase velocities are those for which the left and right sides of 

equation 6 are equal for a given phase velocity. From equation 

6, the normal mode phase velocity depends on the acousticwave frequency, the P- and S-wave speed of the sample, and 

the sample’s geometry.

Figure 3 shows an example of calculating phase velocity 

using equation 6. Figure 3A shows phase velocities for the first 

three P-wave normal modes in a 35-mm-diameter aluminum 

cylinder at two different frequencies, calculated by determining the locations of intersections of the left and right sides 

of equation 6. To develop the complete dependence of phase 

velocity on frequency, the process illustrated in figure 3A must 

be repeated for all frequencies in the range of interest. For the 

fundamental mode, the result is plotted in figure 3B. 

The form of equation 6 is not practical for computing 

normal mode phase velocities. For the sake of simplicity at 

high frequencies, the measured dispersion relation can be 

approximated by setting either J la 1 ( ) = 0 or J la 0 ( ) = 0 in 

equation 6 for the P-wave normal modes. When using the 

approximate dispersion relation, the phase velocity (Ca) of a 

dispersive P-wave normal mode (DPNM) at high frequency 

is given by incorporating the high-frequency approximation 

of the Bessel functions, J la al al 0 ( ) / ( ) cos( / ) = 2   − 4 and

J la al al 1 ( ) / ( ) cos( / ) = 2  − 3 4  (Watson, 1966), yielding:

C a

a 







  = −

2 2

2 2 2 2 Λ , (7)

with Λ = + ( / ) 3 4  n , if J la 0 ( ) is set to zero, or 

Λ = + ( / ) 5 4  n , if J la 1 ( )is set to zero. Results from both 

approximations are given in figure 3B. The exact fundamental 

mode phase velocities calculated from equation 6 (solid curve) 

are close to the approximate solution derived from setting 

J la 0 ( ) = 0 with n = 0 (dashed curve), and the approximation 

improves with increasing frequency. The approximation solution using J la 1 ( ) = 0 with n = 0 (dotted curve) is much higher 

for all frequencies. The slope of the J la 1 ( ) = 0 curve more 

closely approximates the slope of the exact solution, which is 

an important characteristic for estimating the group velocity.

Group Velocity

Group velocities (G) are determined from the phase 

velocity (Ewing and others, 1957) according to:

 G d

dk

C k dC

dk

= = +

 (8)

Using the equation 7 approximation, the P-wave normal mode 

group velocity, Ga, is: 

G d

dk a 

 









 = = − = − 1  1

2 2

2 2

2

2

Λ Λ

Π , (9)

with Π = a / .

The dispersion relation for the DPNM requires both Bessel function approximations because the J la 0 ( ) term in the 

determinant dominates the J la 1 ( ) term in certain frequency 

ranges, the J la 1 ( ) term dominates the J la 0 ( ) term in other 

frequency ranges, and there are some frequency ranges for 

which neither term dominates.

S-wave normal mode phase velocities can be derived by 

requiring the normal mode phase velocity to be less than the 

P-wave velocity but greater than the S-wave velocity. This 

implies l is an imaginary number and m is a real number. 

Although l is imaginary, there exists a real solution for k. In 

contrast to the DPNM, the dispersion relation for dispersive 

shear-wave normal modes (DSNM) is better approximated 

using J1(ma) = 0, and this approximation is appropriate for frequencies greater than about 0.4 MHz for the 35-mm-diameter 

aluminum cylinder.

Normal Mode Amplitude

Amplitude relations provide an independent check on 

the normal mode technique for characterizing the waveform 

features shown in figure 2. Amplitude relations are also particularly useful when differentiating between the fundamental 

normal modes and the body waves used to determine P- and 

S-wave speeds in materials. Normal mode amplitudes relative 

to each other can also be calculated from equations 2 and 4. 

With P- and S-wave sources at the origin, constants A and B in 
6 High-Frequency Normal Mode Propagation in Aluminum Cylinders

equations 2 and 4 are given by:

A (k) A = (S1a22 − S2 a12 ) / ∆ ≡ P

∆(k) (10) 

A (k) B = (−S a + S a ) / ≡ S

1 21 2 11 ∆

∆(k)

In the following amplitude analysis, only P-waves are 

considered. Substituting from equations 4 and 5 into equation 

10: 

A (k) = −V 2(2k2

2

−2)

2

J (ma)H(2)(la) P o 1 0

−V 22

2 2 2

o 1 ( k 2 −2

)lJ1(ma)H(2)

1 (la) / a

−V 2 2 4 2

o 4k   lH1 (la)[mJ0 (ma) − J1(ma) / a] (11)

− 2iS02

2

k(2k2

2

−2)J1(ma)[mH2

0 (ma)

− H2

1 (ma) / a]

