U.S. Department of the Interior

U.S. Geological Survey

Scientific Investigations Report 2010–5013

S-Wave Normal Mode Propagation in Aluminum Cylinders

S-Wave Normal Mode Propagation in 

Aluminum Cylinders

By Myung W. Lee and William F. Waite

Scientific Investigations Report 2010–5013

U.S. Department of the Interior

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

KEN SALAZAR, Secretary

U.S. Geological Survey

Marcia K. McNutt, Director

U.S. Geological Survey, Reston, Virginia: 2010

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Suggested citation:

Lee, M.W., and Waite, W. F., 2010, S-wave normal mode propagation in aluminum cylinders: U.S. Geological Survey 

Scientific Investigations Report 2010–5013, 18 p.
iii

Contents

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

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

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

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

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

Phase Velocity.......................................................................................................................................5

Group Velocity .......................................................................................................................................6

Amplitude ...............................................................................................................................................6

Amplitude of the Dispersive S-Wave Normal Mode..............................................................7

Amplitude of the Body Wave .....................................................................................................7

Analysis of Measured Waveforms..............................................................................................................8

Velocity Analysis...................................................................................................................................8

Amplitude Analysis.............................................................................................................................10

Discussion.....................................................................................................................................................11

Phase Velocity and Normal Mode ...................................................................................................11

Cut-Off Frequencies............................................................................................................................11

Basis of Normal Mode Interpretation .............................................................................................12

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

Acknowledgments.......................................................................................................................................13

References Cited..........................................................................................................................................13

Appendix 1. Approximate Phase Velocity ............................................................................................15

Appendix 2. Torsional Waves .................................................................................................................16

Figures

1. Measurement system ..................................................................................................................2

2. S-crystal waveforms and normalized group velocities measured using endcap

set no. 1 ..........................................................................................................................................3

3. S-crystal waveforms and normalized group velocities measured using endcap 

set no. 2 ..........................................................................................................................................3

4. Normalized phase and group velocity.......................................................................................5

5. Exact and approximate normalized phase and group velocities..........................................9

6. Measured power spectra for the 200-mm-long aluminum cylinder ....................................9

7. Calculated amplitudes for body wave and for fundamental- and first-order

dispersive S-wave normal mode..............................................................................................10

8. Calculated group velocity for fundamental-order extensional mode................................12

1–1. Exact and approximate normalized phase and group velocities........................................15

1–2. Calculated amplitudes for body wave and for fundamental- and first-order

dispersive S-wave normal mode..............................................................................................16

2–1. Calculated exact and approximate phase velocities for torsional waves........................17

Abstract

Large amplitude waveform features have been identified in pulse-transmission shear-wave measurements through 

cylinders that are long relative to the acoustic wavelength. The 

arrival times and amplitudes of these features do not follow 

the predicted behavior of well-known bar waves, but instead 

they appear to propagate with group velocities that increase as 

the waveform feature’s dominant frequency increases. To identify these anomalous features, the wave equation is solved in a 

cylindrical coordinate system using an infinitely long cylinder 

with a free surface boundary condition. The solution indicates 

that large amplitude normal-mode propagations exist. Using 

the high-frequency approximation of the Bessel function, 

an approximate dispersion relation is derived. The predicted 

amplitude and group velocities using the approximate dispersion relation qualitatively agree with measured values at high 

frequencies, but the exact dispersion relation should be used to 

analyze normal modes for full ranges of frequency of interest, 

particularly at lower frequencies.

Introduction

The transmission of shear waves through soil is strongly 

influenced by the stiffness of the intergranular contacts (Santamarina and others, 2001). Crystalline solids, such as ice or 

gas hydrate, forming in the pore spaces of an unconsolidated 

sediment can stiffen the intergranular contacts by cementing 

grains together or by forming as load-bearing components of 

the sediment (Sloan and Koh, 2007). Particularly in the case of 

methane gas hydrate, acoustic measurements have been used 

to quantify the pore-space saturation of this naturally occurring hydrate in marine and beneath permafrost sediments (for 

example, Dai and others, 2008; Helgerud and others, 1999; 

Lee and Collett, 2005). To calibrate the relation between the 

pore-space hydrate saturation and the acoustic wave velocity, 

laboratory wave-velocity tests have been conducted on granular samples containing known hydrate saturations (Berge and 

others, 1999; Waite and others, 2004; Priest and others, 2005; 

Yun and others, 2005).

When the sample length is long compared to the acoustic wavelength, the appearance of multiple, large-amplitude 

waveform features can obscure the shear wave arrival. This 

phenomenon first becomes apparent while making measurements using the gas hydrate and sediment test laboratory 

instrument (GHASTLI) (Winters and others, 2000). Because 

GHASTLI was designed for making triaxial measurements 

of soil strength, cylindrical samples with lengths 2–2.5 times 

greater than their diameters are required (American Society 

for Testing and Materials, 2003). Because of the large sample 

lengths required for use with GHASTLI, Waite and others 

first encountered these large-amplitude waveform features 

during their investigation in 2008 and determined that they 

typically occur when the sample diameter was approximately 

10 times longer than the shear wave wavelength (Waite and 

others, 2008).

To identify these features and distinguish between the 

geometric and porous media controls on waveform features, 

a series of acoustic measurements on aluminum cylinders of 

varying length and diameter were carried out. Kolsky (1963) 

and Graff (1975) have extensively analyzed wave propagation 

in thin cylinders, but their low-frequency, bar-wave formulations are not appropriate for the high-frequency measurements discussed here. Based on the relationship between the 

velocity, frequency, and amplitude of the waveform features, 

we propose to identify the features using a normal mode at 

high frequency.

Normal mode solutions have been extensively investigated in the context of layered Earth models (Ewing and 

others, 1957; Brekhovskikh, 1960; Aki and Richards, 1980). 

Thurston (1978) reviewed the wave propagations in rods 

and clad rods using various normal mode solutions such as 

extensional and flexural modes. Recently, the normal mode 

solution in a cylindrical coordinate system applicable to measurements made in GHASTLI at frequencies in the megahertz 

range has been presented in Lee and Waite (2009). In Lee 

and Waite (2009), the analysis focused on the P-wave normal 

mode propagation using an approximated dispersion relationship appropriate for high frequencies. This report extends the 

theory developed by Lee and Waite (2009) to analyze anomalous shear waveforms. 

S-Wave Normal Mode Propagation in Aluminum Cylinders

By Myung W. Lee and William F. Waite
2 S-Wave 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 displacements, is an 

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

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. For this discussion, only waveforms 

generated and received by the S-wave crystals are considered 

here. The P-waveforms are presented and discussed in Lee and 

Waite (2009).

 A Panametrics Model 5800 Pulser-Receiver generates the 

50-volt pulse used to excite the P- or S-crystal in one endcap. 

