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Sound pulses in thin non-uniform rods
SOUND PULSES IN THIN NON-UNIFORM by J. F. W.
Experiments
have
pulses in smooth
been
carried
and corrugated
BELL*
out
the
pulses
the inertia Further
used. term
structures
were very
The agreement of
in compliance
corrugations.
propagation
small
enough
much higher
between
Measurements
simple
smooth
rods
were
made
and
applications it is necessary to reduce the velocity of electromagnetic waves in transmission lines and waveguides. The travelling wave amplifier and the waveguide particle accelerator which uses a corrugated waveguide are examples of the use of this reduction. The acoustic counterpart of the corrugated waveguide, the corrugated rod, has been investigated by the authors. It was found that if corrugations were machined on a smooth rod, longitudinal or torsional pulses could be delayed by as much as 20°7! without any great distortion of shape or loss of amplitude, Beyond this figure resonances in the structure become significant and the pulse shape is lost. The investigation consisted of measuring the acoustic resistance and velocity for rods with different degrees of corrugation. This enabled the compliance and inertia terms to be determined. The observed results were then compared with calculated values. Measurements were first made on smooth rods to establish the technique and establish the method of measurement.
INITIAL EXPERIMENTS
The expressions describing the reflection and transmission of sound at a boundary between two media are well known. For example, when a pulse of amplitude A strikes the
torsional
frequencies
in the spectra of
experiment
was good
for
in the compliance a single of full
of the velocity
notch
strain
indicated
that
at the corners
of propagation
degrees of corrugation
I n many
and
to the diameter
the resonant
the frequencies
theory
containing
and inertia terms were evaluated
*Royal Naval College,Greenwch, England.
of longitudinal
to make
than
was due to the absence
resistance of metal rods with different the compliance
the
but there was a large discrepancy
investigation
discrepancy
and B. P. DOYLE*
rods. The pulses used were long compared
of the rods and the corrugations of these elementary
on
RODS
the
of the
and the acoustic
and from these measurements
and compared
with the theoretical
values
boundary of two media of resistances rr and r2 some of the incident pulse energy is reflected and the rest is transmitted. The amplitude reflection coefficient R, the ratio of the reflected amplitude j to A is given by: i/A _m” 2..
rf +
.. .... ..
. . . .(I)
rl
The power (intensity) ratio will be (j/A)“. The power transmitted is 1 - (j/A)2. From Equation I it follows that if the second medium has the higher resistance, the polarity of incident and reflected pulses will be the same. If r2 < r1 the polarity will be reversed. The polarity of the transmitted pulse will always be the same as that of the incident pulse. The nickel tube transducer system used in delay lines1 and by the authors2 to measure the velocity of longitudinal and torsional pulses in solids can be used to illustrate these effects. Fig. 1 shows the echo patterns for longitudinal and torsional pulses for rl > r2 and r1 < r2. The echo beyond the end echo is a multiple path echo from the junction and is always of opposite polarity to the junction echo. It will be noticed that the echoes are not precisely the same shape. This is because the junction acts as a filter, removing the higher frequency components of the pulses. The acoustic resistance of a rod is given by (pc,,)rra2 for longitudinal pulses and &(pc,)na4 for torsional pulses, a being the radius of the rod and pc the specific acoustic resistance of the material of the rod.3
ULTRASONICS
/Junuary-March
1964
39
Table LONGITUDINAL:
MATCHING
1
TORSIO’NA:
3,16
TO
IN
BRASS
ROD
LONGlTClDlNAL E
(I,,
in
i. V
u2, in
I0.188 0.188 0,188 0,188 0.188
0.07950,109 0,125 0.203 0,156
’
12.3
17.3 17.3 17.3 17.2 16.8
8.3 6.7 2.8 1.3
~
0.41 0.59 0.66 0.85 I .08
1 !