+ 2iS 2

2 ( 2

0 k 2k 2

−2

)H1(ma)[mJo (ma)

− J1(ma) / a]

Substituting A into the integral equation shown in equation 2, the potential for the P-wave solution is given by the 

following equation (for example, Pekeris, 1948; Ewing and 

others, 1957; Båth, 1968) in the lower half of the complex 

k-plane:∞

Φ(k) = ∫ ∞A(k)J −ikz (2) −ikz

0 (lr)e dk + Vo H0 (lr)e dk −∞ ∫−∞

= ∫ [A(k)J (lr) + V H(2)

(lr)]e−ikzdk − 2i∑RP

0 o 0 n (12) Branch cut

= ΦB + ΦDPNM

where Branch cut is a branch cut integration around the branch 

points k P

 =  /  and k =  /  ; Rn is the residue of integration for a dispersive P-wave normal mode; ΦB and ΦDPNM

are the P-wave potentials for a body wave and a dispersive 

P-wave normal mode, respectively. 

The dispersive P-wave normal mode amplitude can be 

calculated by substituting the wave number, k, of interest into 

equation 12. The DPNM solution from the P-wave source is 

given by:

Figure 3. Numerical derivation of theoretical phase velocities for a 35-mm-diameter aluminum rod with an 

assumed P-wave velocity, α, of 6,380 m/s and S-wave velocity, β, of 3,180 m/s. A, For 0.585 and 0.589 MHz, the 

fundamental and first- and second-order normal mode phase velocities relative to a (radius of a cylinder) are, 

respectively, given by the first, second, and third intersections between the left and right hand sides of equation 6. 

B, Phase velocities of the fundamental order P-wave normal mode calculated from equation 6 are compared with 

those calculated from the approximate dispersion relations with n = 0.

A B

NORMALIZED PHASE VELOCITY, Cα /α

NORMALIZED PHASE VELOCITY, Cα /α

FREQUENCY, IN HERTZ

RIGHT AND LEFT SIDE OF EQUATION 6

1.00

–4 0.8

0.9

1.0

1.1

1.2

–2

0

2

4

6

1.25 1.50 1.75 4×105 6×105 8×105 1×106

Phase velocity for fundamental order

Calculated for P200–2

Exact equation

Left side of equation 6

Right side of equation 6 for f = 0.585 mHz

Right side of equation 6 for f = 0.589 mHz

Fundamental order

First order

Second order

Approximation using J1

(la) = 0

Approximation using J0

(la) = 0
Theory 7

Φ

Δ DPNM

P

ikz

k solution of equation

iA k J lr e

k = −

∂ ∂

−

=

2 0

6

 ( ) ( )

/ ] , (13)

where ∆ is given in equation 5 and AP is given in equation 11. 

Amplitude of P-Body Wave

The branch cut integrations shown in equation 12 yield 

body-wave motions inside the solid. The branch cut integration can be evaluated using the steepest descent method (for 

example, Morse and Feshbach, 1953; Båth, 1968; Greenfield, 

1978) if the observation point is in the far field, meaning the 

wave travels a distance much longer than its wavelength. 

Using Hankel functions, equation 12 can be written as: 

ΦB

ikz

o

ikz A k J lr e dk V H lr e dk

A k

( ) ( ) ( ) ( )

[ (

( )  = +

=

−∞

∞ − −

−∞

∞

∫ ∫

∫

0 0

2

1

2 ) ( ) ( ( ) ) ( )] . ( ) ( ) H lr A k V H lr e dk ikz

0

1

0 0

2 + + 2 −

The integration using the steepest descent method yields: 

ΦB

o

i R iA k R

R

iV e

R ≈ +

− 2 ( cos )cos( / ) 2 / ‘



     (14)

where k =  / ,  = − tan ( / ) 1 r z and R z r = +2 2 .

The displacements can be written as:

U r

i rA k R

R

rV e

R r

B o

i R

= ∂

∂ ≈ − +

− Φ 2 2 2 2

   









  ( cos )sin( / ) /

 (15)

U

z

i zA k R

R

zV e

R z

B o

i R

= ∂

∂ ≈ − +

− Φ 2 2 2 2

   









  ( cos )sin( / ) .