The pulse traverses the sample, is received by the S-crystal in 

the second endcap, is displayed on a Tektronix TDS-340 oscilloscope, and is stored to disk. For a given pair of S-crystals, 

the waveform is essentially independent of which crystal is the 

signal transmitter and which is the receiver. For the purpose 

of deriving a tractable mathematical model of this system, 

the S-wave signals are assumed to be generated from a point 

source on the cylindrical axis, with their radial motion measured across the full diameter of the receiving crystal.

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 dividing by the sample length to obtain the velocity of 

a given waveform feature through the sample. 

Waveforms were measured using two different endcap 

sets on 35-mm-diameter aluminum cylinders having three 

different lengths. Aluminum cylinders are designated as 100-, 

150-, and 200-mm aluminum cylinders (actual lengths are 

101.2952, 149.53, and 202.5523 mm), and the endcaps are 

denoted endcap set no. 1 and no. 2. Figure 2A shows waveforms measured using endcap set no. 1, for which two distinct 

arrivals with large, comparable amplitudes can be easily 

observed for each cylinder. In figure 2A, S100-1 and S100-2 

represent the first and second large amplitude arrivals in a 

100-mm aluminum cylinder, with corresponding labels given 

for the 150- and 200-mm cylinders.

Figure 3A shows the measured waveforms using endcap 

set no. 2, with S200-1 representing the first arrival feature 

with high-frequency content (blue), and S200-2 representing 

the low-frequency feature following S200-1 (red). In contrast 

Titanium Endcaps

 Diameter: 72 mm

 Height: 32 mm

Transducer 

assembly mounted 

inside the endcaps, 

behind the wear

plate

Transducer 

Assembly

S-wave 

crystal P-wave crystal

10.8 mm

11.4 mm

18.7 mm

C-Clamp

Aluminum

Cylinder

Wear 

plate 

A

B

Titanium Endcap

Electrical feedthrough for 

acoustic transducers

S-wave

crystal P-wave crystal

Figure 1. Measurement system. Within each endcap (A), the 

shear (S-wave) and compressional (P-wave) transducer crystals, 

shown schematically (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 end cap wear 

plates. A C-clamp holds the system together. 

to the measured waveforms shown in figure 2A, there are no 

distinct arrivals for these two waveform features in figure 3A, 

rather the high-frequency and low-frequency features make 

one continuous waveform. One marked difference between 

waveforms shown in figure 2A and figure 3A is the broader 

frequency content of the figure 3A waveforms.

Use of the normal mode theory derived below to identify 

features such as those shown in figures 2A and 3A requires the 

group velocity and dominant frequency of each feature. The 

group velocity is calculated from the feature’s arrival time, 

accounting for the system delay time and sample length as 

discussed above. The dominant frequency is calculated from a 

spectral analysis of the waveform feature.
Measuring Waveforms in Aluminum Cylinders 3

A B

Figure 2. S-crystal waveforms and normalized group velocities measured using endcap set no. 1. 

A, Waveforms for 100-, 150-, and 200-mm-long aluminum cylinders. In the graph levels, the prefixes 100, 

150, and 200 denote the length of the cylinder, and the suffixes 1 and 2 denote the first (blue) and second 

(red) large amplitude features, respectively. B, Measured normalized group velocities and approximated 

normalized group velocities derived from J1(ma) = 0 with n = 0 and n = 1. DSNM, dispersive shear wave 

normal mode.

A B

Figure 3. S-crystal waveforms and normalized group velocities measured using endcap set no. 2. 

A, Waveforms for 100-, 150-, and 200-mm-long aluminum cylinders. In the graph levels, the prefixes 100, 

150, and 200 denote the length of the cylinder, and the suffixes 1 and 2 denote the first (blue) and second 

large amplitude features (red), respectively. B, Measured normalized group velocities and approximated 

normalized group velocities derived from J1(ma) = 0 with n = 0 and n = 1. DSNM, dispersive shear wave 

normal mode.
4 S-Wave Normal Mode Propagation in Aluminum Cylinders

U

r z

U

z r r r = z

∂

∂ − ∂

∂ = ∂

∂ +

∂

∂ +

Φ Ψ Φ Ψ Ψ 

Theory

Identifying the waveform features shown in figures 2A 

and 3A requires a theoretical relationship 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 relationship to obtain the normal 

mode phase velocity dependence on frequency.

3. Use the approximate or exact phase velocity at high 

frequency to calculate the group velocity for each 

normal mode.

As an additional check on the normal mode approach, we also 

derive the relative amplitudes of the normal modes as well as 

that of the S-body waves.

Solution of Wave Equation

Under the assumption of axial symmetry, differential 

equations for wave propagation inside an infinite cylinder in a 

cylindrical coordinate system (r,,z) can be written as:

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

Jn and Hn

(2) are the Bessel functions of order n and the second 

kind Hankel function of order n, respectively, and A and B are 

constants to be determined by the boundary conditions. Note 

that uniformly expanding P- and S-wave sources are assumed 

for the wave equation solution. 

The radial (Ur) and axial (Uz) displacement can be calculated by

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

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

velocity, and time, respectively (Biot, 1952; Lee and Balch, 

1982). When the P-wave source is located at the origin with 

the strength of V0 and the S-wave source is located at the origin with the strength of S0

, the formal solution of equation 1 in 

the frequency domain can be written as follows

(1)

(2)

for ω/α > k,

for ω/α < k,

for ω/β > k,

for ω/β < k,

with 

and 

k = ω/c

and 

(3)

In order to solve equation 1, two boundary conditions are utilized: vanishing radial stress, prr = 0, and vanishing tangential 

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

(4)

where λ and μ are the Lamé parameters. Substituting these 

two boundary conditions into equation 3, the following matrix 

equation can be derived for a cylinder of radius a: 

(5)

with 

and 

where ρ is the density of a cylinder.

∂

∂ +

∂

∂ +

∂

∂ = ∂

∂

∂

∂ +

∂

∂ − +

∂

∂ =

2

2

2

2 2

2

2

2

2 2

2

2 2

1 1

1 1

Φ Φ Φ Φ

Ψ Ψ Ψ Ψ

r r r z t

r r r r z

α

β

∂

∂

2

2

Ψ

t

Φ

Ψ

= ∫ + ∫

= ∫ +

− −

−

AJ lr e dk V H lr e dk

BJ mr e dk S H

ikz ikz

ikz

0 0 0

2

1 0

( ) ( )

( )

( )