0.188 0.188
0.219 0.234
2.3 3.9
TORSIONAL:
(1,.
in
0,188 0.188 0.188 0.188 0.188 0.188 0.188
cr2,
in
0,109 0.125 O-141 0.172 0,203 0.219 0,234
MATCHING
j
16.8 17.2
TO
I.15 I ,26
3’ 16 IN
BR4SS
)
0.423 0.580 0.665 0.830 I.080 I.168 I.25
ROD
A.V ,(,4” ;j
.V
4.0 3.35 2.8 0.9 -- 1 ,o _~~, .7 --2.2
5.05 5.25 5.1 4.9 4.8 5.0 4.9
0.584 0.686 0,735 0,911 1,112 1.194 I .27
0.580 0.665 0.750 0,915 I.080 1,168 I .25
Table 1 shows the quantitative results of reflection experiments. Equation 1 reduces to Equation 2 for longitudinal pulses. 1 . . . . . . . . .._..... and to Equation
3 for torsional
pulses . . . . . . . . . . . . . . . .
Fig. I. Echo patterns of pulses transmitted into one end of a plain rod. Pulses enter from the left. From left to right the echoes are junction echo (J), end echo (E) and a multiple echo (M) which has travelled on the “end-junction-end” path
The agreement between observed and calculated values of cr.Ju, verifies the assumptions implicit in the experiment, e.g., that echo heights arc a true measure of signal amplitude and that non linearity and changes in echo spectrum from echo to echo are negligible. The authors have used echo height comparison to measure the attenuation of sound in solids at high temperatures. The specimen rod is welded or otherwise
LONGlTCiD’NAL
Plain rod 5.00 in long
0.156in
:ORSIONAL
in diameter
Lightly corrugated rod 5.00 in long x: 0.156in in diameter. to 0.0.110 in diameter 0.030 in wide every 0.20 in
Slots cut
t -4-4 J
Heavily corrugated rod 5.00 in long Slots cut to 0.110 in diameter 0.030
0,156in in diameter in wide every 0.10 in
1
Fig. 2. Junction and end echoes from rods with various dcgrces of corrugation attached IO a slightly heavier rod. The height of the junction echo gives the resistance and the interval between echoes gives the vclocit). The time scales are 20~s and 40~s per division for longitudinal and torsional paths respectively
40
rll:l-~.4so~1~s/Jurzuur~-Murch
I964
Table 2 LONGITUDINAL
A, V
i, V ~
Plain rod, diameter 0.156 in
PULSES
i-z-’
t, CLS
_~ 1 0.650
0.806
1
Lightly corrugated rod _____Heavily corrugated rod
3.0
10.2
0.545
0.738
/
4.4
10.8
0.421
0.649
Plain rod, diameter 0,125 in
4.3
10.3
0,411
0.641
1.8
TORSIONAL
~~._~
~~~~~...
~
,_~__._ ~ 14.8
Lightly corrugated
rod
--p,-Gp,-o_iioi 8.1
Heavily corrugated
rod
9.7
14.7
0.674
9.6
14.7
0.677
(,j/A)2 1
. . . . . . . . . . . . * *(4)
The attenuation in the specimen is 20 log (e,/e) dB where e is the observed echo height. This method is of particular value at high temperatures where the attenuation is high but lacks sensitivity when the material has a Q-factor above about IO.