/

The far-field solution shown in equation 14 requires the Bessel 

and Hankel function argument to be large: lr >> 1. However, 

the geometry of our measurement does not satisfy the condition because the cylinder radius, a, and observation point 

radius, r, are both small. We can therefore only approximate 

the body-wave amplitude, and we do so by assuming the 

uniformly expanding source waveform can be calculated by 

making a and r large. In other words, we ignore the first term 

of equation 14, which yields the reflections from the cylinder 

boundaries. From equation 14, the body-wave amplitude for 

the P-wave can be approximated by:

Ur ≈ 0

 

U

V e

z z

i z

≈

− 2 0



 /

 and (16)

Normal Mode and Body-Wave Amplitudes for 

P-Waves

The discontinuous phase velocity that complicates the 

group velocity calculation is also a problem when using 

equation 13 to calculate normal mode amplitudes. Figure 4 

shows the wave number calculated using equation 6 for the 

fundamental and first-order DPNM over the measured range 

of dominant frequencies. Like the phase velocities shown in 

figure 3B, the relation between wave number and frequency is 

discontinuous. Using the same technique employed for 

figure 3B, the discontinuous exact solution can be approximated by setting J la 0 ( ) . = 0

The approximation works well for the fundamental and 

first-order modes at high frequencies. Although the general 

trend of the approximation follows the exact relation, the 

approximate solution becomes less accurate with decreasing 

frequency. Numerical analysis indicates the approximation 

becomes worse as the order of the normal mode increases.

Equation 13 requires the phase velocity rather than the 

group velocity to compute normal mode amplitude. Based on 

figure 4, we assume the approximate phase velocities derived 

from J la 0 ( ) = 0 with n = 0 and n = 1 are appropriate to qualitatively analyze amplitudes of the fundamental and first-order 

DPNM. A second option for approximating the phase velocity 

is to set J1(la) = 0, which, as discussed previously, and shown 

Figure 4. Relation between wave number and frequency 

for fundamental and first-order normal modes for a 

35-mm-diameter aluminum cylinder. Approximate relation 

using J0(la) = 0 with n = 0 and 1 are also shown. The 

approximations improve with increasing frequency and 

decreasing order of the normal mode. 

WAVE NUMBER, RADIANS PER METER

ANGULAR FREQUENCY, RADIANS PER SECOND

3 ×106 4 ×106 5 ×106 6 ×106 7 ×106 3 ×106

4 ×106

5 ×106

6 ×106

7 ×106

Fundamental mode from exact equation

First order from exact equation

Approximation using JO(la) = 0 with n=0

Approximation using JO(la) = 0 with n=1
8 High-Frequency Normal Mode Propagation in Aluminum Cylinders

in figure 3B, appears to more accurately fit the slope of the 

exact phase velocity solution (equation 6). Because this slope 

is important in calculating the group velocity in equation 8, 

and since it is the group velocity, not the phase velocity, that 

is easily estimated from the measured waveform, using J1(la) 

= 0 for the approximation rather than J la 0 ( ) = 0 is tempting. 

As demonstrated in figure 5, however, this approximation 

does not reflect the exact solution behavior for normal mode 

amplitudes. 

Figure 5 shows calculated amplitudes of the fundamental 

mode for a 100-mm-long, 35-mm-diameter aluminum cylinder 

using the exact solution, the J la 0 ( ) = 0 , and the J1(la) = 0 

approximations. The exact solution agrees well with the 

J la 0 ( ) = 0 for frequencies away from the spectral holes at the 

phase-velocity discontinuities, but is generally at least an order 

of magnitude below the J1(la) = 0 approximation. Therefore, all P-wave normal mode amplitudes in this paper are 

computed using J la 0 ( ) . = 0 For comparison, the body-wave 

amplitude shown in figure 5 is calculated from equation 16. 

Figure 5 indicates the approximate normal mode amplitudes 

are larger than the body-wave amplitude for all frequency 

ranges. The amplitudes calculated from the exact solution are 

also larger than the body wave except near the spectral holes. 

This finding suggests body waves will likely be difficult to 

detect unless they arrive well in advance of the fundamental 

mode.

S-Wave Amplitude

As mentioned previously, to analyze the shear-wave 

normal mode, we consider phase velocities greater than b and 

less than a. Shear-wave propagation can be calculated using 

the same equations used for the P-wave propagation with the 

following constant: 

B k B k V k ik l J la H la

J la H

S

o ( ) ( ) ( ) [ ( ) ( )

( )

( )

(

≡ = −

−

Δ

2 2 2 2 2 2 2

1 0

2

0 1

   

2

2 4 2

1 1

2

2 2 4

1

4

4

)

( )

( )] /

( ( ) ( ) / ) /

(

la

V ik l J la H la a

S k lJ

o

o

Δ

+ Δ

− {

 

  la mH ma H ma a

k J la H ma

)[ ( ) ( ) / ]

( ) ( ) ( ) /

( ) ( )

( )

0

2

1

2

2 2 2 2

0 1

2 2

− +

 − } Δ

 

 (17)

Using a similar argument for the P-wave, the S-body 

wave amplitude is approximated as: 

 U

i S e

z r

o

i z

≈ −

− 2 



 /

 (18)

 Uz ≈ 0 .