1

( ) 2

∫ ( ) − mr e dk ikz

l k

l i k

m k

= − 



















= − − 



















= −

ω

α

ω ω

α

ω

β

2

2

2

1 2

2

2

2

2

2

/

sgn

2

1 2

2

2





























/

ik ω

l k

l i k

m k

= − 













= − − 



















= −

α

ω ω

α

ω

β

2

2

2

2

2

2

sgn

2

1 2

2

2

2





















= − − 



















/

m isgn ω k ω

β

p r r z

p

r z

rr

rz

= − +

∂

∂ − ∂

∂ ∂





















= ∂

∂ ∂ −

λ ω

α

μ

μ

ω

β

2

2

2

2

2

2

2

2 2

2

Φ Φ Ψ

Φ Ψ − ∂

∂



















 2

2

2

Ψ

z

Q

a a

a a

A

B

S

S = 













































 = 11 12

21 22

1

2

a k J la lJ la a

a ik mJ ma J ma

11

2 2 2

0

2

1

12 0 1

2 2

2

= − +

= −

ρ β ω ρβ

μ

( ) ( ) ( )/ ,

[ ( ) ( )/ ],

( ),

( ) ( ),

(

a

a i k lJ la

a k J ma

S V k

21 1

22

2 2 2

1

1 0

2 2

2

2

2

=

= −

= −

μ

ρ β ω

ρ ω β2

0

2

0

2

1

2

0 0

2

1

2

2

) ( ) ( )/

[ ( )

( ) ( )

(

H la V lH la a

ik S mH ma H

−

− −

ρβ

 μ 2

2 0 1

2

0

2 2 2

1

2 2 2

)

( )

( )/ ],

( ) ( / ) ( ),

ma a

S V = − μ iklH la − − S k μ ω β H ma
Theory 5

Phase Velocity

The dispersion relation for the normal modes, which 

gives the phase velocity dependence on frequency, is given 

by setting the determinant of equation 5 equal to zero (Ewing 

and others, 1957). The determinant of matrix Q, Δ, is given by 

a11a22 – a12a21: 

which agrees with Graff (1975). When the wave equation is 

solved using the dispersion relation derived from Δ = 0, it is 

called the normal mode solution. The general solution is a 

superposition of each normal mode, with each normal mode 

having a unique phase velocity with respect to frequency. The 

dispersion relation for the dispersive S-wave normal mode 

(DSNM) is derived by requiring the phase velocity greater 

than the S-wave velocity and less than the P-wave velocity. 

In this case, l is purely imaginary and m is a real number. 

Although l is an imaginary number, there exist real values for 

k that solve equation 6. Equation 6 cannot be solved analytically, but the dispersion relationship can be numerically calculated following a similar method to that given for the P-wave 

(7)

A B

normal modes by Lee and Waite (2009). The phase velocity of 

an S-wave normal mode (Cβ) is given by Δ = 0 and this can be 

written as:

Figure 4. Normalized phase and group velocity. A, Numerically calculated phase and group velocities 

for the fundamental-order dispersive shear wave normal mode (DSNM) for an aluminum cylinder with a 

35-mm diameter, P-wave velocity of 6,390 m/s and S-wave velocity of 3,180 m/s. B, Numerical solution of 

equation 7 showing phase velocities at 0.128 and 0.14 MHz. 

1 2

4

2

2

3

2

1 0 − 1

























=

− 



 β

β

α

β

β ω

β

C

C

l J la m J ma

J ma

a

* * ( ) ( ) ( )











− 





















 +







 J ma

C

J la

l J la

a 1

2

2 0

2 2 1 1 2 ( ) ( ) ( ) * β β

 β αω











∆ = − +

−

4

2

2 2 4

1 0 1

2 2 2 2

1

k lJ la mJ ma J ma a

k J m

ρ β

ρ β ω

( )[ ( ) ( )/ ]

 ( ) ( a k J la

lJ la a

)[( ) ( )

( )/ ],

2

2

2 2 2

0

2

1

β ω

β

− +

 

Allowable phase velocities are those for which the left side 

of equation 7 is the same as the right side of equation 7 for a 

given phase velocity. As demonstrated in equation 7, the phase 

velocity of a given normal mode depends on the acoustic 

frequency, ω, the P- and S-wave velocities, α and β, and by the 

cylinder radius, a.

Figure 4A shows phase and group velocities, derived 

below, for the fundamental-order DSNM calculated for a 

35-mm-diameter aluminum cylinder. As frequency increases, 

the phase velocity decreases continuously, reaching a minimum near 0.13 MHz. For frequencies greater than about 

0.13 MHz, the phase velocity decreases monotonically with 

increasing frequency. The phase velocity discontinuity near 

(6) l C m C * * = −1 / = −1 / 2 2 2 2 α β β β where and .
6 S-Wave Normal Mode Propagation in Aluminum Cylinders

The approximate solution using J m0 ( )a = 0 and its implication in the normal mode analysis is given in appendix 1.

When the phase velocity is less than the S-wave velocity, both l and m become imaginary. At high frequencies, the 

dispersion relationship can be written as 

Λβ = + ( / 5 4 π πn ) Πβ = aω β/ .

0.13 MHz can be explained from figure 4B, where numerical 

solutions of phase velocities at two different frequencies just 

above and below 0.13 MHz are shown. The phase velocity of 

the fundamental order at 0.128 MHz is Cβ ≈ β, whereas the 

phase velocity at 0.14 MHz is Cβ ≈ 2β. The abrupt change 

of the phase velocity is not a numerical artifact, but instead 

marks the maximum frequency for bar wave propagation 

(indicated in fig. 4A by the shaded region).

The phase velocities at higher frequencies can be approximated using either J0

(ma) = 0 or J1(ma) = 0. If the approximate solution J1

(ma) = 0 is used for the phase velocities, the 

DSNM phase velocity is given by the following equation by 

incorporating the high frequency approximation of the Bessel 

function ( ( J ma a ) /( ) m a cos( m / ); 1 = 2 p − 3p 4 Watson, 

1966):

Group Velocity

Group velocity (G) is given by Ewing and others (1957) 

as:

 (8)

with and

C a

a β

β

β ω

ω β = −

2 2

2 2 2 2 Λ

4

2

2 1 0

2 2

2 2 2 mJ 0 1 la J ma

C

k C J la J ma

l

( ) ( )

/

( / ) ( ) ( )

. − = −

b

b

b

b

Because J0(la)J1(ma) ≈ J1(la)J0(ma) as l and m become pure 

imaginary numbers, equation 9 becomes:

(9)

4 1 (10)

2

2

1

2 2

2 2

2 2

2 2

−

− = −

−

C

C

C

C

β

β

β

β

β

β

β

α

/

/

/

/

This is an identical form of Rayleigh wave dispersion relationship given in White (1965). In other words, a non-dispersive 

S-wave normal mode exists inside an aluminum cylinder 

if the wavelength is much smaller than the radius of the 

aluminum cylinder.

In this report, the normal mode’s order is defined from 

Λβ = (5π/4 + nπ). If the phase (or group) velocity of a waveform feature approximately agrees with equation 8 when n = 0, 

it is called the fundamental-order normal mode. If the phase 

(or group) velocity agrees with equation 8 when n = 1, it is 

called the first-order normal mode. This naming convention 

differs from that of Kolsky (1963) and Graff (1975). Their 

naming convention is based on the order of the numerical solution of the phase velocity (equation 7) in such a way that for 

a given frequency, the first solution (lowest phase velocity) is 

the first order and the second solution is the second order, and 

so forth.