CORRUCiATED
1.126
0.151
1.232
0.142
lx0.115
/
0.465
attached to a probe rod and located in the isothermal region of a furnace. By measuring A and j the end echo height e,, in the absence of attenuation can be calculated 1 -
81.8 ____89.6
, :
5.4
e, = A
0.127
i
PULSES
Plain rod, diameter 0.156 in
Plain rod, diameter 0.125 in
0.143
RODS
An extension of the simple theory has been used to deduce the compliance and inertia of corrugated rods for conditions limited to geometries where the resonant frequencies are high compared to frequencies in the pulse spectrum. This means that the depth of the corrugations must be a small fraction of the diameter and the separation small compared to the pulse length. Fig. 2 shows the longitudinal and torsional echo patterns from three brass rods each of the same length attached to a second, slightly larger diameter, brass rod. On each oscillogram the first echo is from the junction and the second is from the free end of the rod. The time scales are 20~s and 40~s per division for the longitudinal and torsional patterns respectively. The features apparent are, first the fall in acoustic resistance with greater corrugation, shown by the increase in junction echo, and second, the increase in time delay with greater corrugation. Both changes are greater for torsional than for longitudinal pulses. It will be noticed that there is very little echoing from the notches along the length of the rod. Table 2 shows the measurements corresponding to Fig. 2:d,,,,, is the diameter deduced from echo heights. It is
1 0.124
~ 209-6
~ 1.634 I I
i
0,141
0.110
I
the diameter a plain rod would have if it gave the same junction echo height. r is the ratio of velocity in a plain rod to that in a corrugated rod. These observations enable the inertia and compliance terms of the rods to be separated. (acoustic resistance)2 = inertia/compliance l/c2 = inertia x compliance. Hence, for longitudinal pulses d, = dObar”’and d, = d,,,,r I/”. . . . . . . . . . . . (5) For torsional pulses d, = d,,bsr’.4 and d, = dol,,Tr’ I. . . . . . . . . . .(6) The equivalent plain rod diameters which would give the same inertia and compliance as the corrugated rods, d,,, and d, are also shown in Table 2. Theoretical support for these observations has been attempted. The expression for the velocity of a long pulse in a uniform rod is L’= z/E/p. The absence of rod diameter in the equation arises from the cancellation of the diameter in the expressions for compliance and inertia. In a corrugated rod the compliance is 4 S/d2 z-E Zl where 1 is the length of the region of diameter is given by pi 4 Hence c2 = As
d. The inertia
Z1d2 ZI
E (W2 -p Zl/d2Zld2
(W2 .~< 1, any irregularities Zl/d2 Zld2
the velocity. The experimental values of d,, and Table 3 beside the calculated values.
.(7) in a rod will reduce
d, are
ULTRAsoNrCsfJunuury-Murch
i9fi4
shown
in
41
Table 3 LOi’GGLTUDINAL d,,,,
Lightly
corrugated
Heavily
corrugated
Fig.
3.
The
0.10
time an long
exptl
talc
exptl
talc
cvptl
talc
0.151
0.150
0.135
0.145
0,154
0.151
0.131
0.142
0.142
0.144
0.115
0,137
0.141
0.146
0.1 IO
O-133
06
delay
caused
(Equotlon
by
the
IO
08 9
double
passage
of
a
pulse
past
II
plotted against F -Measured
values
-Theoretical
cuwc
from
Equation
X
DISCONTINUITY
To test this effect, the analysis was applied to the delay ,/ I produced by a single notch of length I cut to a diameter ‘I, on a rod of diameter ~1,. [‘.I Where c is the velocity longitudinal pulses
IF
. . . . . . . . . . . .(8)
2c
of the
pulse
and
where.
for
.(9) For torsional pulses the indices 2 are replaced by indices 4. Two sets of measurements were carried out. The first
43
(I,, in
cl,,,, in
talc
The results show good agreement between experimental and calculated values of the inertia terms for both longitudinal and torsional propagation. There is a large disbetween the compliance terms. A physical crepancy picture is that the large diameter sections follow the movement of the pulse and make a full contribution to inertia. These sections are not fully stressed at the edges near the section junctions and the experimental compliance will be less than the calculated value. This implies that there is an end correction for each notch.
NOTCH
in
exptl
04
02
F
notch
cl,,
in
_ ~-.I._.__l__.._L+
I--.
0
TORSION41
I ILTRASONICS / Jrmuury-Much
1964
set is shown in F-ig. 3 where the variation of delay due to a double passage past the notch is plotted as a function of F-. A notch of width 0.10 in on a rod of diameter 0.156 in was used. The theoretical line is also shown. The discrepancy between theoretical and experimental values of I gives the end correction. The second set of measurements, shown in Fig. 4, gives the variation of delay as a function of notch width for two diameters of the notch. It is at once evident that the end correction is a true concept. The end corrections for the two diameters are 0.16 ps and 0.28 ;Ls. From the results shown in Fig. 3 the corrections are 0.15 ;Ls and 0.32 IIS respectively. The agreement is within the limits of accuracy of the observations. Thus, the use of mean values of inertia and compliance to describe the propagation of pulses in corrugated rods is supported by experiment. The discrepancies have been brought together in terms of an end correction. but further theoretical work would be of value to the authors.