Discussion

To demonstrate the utility of the normal mode approach, 

we identify waveform features by comparing their group 

velocities and dominant frequencies (see table 1) to theoretical 

values for each normal mode. Body waves are identified on 

the basis of the arrival time and amplitude comparisons to the 

fundamental mode amplitudes. We begin with considering 

waveforms from cylinders of different length, but identical 

radii. Later in this report, we consider waveforms from 

cylinders of identical length but different radii.

Figure 5. Calculated amplitude of the dispersive P-wave 

fundamental order normal mode (fundamental order DPNM) 

for a 35-mm-diameter, 100-mm-long aluminum cylinder. 

Between the spectral holes caused by discontinuities in the 

phase velocity near 0.65, 0.75, and 0.86 MHz. Setting J0(la) = 

0 in equation 13 provides a close approximation to the exact 

amplitude calculation. RELATIVE AXIAL DISPLACEMENT

FREQUENCY, IN HERTZ

6 ×105 103

104

105

106

107

7 ×105 8 ×105 9 ×105

Body wave

Fundamental order DPNM with J0

(la) = 0

Fundamental order DPNM with J1

(la) = 0

Fundamental order DPNM with exact

 dispersion relation
Discussion 9

P-Wave Group Velocity

For the 35-mm-diameter aluminum cylinders, figure 6 

shows the measured group velocity with solid stars indicating the measured waveform features labeled in figure 2. Open 

stars denote group velocities calculated using equation 6 for 

the first-order normal modes P100-2 and P200-2 and the second-order normal mode P200-3. Approximate group velocities 

derived from equation 9 are shown for the approximations 

J la 0 ( ) = 0 (dotted curves) and J1

(la) = 0 (solid curves). Most 

measured group velocities agree with those approximated by 

setting J1(la) = 0, implying for these cylinders that matching 

the phase velocity slope in figure 3B is more important than 

matching the phase velocity values when calculating group 

velocities.

There are some discrepancies between the exact group 

velocity calculations and the measured values they attempt to 

predict. Precise group velocity predictions using normal mode 

analysis are hampered by the complex frequency behavior of 

the phase velocity. In general, the phase velocity increases 

with decreasing frequency. The group velocity, however, 

generally decreases with decreasing frequency. The highfrequency components of the wave, therefore, generally arrive 

earlier than the low-frequency components. However, for frequencies between phase velocity jumps, for example between 

0.65 and 0.76 MHz, the group velocity at the high-frequency 

end is smaller than that at the low-frequency end. For components with closely spaced frequencies, the low-frequency 

wave components can arrive earlier than higher frequency 

components.

This complex behavior makes precise group velocity 

predictions difficult because the measured group velocity is 

an averaged value based on all frequencies contained in each 

normal mode, whereas the calculated group velocities are 

evaluated at a single frequency. Considering the rapid variation of phase velocities with frequency shown in figure 3B, 

the discrepancy between the measured and calculated group 

velocities is not surprising.

For the purposes of correctly identifying measured 

waveform features, high-precision group velocity predictions 

are not necessary, and the approximation presented here can 

be used effectively. As an example, we compare waveform 

features P100-3 and P200-3 from figure 2. Considering P100-3 

first: in the 100-mm-long cylinder in figure 6, it is suggested 

that both P100-2 and P100-3 are first-order P-wave normal 

modes. However, there can only be one P-wave normal mode 

of each order in a given waveform. As we showed previously, 

P100-3 is more likely a dispersive shear-wave normal mode 

propagation (DSNM) rather than a P-wave normal mode.

The P200-3 arrival, like the P100-3 arrival, is the third 

high-amplitude feature in the waveform. Unlike P100-3, however, P200-3 has a relatively high frequency content in spite of 

its slow group velocity, and agrees with the predicted relation 

between dominant frequency and group velocity for a secondorder dispersive P-wave normal mode (DPNM).

To reiterate, in comparing the 100- and 200-mm-long 

cylinder waveforms, we find the high-amplitude feature arriving third in the short-cylinder waveform is not at all the same 

as the third arrival in the long-cylinder waveform. For the 

short cylinder, the arrival is a first-order S-wave normal mode, 

but in the long cylinder, the third arrival is a second-order 

P-wave normal mode. Correctly identifying high-amplitude 

features cannot be done on the basis of arrival sequence or 

dominant frequency alone.

S-Wave Group Velocity

Because complex acoustic interference patterns give rise 

to shear normal modes that occur later than the times shown in 

figure 2, a second set of waveforms were measured for three 

35-mm-diameter aluminum cylinders in 2007 (fig. 7A). An 

alteration of one endcap between the 2006 and 2007 measurements slightly decreased the transducer output frequency. As 

shown in figure 7A, each cylinder has two dispersive S-wave 

Figure 6. Group velocities for 35-mm-diameter aluminum 

cylinders. The velocities are normalized by the P-wave 

speed through aluminum, α. Calculated group velocities are 

plotted using the exact equation 6 (open stars), using the 

approximate solutions (dotted and solid lines) for the normal 

modes (n = 0 to 3), and using measured group velocities 

(solid stars).