G d

dk

C k dC

dk = = +

w (11)

Therefore, the S-wave normal mode group velocity (Gβ) is: 

G d

dk a β

β β

β

ω β β

ω = = 1− = β 1−

2 2

2 2

2

2

Λ Λ

Π . (12)

Figure 4A also shows the numerically calculated 

group velocities using the exact dispersion relation for the 

fundamental-order DSNM. At higher frequencies greater 

than about 0.2 MHz, the group velocity is a monotonically 

increasing function of frequency. However, below 0.2 MHz 

there exists a local minimum group velocity near 0.1 MHz 

and a local maximum group velocity near 0.14 MHz. At near 

zero frequency, the phase and group velocities are equal and 

the group velocity is about Cβ ≈ 1.63β , the bar wave velocity shown in Kolsky (1963). Group velocities for frequencies 

up to 0.1 MHz follow the dispersion relation for the bar wave 

given in Kolsky (1963), which was derived from low frequency Bessel function approximations. The phase and group 

velocities approximated using equations 8 and 12 are valid for 

frequencies greater than about 0.4 MHz for this investigation, 

as shown later.

The low-frequency dispersion relation is different from 

that at high frequencies. For bar waves, which are governed 

by the low-frequency dispersion relation, the low-frequency 

component of wave feature arrives before the high-frequency 

component. For DSNM, which are governed by the highfrequency dispersion relation, the high-frequency component 

of wave feature arrives before the low-frequency component. 

Consequently, the bar wave and DSNM described in this paper 

have markedly different waveforms. 

Amplitude

Amplitude relations provide an independent check on the 

normal mode interpretation for characterizing the waveform 

features shown in figures 2A and 3A. Relative amplitudes are 

also particularly useful when differentiating between the fundamental normal modes and the body waves used to determine 

S-wave speeds in materials. Normal mode amplitudes relative 

to each other can be calculated from equations 2 and 5. With 

P- and S-wave sources at the origin, the constants A and B in 

equations 2 and 5 are given by:

A S a S a

A k

k

B S a S a

A k

k

p

S

= − ≡

= − + ≡

( )/ ( )

( )

( )/ ( )

1 22 2 12

1 21 2 11

∆

∆

∆

∆

 (13)
Theory 7

In the following amplitude analysis, only the S-wave amplitude is considered. Then, The integration using the steepest decent method yields 

A k V k (14) ik l J la H la

J

s( ) ( ) [ ( ) ( )

(

( ) = −

−

0

2 2 2 2 2

1 0

2

0

2 2 β ω ρ β

 la H la

V ik l J la H la a

) ( )]

( ( ) ( )/ )

( )

( )

1

2

0

2 4 2

1 1

2 + 4 ρ β

 

 

− S k 0 lJ la mH ma − H ma a 2 2 4

1 0

2

1

2 ρ β { ( 4 )[ ( ) ( )/ ] ( ) ( )

 + − ( ) ( ) ( )} ( ) 2 2 2 2 2

0 1

2 k J β ω la H ma

Substituting B into the integral equation shown in equation 2, the potential for the S-wave solution is given by the following equation in the lower half of the complex k-plane (for 

example, Pekeris, 1948; Ewing and others, 1957; Båth, 1968):

Ψ( ) ( ) ( ) ( )

[

( ) k B k J mr e dk S H mr e dk ikz ikz = ∫ + ∫

=

−

−∞

∞

−∞

∞ −

1 0 1

2

 B k J mr S H mr e dk i R ikz

n

S

Branch cut ( ) ( ) ( )] ( )

1 0 1

2

∫ + − 2 ∑

=

− p

 Ψ Ψ B D + SNM

(15)

where Branch cut is a branch cut integration around the branch 

points kα = ω/α and kβ = ω/β, RS is the residue of integration 

for dispersive S-wave normal modes, and ΨB and ΨDSNM are 

S-wave potentials for body wave and dispersive S-wave normal modes, respectively. 

Ψ

∆ DSNM

S

ikz

k

iA k J mr e

k = −

∂ ∂

−

=

2p 1 ( ) ( ) ] solution of equation 6 (16)

Amplitude of the Dispersive S-Wave 

Normal Mode

The amplitude of the dispersive S-wave normal modes 

can be calculated by substituting solutions of the dispersion 

relation into equation 14. The solution for DSNM is given by

Ψ( ) ( ) ( ) ( )

[

( ) w = ∫ + ∫

=

−

−∞

∞ − B k J mr e dk S H mr e dk

B

ikz ikz

1 0 1

2

1

2 ( ) ( ) ( ( ) ) ( )] . ( ) ( ) k H mr B k S H mr e dk ikz

1

1

0 1

2 2 −∞

∞ −

∫ + +

(17)

ΨB

i R B k R

R

S e

R

k r z

≈ − −

= =

−

−

2 2 0

1

( cos )cos( / )

/ tan ( / )

/ β φ ω β

ω β φ

ω β

β , , R z = + r 2 2 .

The displacements can be written as:

k r z R z r β = = ω β φ = + − / t , an ( / ), . 1 2 2 where and

U r r

rB k R

R

i rV e

z

B B

i R

= ∂

∂

+ ≈

−

−

Ψ Ψ

 2 2

2

0 ω φ ω β

β

β ω ω ( cos ) sin( / ) /φ

β

ω φ

β

φ ω β

ω

R

B k R

rR

S e

rR

U

z

zB

i R

r

B

2

0 2 2

2

 − −

= − ∂

∂ ≈

− ( cos )cos( / ) /

Ψ ( cos ) sin( / ) / k R

R

i zV e

R

i R

β

ω β φ ω β

β

ω

β 2

0

2

2 −

−

 The far-field solution shown in equation 17 requires that 

mr >> 1. The geometry of our measurement, however, does 

not satisfy this condition, because the cylinder radius, a, and 

hence, the observation point radius, r, are both small. The 

body wave amplitude can, therefore, only be approximated by 

assuming the uniformly expanding source waveform can be 

calculated by making a and r large. In other words, the first 

term of equation 17, because this term yields the reflections 

from the cylinder boundaries, can be ignored. From equation 17, the amplitude of S-wave body wave generated by a 

S-wave source can be approximated by:

(18)

U i S e

z

U

r

i z

z

=

≈

− 2

0

ω 0

β

ω β/

 

(19)

Because only shear waves generated by the S-wave source (S0) 

were considered for the following amplitude analysis, V0 can 

be set to zero in equation 14.