Experiments
have
pulses in smooth
been
carried
and corrugated
BELL*
out
the
pulses
the inertia Further
used. term
structures
were very
The agreement of
in compliance
corrugations.
propagation
small
enough
much higher
between
Measurements
simple
smooth
rods
were
made
and
applications it is necessary to reduce the velocity of electromagnetic waves in transmission lines and waveguides. The travelling wave amplifier and the waveguide particle accelerator which uses a corrugated waveguide are examples of the use of this reduction. The acoustic counterpart of the corrugated waveguide, the corrugated rod, has been investigated by the authors. It was found that if corrugations were machined on a smooth rod, longitudinal or torsional pulses could be delayed by as much as 20°7! without any great distortion of shape or loss of amplitude, Beyond this figure resonances in the structure become significant and the pulse shape is lost. The investigation consisted of measuring the acoustic resistance and velocity for rods with different degrees of corrugation. This enabled the compliance and inertia terms to be determined. The observed results were then compared with calculated values. Measurements were first made on smooth rods to establish the technique and establish the method of measurement.
INITIAL EXPERIMENTS
The expressions describing the reflection and transmission of sound at a boundary between two media are well known. For example, when a pulse of amplitude A strikes the
torsional
frequencies
in the spectra of
experiment
was good
for
in the compliance a single of full
of the velocity
notch
strain
indicated
that
at the corners
of propagation
degrees of corrugation
I n many
and
to the diameter
the resonant
the frequencies
theory
containing
and inertia terms were evaluated
*Royal Naval College,Greenwch, England.
of longitudinal
to make
than
was due to the absence
resistance of metal rods with different the compliance
the
but there was a large discrepancy
investigation
discrepancy
and B. P. DOYLE*
rods. The pulses used were long compared
of the rods and the corrugations of these elementary
on
RODS
the
of the
and the acoustic
and from these measurements
and compared
with the theoretical
values
boundary of two media of resistances rr and r2 some of the incident pulse energy is reflected and the rest is transmitted. The amplitude reflection coefficient R, the ratio of the reflected amplitude j to A is given by: i/A _m” 2..
rf +
.. .... ..
. . . .(I)
rl
The power (intensity) ratio will be (j/A)“. The power transmitted is 1 - (j/A)2. From Equation I it follows that if the second medium has the higher resistance, the polarity of incident and reflected pulses will be the same. If r2 < r1 the polarity will be reversed. The polarity of the transmitted pulse will always be the same as that of the incident pulse. The nickel tube transducer system used in delay lines1 and by the authors2 to measure the velocity of longitudinal and torsional pulses in solids can be used to illustrate these effects. Fig. 1 shows the echo patterns for longitudinal and torsional pulses for rl > r2 and r1 < r2. The echo beyond the end echo is a multiple path echo from the junction and is always of opposite polarity to the junction echo. It will be noticed that the echoes are not precisely the same shape. This is because the junction acts as a filter, removing the higher frequency components of the pulses. The acoustic resistance of a rod is given by (pc,,)rra2 for longitudinal pulses and &(pc,)na4 for torsional pulses, a being the radius of the rod and pc the specific acoustic resistance of the material of the rod.3
ULTRASONICS
/Junuary-March
1964
39
Table LONGITUDINAL:
MATCHING
1
TORSIO’NA:
3,16
TO
IN
BRASS
ROD
LONGlTClDlNAL E
(I,,
in
i. V
u2, in
I0.188 0.188 0,188 0,188 0.188
0.07950,109 0,125 0.203 0,156
’
12.3
17.3 17.3 17.3 17.2 16.8
8.3 6.7 2.8 1.3
~
0.41 0.59 0.66 0.85 I .08
1 !