NORMALIZED GROUP VELOCITY, G/

α1

FREQUENCY, IN HERTZ

4 ×105 5 ×105 6 ×105 7 ×105 8 ×105 9 ×105 1 ×106

0

0.2

0.4

0.6

0.8

1.0

Measured group velocity

P100–3

P100–2

P100–1 P200–1

P200–2

P200–3

n=0

n=1

n=2

Numerical solution of equation 6

Approximation using J1

(la) = 0

Approximation using J0

(la) = 0
10 High-Frequency Normal Mode Propagation in Aluminum Cylinders

normal modes (DSNM), identified as S100-1, S100-2, S150-

1, S150-2, S200-1, and S200-2. The group velocities of the 

fundamental and first-order DSNM for 100- and 200-mm-long 

aluminum cylinders are shown in figure 7B, along with waveform feature P100-3 from figures 2 and 6. The group velocities 

of P100-3 and S100-2 follow the first-order DSNM curve. 

Therefore, the P100-3 shown in figure 2 is probably the axial 

component of the first-order DSNM propagation, designated 

as S100-2 in figure 7A.

P-Wave Amplitudes

Here we qualitatively compare amplitudes of waveform 

features in the frequency domain. The body-wave and DPNM 

amplitudes from a P-wave source in a 200-mm-long aluminum 

cylinder with a 35-mm diameter are plotted in figure 8. The 

shaded regions cover the dominant frequency ranges measured 

for the fundamental and first-order normal modes, and are 

centered on the predicted relative amplitude for that phase. 

Figure 8 indicates the fundamental and first-order normal 

modes have similar amplitudes, with the fundamental mode 

having a slightly higher amplitude than the first-order mode, 

in agreement with the measured waveforms shown in figure 2. 

The calculated amplitude of the body wave is much smaller 

than either normal mode.

Because the arrival times of the fundamental order 

DPNM are so close to those of the first arrival P-body wave 

for the 100- and 200-mm-long aluminum cylinders, the 

body wave is indistinguishable from the fundamental normal 

mode. Two consequences for these cylinders are (1) using the 

fundamental order DPNM as a proxy for the body wave in 

determining the P-wave speed in aluminum underpredicts the 

P-wave speed of 6,380 m/s by only 0.3 percent, and (2) the 

measured fundamental order DPNM is a mixture of normal 

mode and body-wave arrivals. The body-wave amplitude is 

much smaller than that of the fundamental order DPNM for 

these cylinders, but as the cylinder length decreases, the bodywave amplitude varies as 1/(cylinder length), whereas the 

normal mode amplitude is independent of the cylinder length. 

Thus, the body wave becomes more distinct as the normal 

mode interference diminishes with decreasing cylinder length.

Figure 7. Dispersive shear-wave normal mode analysis. A, Shear-wave waveforms for 35-mm-diameter 

aluminum cylinders (AL) of nominal length 100, 150 and 200 mm, all measured in 2007. S100-1, S150-1, and S200-1 

are fundamental order dispersive S-wave normal modes (DSNM) observed for 100-, 150- and 200-mm-diameter 

cylinders, respectively. Similarly, S100-2, S150-2, and S200-2 are the first-order DSNM. B, Measured and calculated 

group velocities for S100-1, S100-2, S200-2, and P100-3. P100-3 is likely the axial component of the first-order shear 

normal mode. Group velocities are normalized to the shear-wave speed in aluminum, β. NORMALIZED GROUP VELOCITY, Gβ

/β

RELATIVE AMPLITUDE

FREQUENCY, IN HERTZ

From approximation J1(ma) = 0

From S–transducer

From P–transducer

TIME, IN MICROSECONDS

15 30 45 60 75 90 –3,000

3,000

6,000

9,000

0

200-mm AL

150-mm AL

S100–1

S100–2

S100–2

S200–1

S100–1

S200–2

n = 0

n = 1

S150–2

S150–1

First–S arrival

S200–1

S200–2

100-mm AL

2 ×105 4 ×105 6 ×105 0

0

0.2

0.4

0.6

0.8

A 1.0 B
Discussion 11

S-Wave Amplitudes

Figure 9 shows the calculated radial amplitudes for various arrivals for a 200-mm-long, 35-mm-diameter cylinder 

from an S-wave source. The shaded region spans the dominant frequencies of the measured first-order and fundamental 

modes. In agreement with figure 7A, the radial displacements 

of the fundamental and first-order DSNM are similar. Because 

normal mode amplitude is independent of the cylinder length, 

the amplitude relations between the fundamental and firstorder DSNM remains similar for all three aluminum cylinders. 