Theoretical amplitudes presented here are displacements of the particles and are proportional to the amplitudes 

of measured waveforms. Because we are only interested in 

comparing amplitude within a single waveform, the fact that 

displacement and amplitude are proportional means that they 

can be used interchangeably. 

n

where Δ is given in equation 6 and AS(k) is given in 

equation 14.

Amplitude of the Body Wave 

The branch cut integrations shown in equation 15 yield 

body wave motions inside the solid. The branch cut integration can be evaluated using the steepest decent method if the 

observation point is in the far field, meaning the wave travels 

a distance much longer than its wavelength (for example, 

Morse and Feshbach, 1953; Båth, 1968; Greenfield, 1978). 

Using Hankel functions, equation 15 can be written as 
8 S-Wave Normal Mode Propagation in Aluminum Cylinders

Analysis of Measured Waveforms

The waveform features shown in figure 2A are interpreted as the fundamental and first-order DSNM. For example, 

S150-1 is the fundamental-order DSNM and S150-2 is the 

first-order DSNM for the 150-mm aluminum cylinder. The 

measured group velocities of the fundamental and first-order 

DSNM for the 100-, 150-, and 200-mm aluminum cylinders 

are shown in figure 2B along with the calculated group velocities derived from J1

(ma) = 0 with n = 0 and n = 1. The arrival 

times of the fundamental-order DSNM and S-wave body 

wave are close, thus the waveforms shown as S100-1, S150-1, 

and S200-1 in figure 2A are mixtures of the body wave and 

DSNM. However, for the 200 mm-aluminum cylinder, the 

slightly faster body S-wave is able to separate from the 

fundamental-order normal mode and is observed as a small 

amplitude wave with high frequency content arriving earlier 

than S200-1 (fig. 2A). This waveform clearly demonstrates 

that the body wave and fundamental-order dispersive normal 

mode travel at nearly the same speeds, and the interpretation 

of the large amplitude waveform features such as S100-1 and 

S100-2 as the fundamental- and first-order DSNM appears to 

be sound.

The dominant frequencies of the fundamental- and 

first-order DSNM are very similar for the endcap set no. 1 

waveforms (fig. 2A). This is not the case for the waveforms 

measured using endcap set no. 2, as shown in figure 3A. Two 

interpreted waveform features are shown as blue and red 

colors for the 150- and 200-mm-long cylinders. The dominant 

frequencies of the first part of the arrivals, shown in blue as 

S100-1, S150-1, and S200-1, range from 0.7 to 0.9 MHz, with 

the longer cylinder displaying a higher dominant frequency. 

The dominant frequencies of the second part of the arrivals, shown in red as S150-2 and S200-2, range from 0.28 to 

0.4 MHz. As can be seen in figure 3A, the frequency content 

changes abruptly for S150-2 and S200-2 relative to S150-1 

and S200-1. Figure 3B shows the measured group velocities 

along with the approximate group velocities calculated from 

J1(ma) = 0 with n = 0 and n = 1. Contrary to the results using 

endcap no. 1, all of the measured group velocities for endcap 

no. 2 appear to follow the same trend as the calculated group 

velocities of fundamental-order DSNM.

Velocity Analysis

Relating the group velocity to the dominant frequency 

of a waveform feature using the approximated group velocity 

given by equation 12 appears to work well when identifying 

waveform features generated by endcap set no. 1 (fig. 2B). 

Normal mode identification is less obvious for features S150-2 

and S200-2 when using endcap set no. 2 (fig. 3B), partly due 

to analyzing features with frequencies below 0.4 MHz using 

equation 12, which is intended for group velocities at high frequencies. Here we use the exact dispersion and group velocity 

equations (eq. 7 and 11) to investigate the possibility that the 

S150-2 and S200-2 features in figure 3 are combinations of the 

fundamental and first-order normal modes, rather than being 

purely first-order normal modes. 

Figure 5A shows the exact and approximated phase and 

group velocities for the fundamental-order DSNM, whereas 

figure 5B shows the first-order DSNM. The symbols indicate 

the measured results taken from figures 2B and 3B and plotted 

for comparison with the theoretical curves. The black symbols 

are the fundamental-order normal-mode waveform features, 

whereas the white symbols are first-order normal-mode features. The split-color symbols are the waveform features that 

appear to be superposition of both the fundamental and firstorder normal modes, as discussed below.

Figure 5 indicates that the approximated phase and group 

velocities derived from J1(ma) = 0 appear to be appropriate for frequencies greater than about 0.4 MHz, both for the 

fundamental-order DSNM features identified in figure 5A and 

the first-order DSNM in figure 5B, because of similar trends. 

For frequencies less than about 0.4 MHz, the approximated 

group velocity diverges from the exact solution for the firstorder DSNM (fig. 5B), indicating that the Bessel function 

approximation should not be utilized below 0.4 MHz. Therefore, the approximate group velocity curves shown in figure 

3B or in figure 5 should not be expected to provide insight for 

the split-color features S150-2 and S200-2 measured using 

endcap set no. 2. 

These group velocities shown by the split-color symbols 

fall reasonably close to the first-order group velocity predicted 

using the exact solution (fig. 5B) and can be characterized as 

first-order DSNM. Additional insight into these arrivals can 

be gained by examining their power spectra (fig. 6). Figure 6 

compares the power spectra for S200-1 and S200-2 measured 

using endcap set no. 1 in figure 6A with those using endcap set 

no. 2 in figure 6B. The first characteristic observed in figure 6 

is that the power spectra for the endcap set no. 2 data are complex relative to that of the first-order DSNM using endcap set 

no. 1. Also, S200-2 measured using endcap set no. 2, unlike 

the other features in figure 6A, has two prominent frequency 

components: one at 0.31 MHz and one at 0.525 MHz.

Figure 5B indicates that the measured group velocity of 

S200-2 measured using endcap set no. 2 at the lower dominant 

frequency, 0.31 MHz, agrees with the exact group velocity of 

the first-order DSNM. At the higher frequency component, 

0.525 MHz, the measured group velocity agrees with the exact 

group velocity of the fundamental-order DSNM. This can 

be seen in figure 5A by shifting the split-colored symbol at 

0.31 MHz across to 0.525 MHz, maintaining its plotted group 

velocity. Based on this agreement between the high-frequency 

component of S200-2 and the exact solution prediction of 

the fundamental-order DSNM group velocity, combined with 

the agreement between the low-frequency component and 

the exact solution prediction of the first-order DSNM group 

velocity, we conclude that S200-2 is a combination of the fundamental- and first-order normal modes. The same situation 

holds for S150-2, as measured by endcap set no. 2. 
Analysis of Measured Waveforms 9

A B

A B

Figure 5. Exact and approximate normalized phase and group velocities. A, Fundamental order. B, First 

order. Measured group velocities of both orders are plotted in both A and B for comparison. Symbols are 

the same as for figures 2B and 3B, with fundamental modes as black symbols, the first-order normal modes 

as white symbols, and combined fundamental and first-order normal modes as split-color symbols. DSNM, 

dispersive shear wave normal mode.