0.188 0.188
0.219 0.234
2.3 3.9
TORSIONAL:
(1,.
in
0,188 0.188 0.188 0.188 0.188 0.188 0.188
cr2,
in
0,109 0.125 O-141 0.172 0,203 0.219 0,234
MATCHING
j
16.8 17.2
TO
I.15 I ,26
3’ 16 IN
BR4SS
)
0.423 0.580 0.665 0.830 I.080 I.168 I.25
ROD
A.V ,(,4” ;j
.V
4.0 3.35 2.8 0.9 -- 1 ,o _~~, .7 --2.2
5.05 5.25 5.1 4.9 4.8 5.0 4.9
0.584 0.686 0,735 0,911 1,112 1.194 I .27
0.580 0.665 0.750 0,915 I.080 1,168 I .25
Table 1 shows the quantitative results of reflection experiments. Equation 1 reduces to Equation 2 for longitudinal pulses. 1 . . . . . . . . .._..... and to Equation
3 for torsional
pulses . . . . . . . . . . . . . . . .
Fig. I. Echo patterns of pulses transmitted into one end of a plain rod. Pulses enter from the left. From left to right the echoes are junction echo (J), end echo (E) and a multiple echo (M) which has travelled on the “end-junction-end” path
The agreement between observed and calculated values of cr.Ju, verifies the assumptions implicit in the experiment, e.g., that echo heights arc a true measure of signal amplitude and that non linearity and changes in echo spectrum from echo to echo are negligible. The authors have used echo height comparison to measure the attenuation of sound in solids at high temperatures. The specimen rod is welded or otherwise
LONGlTCiD’NAL
Plain rod 5.00 in long
0.156in
:ORSIONAL
in diameter
Lightly corrugated rod 5.00 in long x: 0.156in in diameter. to 0.0.110 in diameter 0.030 in wide every 0.20 in
Slots cut
t -4-4 J
Heavily corrugated rod 5.00 in long Slots cut to 0.110 in diameter 0.030
0,156in in diameter in wide every 0.10 in
1
Fig. 2. Junction and end echoes from rods with various dcgrces of corrugation attached IO a slightly heavier rod. The height of the junction echo gives the resistance and the interval between echoes gives the vclocit). The time scales are 20~s and 40~s per division for longitudinal and torsional paths respectively
40
rll:l-~.4so~1~s/Jurzuur~-Murch
I964
Table 2 LONGITUDINAL
A, V
i, V ~
Plain rod, diameter 0.156 in
PULSES
i-z-’
t, CLS
_~ 1 0.650
0.806
1
Lightly corrugated rod _____Heavily corrugated rod
3.0
10.2
0.545
0.738
/
4.4
10.8
0.421
0.649
Plain rod, diameter 0,125 in
4.3
10.3
0,411
0.641
1.8
TORSIONAL
~~._~
~~~~~...
~
,_~__._ ~ 14.8
Lightly corrugated
rod
--p,-Gp,-o_iioi 8.1
Heavily corrugated
rod
9.7
14.7
0.674
9.6
14.7
0.677
(,j/A)2 1
. . . . . . . . . . . . * *(4)
The attenuation in the specimen is 20 log (e,/e) dB where e is the observed echo height. This method is of particular value at high temperatures where the attenuation is high but lacks sensitivity when the material has a Q-factor above about IO.