The measured amplitude relation between the S-body wave 

and fundamental order DSNM for the 200-mm-long aluminum 

cylinder also agrees with the calculated relation. 

The shear crystal traces in figure 2 show the radial displacement of the P-wave normal modes P100-1 and P100-2, 

indicating the radial amplitude of P100-2 is larger than that 

of P100-1. Figure 10 shows the calculated radial displacement for fundamental and first-order DPNM, indicating the 

radial displacement of P100-2 is larger than that of P100-1, in 

qualitative agreement with the observed amplitude variation. 

Because the P-wave normal mode amplitude is independent 

of the length of the aluminum cylinder, the calculated and 

measured radial displacements of P200-2 are also larger than 

those of P200-1.

As was the case for P-waves, arrival times of the fundamental order DSNM and S-body wave are close. The 100-1, 

S150-1, and S200-1 waveforms in figure 7A are therefore mixtures of the fundamental normal mode and the body wave. In 

the 200-mm-long cylinder, however, a small amplitude event 

with high-frequency content arrives earlier than S200-1 and 

is separated from the fundamental order DSNM. We interpret 

this earlier arrival as the first arrival S-body wave. This waveform demonstrates how the nearly identical body-wave and 

fundamental DSNM-wave speeds mean the body wave can 

only separate from the fundamental mode in long cylinders. 

For the 200-mm-long cylinder, interpreting the largest feature, 

S200-1, as the body wave would underpredict the shear-wave 

Figure 8. Predicted relative axial amplitudes for the P-body 

wave and fundamental and first-order P-wave normal modes 

in a 35-mm-diameter, 200-mm-long aluminum cylinder. 

Fundamental and first-order normal mode amplitudes are 

similar, and both are much larger than the body-wave 

amplitude.

Figure 9. Predicted relative radial amplitudes of S-wave 

arrivals for a 35-mm-diameter, 200-mm-long aluminum 

cylinder. The fundamental and first-order normal modes 

have comparable amplitudes, and both are significantly 

larger than the S-body wave, in agreement with figure 7A.

RELATIVE AXIAL AMPLITUDE

First-order DPNM with J1(la) = 0

Body wave

Dominant frequency of fundamental order

Dominant frequency of first order

r = 0 cm

Fundamental order DPNM with JO(la) = 0

4 ×105 103

104

105

106

107

5 ×105 6 ×105 7 ×105 8 ×105

FREQUENCY, IN HERTZ

RELATIVE RADIAL DISPLACEMENT

First-order DSNM using J1(ma) = 0

Body wave

Dominant frequencies of fundamental 

and first-order DSNM

Fundamental order DSNM with J1

(ma) = 0

FREQUENCY, IN HERTZ

2 ×105 4 ×105 6 ×105 8 ×105 1 ×106 103

104

105

106
12 High-Frequency Normal Mode Propagation in Aluminum Cylinders

Figure 10. Predicted relative radial amplitudes of the 

fundamental and first-order P-wave normal modes for a 

35-mm-diameter, 100-mm-long aluminum cylinder. In agreement 

with the shear-wave traces in figure 2, the first-order normal 

mode amplitudes are greater than the fundamental mode 

amplitudes.

speed of 3,180 m/s by 1 percent. As shown in figure 2, the 

high wave speed of P-waves relative to S-waves, and the close 

agreement between the P-body wave and fundamental normal 

mode, mean the P-body wave does not separate from the fundamental mode, even in the 200-mm-long cylinder.

P-Wave Group Velocity: Dependence on 

Cylinder Radius

Figure 11A compares measured waveforms in 35- and 

25.4-mm-diameter, 200-mm nominal-length aluminum 

cylinders. The waveforms, generated and measured with the 

P-wave crystals, contain two notable differences in the thin 

cylinder: (1) the first-order DPNM arrives more quickly than 

in the thick cylinder, and (2) the first-order DPNM is at least 

twice the amplitude of the fundamental mode.

The normal mode group velocities, compared with 

theoretical predictions in figure 11B, further exemplify the RELATIVE RADIAL DISPLACEMENT

First-order DPNM

Dominant frequency of fundamental order

Dominant frequency of first order

Fundamental order DPNM

FREQUENCY, IN HERTZ

4 ×105 5 ×105 6 ×105 7 ×105 8 ×105

102

103

104

105

106

107 control that sample geometry exerts on waveform structure. 

The horizontal axis in 11B has been nondimensionalized so 

results from cylinders of different radii can be compared. 