Figure 6. Measured power spectra for the 200-mm-long aluminum cylinder. A, Measured using endcap set 

no. 1, displaying a single dominant frequency for both the fundamental- and first-order DSNM. B, Measured 

using endcap set no. 2, displaying a single dominant frequency for the fundamental-order DSNM, but two 

significant frequency contributions for the first-order DSNM. DSNM, dispersive shear wave normal mode.
10 S-Wave Normal Mode Propagation in Aluminum Cylinders

The mixture of two normal modes is not evident for the 

100-mm-long aluminum cylinder. This kind of mixture of 

normal modes occurs because there exists a minimum group 

velocity. For example, this minimum group velocity occurs 

at approximately 0.34 MHz for the first-order normal mode 

(fig. 5B). Therefore, it is emphasized that the approximate 

group velocities derived from the Bessel function are valid 

only for high frequencies, meaning frequencies greater than 

the frequency corresponding to the minimum group velocity.

The mixture of normal modes is not seen in endcap set 

no. 1, for which the fundamental and first-order group velocities are distinct enough to produce distinct waveform features, 

as seen in figure 2A. These waveform features each have a 

single dominant frequency, as seen in figure 6A, indicating that 

they are not a superposition of two normal modes. However, 

the group velocities of S150-2 and S200-2 are between those 

of the fundamental- and the first-order normal modes, and as 

the frequency decreases, the group velocity agrees better with 

that of the fundamental order. At present, there are no satisfactory explanations for the abnormal group velocity behavior 

of S150-2 and S200-2, although bad resonance/vibration 

associated with the transmitted signal could have affected the 

analysis of the waveforms (D.R. Hutchinson, written commun., 2006). With this uncertainty, it is assumed hereafter that 

S150-2 and S200-2 measured using endcap set no. 1 are the 

first-order DSNM.

It is emphasized that the approximate group velocities derived from the Bessel function appear to be useful for 

frequencies greater than the frequency corresponding to the 

minimum group velocity. The frequency corresponding to 

the minimum group velocity increases as the order of normal 

mode increases. Therefore, it is more difficult to apply the 

approximate group velocity using the Bessel function when 

analyzing the higher-order normal modes.

Because of the existence of minimum group velocities 

for the DSNM discussed in this report, the application of 

waveform analysis using the approximate group velocities 

is limited. Consequently, the exact solution of the dispersion relation for DSNM should be used to accurately analyze 

measured waveforms. As opposed to the S-wave normal mode 

described in this report, however, the approximate phase (or 

group) velocities derived from J0(ma) = 0 for torsional wave 

modes are very accurate, as shown in appendix 2. For torsional modes, either the exact or approximate phase (or group) 

velocity can be used and the accuracy of the approximated 

velocity increases as the normal mode order and frequency 

increase. Although the torsional wave is not the subject of this 

report, it is included, because the approximate dispersion relation of torsional wave is instructive for understanding various 

normal modes. 

Amplitude Analysis

As an independent check on the normal-mode approach 

used to identify waveform features, we compared the predicted relative amplitudes of the waveform features. Figure 7 

shows a comparison of the amplitudes of the fundamental and 

first-order DSNM calculated with both the exact dispersion 

relation and approximate dispersion relation using J1(ma) = 0 

with n = 0 and n = 1 for the 200-mm-long cylinder. The largest 

amplitude difference occurs near 0.3 MHz. The amplitude 

of the approximate solution is within about 30 percent of the 

exact amplitude for frequencies greater than 0.4 MHz for the 

fundamental-order DSNM. The amplitude of the first-order 

DSNM calculated using the approximation solution is remarkably close to the exact amplitude, within 10 percent of the 

exact solution, for frequencies greater than 0.4 MHz. Although 

the magnitude of the approximate solution is smaller than that 

of the exact solution for the fundamental-order DSNM, the 

amplitude trends with respect to frequency are similar, indicating that the approximate solution can be used to qualitatively 

compare the amplitudes of the different normal modes at 

high frequencies.

Figure 7. Calculated amplitudes for body wave 

and for fundamental- and first-order dispersive 

S-wave normal mode (DSNM) using the exact and 

the approximate dispersion relations.
Discussion 11

Either the approximate or exact dispersion relation can be 

used to calculate the DSNM amplitudes at high frequencies. 

However, calculating the exact dispersion relation numerically 

is much more complicated than calculating the approximate 

relation. Therefore, it is more practical to utilize the approximate solutions for frequencies greater than the frequency of 

the minimum group velocity. When analyzing amplitude for 

frequencies less than or near frequencies of the minimum group 

velocity, however, the exact dispersion relation must be used.

The waveforms in figure 2A, measured using endcap set 

no. 1, have fundamental- and first-order normal modes that are 

of similar magnitudes. By comparison, the waveforms measured with endcap set no. 2 (fig. 3A) each have a fundamental 

mode that is larger than the trailing feature containing the 

first-order normal mode contribution. Because of their very different frequency content, separate analysis was required for the 

endcap no. 1 and endcap no. 2 results.

Two different frequency ranges are highlighted in figure 7 

and are indicated by shaded boxes. The first shaded area in 

figure 7 spans the frequency range between 0.38 and 0.42 MHz 

and contains the dominant frequencies of the fundamental- and 

first-order normal modes for the 200-mm-long cylinder waveforms measured using endcaps no. 1 (fig. 6A). The amplitude 

of the fundamental-order DSNM at the dominant frequency of 

0.42 MHz is comparable to that of the first-order DSNM at the 

dominant frequency of 0.38 MHz. As mentioned previously, 

the amplitude of the fundamental-order DSNM is mixed with 

the amplitude of body wave, which decreases at 1/length and is 

small compared to the normal mode amplitudes. Therefore, the 

normal-mode solution predicts that the amplitudes of the fundamental- and first-order DSNM are comparable and independent 

of the cylinder length. Measured waveforms shown in figure 2A

support this prediction. 

The second shaded area shown in figure 7 covers the frequency range spanned by the fundamental-order normal modes 

observed for 200-mm-long aluminum cylinder using endcap 

set no. 2 (S200-1). Note that the dominant frequencies for the 

S100-1, S150-1, and S200-1 waveform features are 0.71 MHz, 

0.78 MHz, and 0.89 MHz, respectively. Because these frequencies are within the validity range for using the approximate 

solution, the normal mode amplitudes can be calculated using 

the approximate dispersion equation. Because the amplitude of 

the body wave at high frequencies is the same order of magnitude as the fundamental-order DSNM, the amplitude of the 

fundamental-order DSNM could have some contribution from 

the body wave.