The normal mode approximations agree with the measured 

results, but it is the J0(la) = 0 approximation that works for the 

25.4-mm-diameter cylinder, whereas the J1(la) = 0 approximation is more appropriate for the 35-mm-diameter cylinder. As 

predicted by the normal mode approach, the complex internal 

reflection patterns that create these normal modes produce a 

much higher first-order normal mode amplitude relative to 

the fundamental mode in the thin cylinder (fig. 12), but nearly 

equal amplitudes in the thick cylinder (fig. 8).

Conclusions

Laboratory measured elastic waveforms in cylinders can 

contain large amplitude waveform features that can make it 

difficult to identify the nature of each arrival. Sample geometry influences the group velocity and amplitude of each 

feature differently, meaning their arrival sequence and relative 

amplitudes within a waveform can all differ in samples of different size. These waveform features behave as normal mode 

propagations, which we identify and characterize using a wave 

propagation solution in a cylindrical coordinate system.

The primary challenge in correctly identifying the 

observed normal modes is coping with the discontinuous 

frequency dependence of their phase velocities. Identifying 

normal modes via amplitude and group velocity predictions 

can be accomplished by using Bessel functions to approximate 

the phase-velocity dependence on frequency. The accuracy of 

these approximations improves with increasing frequency as 

the situation more closely resembles a small wavelength signal 

traveling along the axis of a large, long cylinder. 

Two key findings from this study are:

1. P- and S-wave crystals each produce a combination 

of P- and S-wave normal modes. A waveform can 

therefore contain both P- and S-wave normal modes, 

with their arrival order depending on the sample 

geometry in a manner that is difficult to predict from 

a visual inspection of the waveform. These arrivals can be identified by analyzing amplitude and 

group velocities in the context of normal mode wave 

propagation.

2. Body waves, which travel at the P- and S-wave 

speeds generally sought after in acoustic measurements, can be obscured by the relatively higher 

amplitude fundamental normal modes. When the 

cylinder is long, or the wave speed is low, the body 

wave can measurably separate from the fundamental 

mode. In these cases, the more easily observed fundamental mode may be mistaken for the body wave, 

leading to an underestimate of the P- or S-wave 

speed.
Conclusions 13

Figure 11. Normal mode dependence on cylinder radii. A, Measured waveforms in 25.4-mm-diameter (thin) and 

35-mm-diameter (thick) aluminum cylinders. B, Measured and calculated group velocities for fundamental and 

first-order P-wave normal modes. Horizontal axis is nondimensionalized so results from both cylinders can be 

compared. Though the J0(la) = 0 approximation best fits the thin cylinder normal modes (solid stars), J1(la) = 0 is a 

better approximation for the thick cylinder (open stars).

A B

TIME, IN MICROSECONDS DIMENSIONLESS PARAMETER, π=aω/α1

RELATIVE AMPLITUDE

NORMALIZED GROUP VELOCITY, G/

α1

30 –500

0

500

1,000

40 50 60 0

0

0.2

0.4

0.6

0.8

1.0

5 10 15 20

Measured group velocity for 35-mm cylinder

Measured group velocity for 25.4-mm cylinder

Approximation using J1

(la) = 0

Approximation using J0

(la) = 0

P200–1

PT200–1

PT200–2

P200–2

n = 0 n = 1

35-mm-thick cylinder

25.4-mm-thick cylinder
14 High-Frequency Normal Mode Propagation in Aluminum Cylinders

Acknowledgments

This work was supported by the Gas Hydrate Project 

within the U.S. Geological Survey’s Energy and Coastal and 

Marine Geology Programs. We thank Peter M. Bratton for his 

assistance in measuring acoustic waveforms.

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for P-wave velocity, anisotropy, and attenuation structure 

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hydrate production research well, in Dallimore, S.R., and 

Collett, T.S., eds., Scientific results from the Mallik 2002 

gas hydrate production research well program, Mackenzie 

Delta, Northwest Territories, Canada: Geological Survey of 

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bore containing a fluid: Journal of Applied Physics, v. 23, 

p. 997–1,009.

Brekhovskikh, L.M., 1960, Waves in layered media: New 

York, Academic Press, 559 p.