The S150-2 and S200-2 waveform features are each a mixture of the first-order normal mode and the tail of the fundamental-order normal mode. The first-order normal mode amplitude 

near 0.3 MHz shown in figure 7 is relatively small compared to 

the fundamental mode. As indicated by the figure 3A waveforms, the amplitudes of the first features are indeed larger 

than the amplitudes of the second features. However, because 

the waveforms show constructive and destructive interference, 

interpreting the mixed waveform amplitudes using the theoretical amplitude relation shown in figure 7 requires caution. 

Discussion

Phase Velocity and Normal Mode

The discontinuous phase and group velocities shown in 

figure 4 are caused by the assumption made to allow phase 

velocities to be greater than the S-wave velocity of the cylinder. There exist two different normal mode propagations, 

depending on the assumptions about phase velocity: 

1. S-wave normal modes (described here) are derived 

assuming phase velocity cannot be less than shear wave 

velocity, and 

2. extensional wave derived assuming that the high-frequency phase velocity can be less than the shear wave 

velocity (Kolsky, 1963). 

As the frequency increases or wavelength decreases, the 

S-wave normal mode velocities shown in figure 4A approach 

the shear-wave velocity of the cylinder, whereas the shear and 

group velocity of an extensional wave or lowest axial-radial 

mode approaches that of the Rayleigh surface wave velocity of 

the cylinder, which is about 0.93β (fig. 8) (Thurston, 1978). 

Two different normal mode propagations appear to 

be generated in the region indicated as “transition zone” in 

figure 4A when the wavelength is a little less than the cylinder radius. One mode develops into the fundamental-order 

DSNM (fig. 4) and one develops into the Rayleigh surface 

wave (fig. 8). The extensional wave propagation that develops 

into the Rayleigh surface wave is the well-known conventional longitudinal wave propagation (Kolsky, 1963; Graff, 

1975; Thurston, 1978). However, it appears the fundamental 

S-wave normal modes described here are rarely observed or 

discussed. Except for the fundamental-order normal mode, 

the higher DSNM described in this report are the same as the 

higher-order extensional modes as long as the phase velocity 

is greater than the S-wave velocity and less than the P-wave 

velocity of the cylinder (Kolsky, 1963; Graff, 1975). 

Cut-Off Frequencies

Equations 11 and 12 indicate there exists a cut-off 

frequency for each normal mode at which the phase velocity becomes infinite or the group velocity becomes zero. 

The exact cut-off frequency can be obtained from the dispersion relation by setting k = 0 (Thurston, 1978). Inserting k = 0 into equation 6, the cut-off frequency satisfies the 

following equation:

J m1 a lJ la a J la 2

1

2 ( )[ ( 2β )/ − = ω 0 ( )] 0

For DSNM, the term in square brackets is never zero, 

meaning the cut-off frequency is given by the solution of 

J1(ma) = 0.This is exactly the same assumption made for 

the deriving the approximate solution. Therefore, the cut 

(20)
12 S-Wave Normal Mode Propagation in Aluminum Cylinders

2. If these are first-order extensional waves at high frequencies, otherwise known as Rayleigh surface waves, their 

group velocities are too slow. The observed fundamentalorder normal mode features have group velocities exceeding 0.95β, whereas the surface Rayleigh wave velocity 

is about 0.93β, which is about 2,962 m/s (fig. 8, light 

blue curve). 

3. If these are second-order extensional waves (Kolsky, 

1963), their group velocities are too slow (fig. 8). The 

group velocity of the second-order extensional wave, 

which is the same as the first-order DSNM, is 3,056 m/s 

at 0.7 MHz, whereas the measured velocity for the S100-1 

waveform feature measured at this frequency exceeds 

3,180 m/s (fig. 3B). The velocity discrepancy becomes 

more pronounced as the frequency decreases (the first 

arrivals shown in figure 2). 

Therefore, considering arrival times together with the amplitudes shown in figures 2 and 3, a logical interpretation of 

S100-1, S150-1 and S200-1 is that they are the fundamentalorder S-wave normal modes described in this report.

Conclusions

Normal mode solutions in a cylindrical coordinate system 

qualitatively explain large amplitude arrivals measured for 

aluminum cylinders with various lengths. Those waveform 

features have amplitudes and arrival times that differ from the 

well-known shear body and bar waves. The following conclusions can be drawn from this investigation:

1. The phase and group velocities of the waveform features can be approximated using the Bessel function of 

the first kind for frequencies greater than the frequency 

corresponding to the minimum group velocity, which is 

about 0.4 MHz in this analysis. For frequencies less than 

0.4 MHz, exact numerical solutions of the dispersion relation should be used to analyze measured waveforms.

2. The normal mode amplitudes calculated using the 

approximated dispersion relation (J1(ma) = 0) are comparable to those calculated from the exact relation, and 

the qualitative analysis of the normal mode amplitude 

can be accomplished using the approximated solution at 

high frequencies.

3. Two remarkably different waveforms were observed 

depending on which endcaps were used in the investigation. Using endcap set no. 1, the fundamental-order 

dispersive S-wave normal mode (DSNM) and the firstorder DSNM are observed as distinct arrivals, whereas 

using endcap set no. 2, single continuous waveforms 

with quite different frequencies were observed. Using 

endcap set no. 1, the amplitudes of the separate fundamental- and first-order DSNM are comparable for the 

100-, 150- and 200-mm-long aluminum cylinders. Using 

off-frequency at short wavelength is given by Λβ = Πβ. For 

example, the cut-off frequency of the first-order DSNM is 

0.20 MHz, when β = 3,180 m/s and a = 0.0175 m. The group 

velocity curve for the first-order DSNM shown in figure 8 (red 

curve) confirms this cut-off frequency. As the normal mode 

order increases, the cut-off frequency also increases. 

Basis of Normal Mode Interpretation

The S100-1, S150-1, and S200-1 waveform features 

shown in figures 2 and 3 are interpreted as the fundamentalorder DSNM, but there are other interpretations possible. 

These features could also be interpreted as shear body waves, 

Rayleigh surface waves, or second-order extensional waves. 

Based on the following arguments, however, these other interpretations cannot be valid, indicating that these features must 

be the fundamental-order DSNM. The reasons are as follows:

1. If these are S-body wave arrivals, their amplitude dependence on distance does not agree with the theory. Body 

wave amplitudes decay at 1/length, but the amplitudes 

of S100-1, S150-1 and S200-1 have almost the same 

amplitude even though the length of the S200-1 cylinder 

is almost twice that of S100-1.