Dvorkin, Jack, Helgerud, M.B., Waite, W.F., Kirby, S.H, 

and Nur, A., 2000, Introduction to physical properties and 

elasticity models in natural gas hydrate, in Max, M.D., ed., 

Oceanic and permafrost environments: Boston, Kluwer 

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Ewing, W.M., Jardetzky, W.S., and Press, F., 1957, Elastic 

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RELATIVE AXIAL DISPLACEMENT

FREQUENCY, IN HERTZ

6 ×105

103

104

105

106

107

7 ×105 8 ×105 9 ×105 1 ×106

Body wave

Dominant frequency of fundamental mode

Dominant frequency of first order

Fundamental order DPNM with J0

(la) = 0

First-order DPNM with J0

(la) = 0

Figure 12. Predicted relative amplitudes for the P-body 

wave and fundamental and first-order normal modes in a 

25.4-mm-diameter, 200-mm-long aluminum cylinder. Both normal 

mode amplitudes are significantly larger than the P-body-wave 

amplitude. Unlike the 35-mm-diameter aluminum cylinder results 

(fig. 8), the first-order normal mode amplitude is larger than that 

of the fundamental mode. This prediction is in agreement with 

the results shown in figure 11.
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16 High-Frequency Normal Mode Propagation in Aluminum Cylinders

This appendix derives two well-known equations using 

the formulas given in the main text; one is the velocity of bar 

waves and the other is the dispersion relation of the Rayleigh 

surface wave. Retaining the first term of power series of the 

Bessel functions, J la 0 ( ) = 1 (Watson, 1966) and equation 5 at 

low frequencies can be approximated by: 

        2 2 2 2 2 2 2 2 2 2 4 2 2 (2k − + l k m k l m )( ) 2 − = −2 (A-1)

Rejecting the solution m = 0, from which the shear-wave 

velocity can be derived (Kolsky, 1963), and casting A-1 in 

terms of Lame’s constants instead of velocities, equation A-1 

can be written as:

( )( ( )2 ) 22 2

2

2

2

2 2 k l − + k l

+ = 





   (A-2)

Rearranging equation A-2, the following phase velocity 

can be derived:

 C

k

E = = +

+ = = −

−

   

      

 

( )

( )

/

/

3 2 3 4

1

2 2

2 2 (A-3)

where E is Young’s modulus. Equation A-3 is the same as the 

bar-wave velocity, equation 3.58 shown in Kolsky (1963).

When a becomes large, the dispersion relation shown in 

equation 5 becomes:

 Δ ≈ +

− =

4

2 0

2 2 2

1 0

2 2 2 2 2

0 1

k lmJ la J ma

k J la J ma

 

  

( ) ( )

( ) ( ) ( )

 (A-4)

or

4

2

2 1 0

2 2

2 2 2 mJ la J ma 0 1

C

k C J la J ma

l

( ) ( )

/

( / ) ( ) ( )

− = −



If the phase velocity C is less than the S- and P-wave 

velocities, l and m become pure imaginary numbers as shown 

in equation 2. When l and m are purely imaginary and a

becomes large, J la J ma J la J ma 0 1 1 0 ( ) ( ) ( ) ( ) ≈ , and equation 

A-4 becomes:

 4 1

2

2

1

2 2

2 2

2 2

2 2

−

− = −

−

C

C

C

C

/

/

/

/









 (A-5)

This is an identical form of the Rayleigh wave dispersion relation given in White (1965).

It is important to note that the low-frequency correspondence between our approach and those of Kolsky (1963), 

Graff (1975), and White (1965) does not extend to the frequency range of interest to our laboratory work. The waveform features shown in figures 2, 7, and 11 are neither the bar 

waves, nor extensional-mode waveforms described by Kolsky 

(1963) and Graff (1975).

Bar waves are generated when the wavelength is much 

larger than the cylinder radius. For example, if P200-1 was a 

bar wave, the low-frequency component of the wave would 

arrive earlier than the high-frequency component, and the 

onset time would be near 40 microseconds. This does not 

agree with the observed onset time or wave-feature characteristics. To eliminate the possibility of a bar-wave-type feature 

in our analysis, we restricted the wavelength of the body wave 

to be less than ~60 percent of the cylinder radius. 

Conventional extensional modes described by Kolsky 

(1963) and Graff (1975) are similar to the S-wave normal 

modes described here, but differ in key respects that can be 

traced to the manner in which our normal modes are defined. 

The normal modes described in this paper are defined as follows: the P-wave normal mode is defined as a guided wave 

with a phase velocity greater than the P-body wave velocity, 

α, and with a group velocity approaching α as the wavelength approaches zero. Similarly, the S-wave normal mode 

is defined as a guided wave with a phase velocity greater than 

the S-body wave velocity, β, but less than α, and with a group 

velocity approaching β as the wavelength approaches zero.

The conventional extensional modes described by Kolsky 

(1963) and Graff (1975) approach the Rayleigh surface-wave 

velocity at high frequencies, but the S-wave normal mode 

defined here has a velocity that approaches β at high frequencies and small wavelengths. This happens because the 

extensional-wave-phase velocity becomes less than β as the 

frequency increases, but the phase velocity is restricted to be 

greater than β for the S-wave normal modes in this report.

Appendix A. Low-Frequency and Large Diameter Approximation
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Lee and Waite—High-Frequency Normal Mode Propagation in Aluminum Cylinders—Scientific Investigations Report 2009–5142

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