Figure 8. Calculated group velocity for fundamental 

(lowest)-order extensional mode, which was calculated 

under the assumption that the phase velocity changes 

continuously from greater than the shear-wave velocity 

to less than the shear-wave velocity, with those for the 

fundamental- and first-order DSNM. When the wavelength 

is small compared to the cylinder radius, the velocity of the 

fundamental extensional wave approaches the velocity 

of the Rayleigh surface wave, whereas the velocity of the 

fundamental-order DSNM approaches the shear wave 

velocity. DSNM, dispersive shear wave normal mode. 
References Cited 13

endcap set no. 2, it appears that the low-frequency part 

of the first-order DSNM and the high-frequency part of 

fundamental-order DSNM arrive at the same time, resulting in a waveform feature that is a superposition of lowand high-frequency energy. Detailed analyses of group 

velocity and relative amplitudes using the exact dispersion 

equation are required to fully understand the nature of 

these complex waveforms. 

4. It is not clear why two distinct and remarkably different waveforms were measured using the two different 

endcaps, though the differing frequency content emitted from the two endcap sets is partly responsible for 

this difference. The observed differences highlight the 

importance of characterizing the transmitted signal 

using well-characterized samples, such as aluminum 

cylinders, prior to utilizing the waveform to characterize 

unknown material. 

Acknowledgments

This work was supported by the Gas Hydrate Project 

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

Coastal & Marine Geology Program. We thank Peter M. 

Bratton for his assistance in measuring acoustic waveforms 

and K. Lewis and W. Winters for many helpful comments 

and suggestions. 

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Appendix 1. Approximate Phase Velocity 15

C a

a

n a

β

β

β β

β ω

ω β

π π ω β

= −

= + =

2 2

2 2 2 2

3 4

Λ

 Λ ( / ) Π / .

Appendix 1. Approximate Phase Velocity

In the main text, the approximate phase (or group) velocity is derived from J1(ma) = 0. If 

J0(ma) = 0 is used to derive the approximate phase velocity, the phase velocity at high frequencies is given by

C a

a

n a

β

β

β β

β ω

ω β

π π ω β

= −

= + =

2 2

2 2 2 2

3 4

Λ

 Λ ( / ) Π / .

 (1–1)

with and

A B

Figure 1–1. Exact and approximate normalized phase and group velocities using equation 1–1. 

A, Fundamental order. B, First order. Measured group velocities of both orders are plotted in 

both A and B for comparison. Symbols are the same as for figures 2B and 3B, with fundamental 

modes as black symbols, first-order normal modes as white symbols, and combined normal-and 

first-order modes as split-color symbols. DSNM, dispersive shear wave normal mode.

If the fundamental order is given by equation 1–1 with n = 1 instead of n = 0, the phase 

and group velocities using equation 1–1 are more accurate than those derived from J1(ma) = 0 

with n = 0. Figure 1–1A shows the phase and group velocity of the fundamental order shown 

in figure 5A using equation 1–1 with n = 1, and figure 1–1B shows the first order shown in figure 5B using equation 1–1 with n = 2. Overall those velocities derived from equation 1–1 with 

one order higher than those derived from equation 8 appear to more closely match the exact 

solution. However, the amplitudes of the normal modes calculated using equation 1–1 (shown 

in fig. 1–2) are less accurate than those calculated using equation 8 (shown in fig. 7). Relative 

to the exact phase velocity, the phase velocity approximation using J0(ma) = 0 is slightly high, 

which appears to significantly affect the predicted amplitude. It is not clear why this is the case 

because, in contrast, the phase velocity approximation using J1(ma) = 0 is slightly low and 

the predicted amplitudes are much more accurate. One drawback of using J1(ma) = 0 as the 

approximate dispersion relation is that the cut-off frequency using this approximation does not 

agree with that of the exact equation. Both approximations are useful, however, as either can be 

used depending on the analysis objective. Because the amplitude of the normal mode appears to 

be more diagnostic for identifying normal-mode arrivals, the approximation using J0(ma) = 0 is 

used in the main text.
16 S-Wave Normal Mode Propagation in Aluminum Cylinders

U D J mr e dk ikz

q = ∫ −1( )

J ma

J ma

ma 0

1 2 ( ) ( ) =

C a

a a

T

β

ω

ω β β β ω β

= − = −

2 2

2 2 2 2 2 2 2 2

1

1   Λ Λ /( )

 (2–1)

(2–2) 

 (2–3)

Figure 1–2. Calculated amplitudes for body wave 

and for fundamental- and first-order dispersive 

S-wave normal mode (DSNM) using the exact 

dispersion relation and the approximation shown 

in equation 1–1.

Appendix 2. Torsional Waves

In the torsional vibrations of a cylinder, each transverse section remains in its own plane 

and rotates about its center. The family of torsional modes results when only the azimuthal 

displacement is assumed to exist and is independent of the azimuthal angle. Therefore, the 

azimuthal displacement (Uθ) can be written as (Kolsky, 1963; Graff, 1975):

The boundary condition for the torsional wave is P U r U r rθ = ∂ μ1( /θ ∂ − θ / ) = 0 and the 

dispersion relation is satisfied by the following equation:

The solution of equation 2–2 can be numerically derived excluding ma = 0 (non-dispersive 

phase velocity). The first six solutions of equation 2–2 are ma = 1.84093, ma =5.33135, 

ma = 8.53623, ma =11.70597, ma = 14.86357, and ma = 18.01549. Therefore, the phase velocity of the torsional wave (CT

) is given by

where 

Lb is the solution of equation 2–2, such as ma = 1.84093. 
Appendix 2. Torsional Waves 17

Equation 2–3 indicates that as the frequency increases, the phase velocity approaches the 

shear-wave velocity, and a cut-off frequency exists for each mode. The cut-off frequency ( ) wc

is given by

C a

a a

T

β

ω

ω β β β ω β

= − = −

2 2

2 2 2 2 2 2 2 2

1

Λ 1 Λ /( )

 (2–4)

(2–5)

Figure 2–1. Calculated exact and approximate 

phase velocities for torsional waves for a 

35-mm-diameter aluminum cylinder. The 

approximate phase velocities are derived from 

J m0 ( )a = 0 .

where Λβ = + ( ) 3 4 π π / n . The difference between the exact and approximated phase velocities is in the difference between 

L L b b . The first six Λβ are 2.355, 5.495, 8.635, 11.775, = + ( ) 3 4 π π / n

14.925, and 18.055 (from n = 0 to n = 5). As the order of the normal mode (n) increases, the 

approximate phase velocity approaches the exact phase velocity. Calculated phase velocities 

using equations 2–3 and 2–5 are shown in figure 2–1. As shown in figure 2–1, the approximate phase velocity can be used for high-frequency normal-mode propagation for all orders of 

normal modes. Also, the approximate phase velocity can be used for normal mode propagation 

above that of the fundamental mode for all frequencies. 

The cut-off frequency increases as 

Lb (or the order of normal mode) increases and as the 

diameter of the cylinder decreases. 

If the dispersion equation is approximated using J0(ma) = 0, the phase velocity is given by

and

ω β c = β a Λ/ / .
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Lee, M.W., and Waite, W.F.—S-Wave Normal Mode Propagation in Aluminum Cylinders—Scientific Investigations Report 2010–5013