CORRUCiATED
1.126
0.151
1.232
0.142
lx0.115
/
0.465
attached to a probe rod and located in the isothermal region of a furnace. By measuring A and j the end echo height e,, in the absence of attenuation can be calculated 1 -
81.8 ____89.6
, :
5.4
e, = A
0.127
i
PULSES
Plain rod, diameter 0.156 in
Plain rod, diameter 0.125 in
0.143
RODS
An extension of the simple theory has been used to deduce the compliance and inertia of corrugated rods for conditions limited to geometries where the resonant frequencies are high compared to frequencies in the pulse spectrum. This means that the depth of the corrugations must be a small fraction of the diameter and the separation small compared to the pulse length. Fig. 2 shows the longitudinal and torsional echo patterns from three brass rods each of the same length attached to a second, slightly larger diameter, brass rod. On each oscillogram the first echo is from the junction and the second is from the free end of the rod. The time scales are 20~s and 40~s per division for the longitudinal and torsional patterns respectively. The features apparent are, first the fall in acoustic resistance with greater corrugation, shown by the increase in junction echo, and second, the increase in time delay with greater corrugation. Both changes are greater for torsional than for longitudinal pulses. It will be noticed that there is very little echoing from the notches along the length of the rod. Table 2 shows the measurements corresponding to Fig. 2:d,,,,, is the diameter deduced from echo heights. It is
1 0.124
~ 209-6
~ 1.634 I I
i
0,141
0.110
I
the diameter a plain rod would have if it gave the same junction echo height. r is the ratio of velocity in a plain rod to that in a corrugated rod. These observations enable the inertia and compliance terms of the rods to be separated. (acoustic resistance)2 = inertia/compliance l/c2 = inertia x compliance. Hence, for longitudinal pulses d, = dObar”’and d, = d,,,,r I/”. . . . . . . . . . . . (5) For torsional pulses d, = d,,bsr’.4 and d, = dol,,Tr’ I. . . . . . . . . . .(6) The equivalent plain rod diameters which would give the same inertia and compliance as the corrugated rods, d,,, and d, are also shown in Table 2. Theoretical support for these observations has been attempted. The expression for the velocity of a long pulse in a uniform rod is L’= z/E/p. The absence of rod diameter in the equation arises from the cancellation of the diameter in the expressions for compliance and inertia. In a corrugated rod the compliance is 4 S/d2 z-E Zl where 1 is the length of the region of diameter is given by pi 4 Hence c2 = As
d. The inertia
Z1d2 ZI
E (W2 -p Zl/d2Zld2
(W2 .~< 1, any irregularities Zl/d2 Zld2
the velocity. The experimental values of d,, and Table 3 beside the calculated values.
.(7) in a rod will reduce
d, are
ULTRAsoNrCsfJunuury-Murch
i9fi4
shown
in
41
Table 3 LOi’GGLTUDINAL d,,,,
Lightly
corrugated
Heavily
corrugated
Fig.
3.
The
0.10
time an long
exptl
talc
exptl
talc
cvptl
talc
0.151
0.150
0.135
0.145
0,154
0.151
0.131
0.142
0.142
0.144
0.115
0,137
0.141
0.146
0.1 IO
O-133
06
delay
caused
(Equotlon
by
the
IO
08 9
double
passage
of
a
pulse
past
II
plotted against F -Measured
values
-Theoretical
cuwc
from
Equation
X
DISCONTINUITY
To test this effect, the analysis was applied to the delay ,/ I produced by a single notch of length I cut to a diameter ‘I, on a rod of diameter ~1,. [‘.I Where c is the velocity longitudinal pulses
IF
. . . . . . . . . . . .(8)
2c
of the
pulse
and
where.
for
.(9) For torsional pulses the indices 2 are replaced by indices 4. Two sets of measurements were carried out. The first
43
(I,, in
cl,,,, in
talc
The results show good agreement between experimental and calculated values of the inertia terms for both longitudinal and torsional propagation. There is a large disbetween the compliance terms. A physical crepancy picture is that the large diameter sections follow the movement of the pulse and make a full contribution to inertia. These sections are not fully stressed at the edges near the section junctions and the experimental compliance will be less than the calculated value. This implies that there is an end correction for each notch.
NOTCH
in
exptl
04
02
F
notch
cl,,
in
_ ~-.I._.__l__.._L+
I--.
0
TORSION41
I ILTRASONICS / Jrmuury-Much
1964
set is shown in F-ig. 3 where the variation of delay due to a double passage past the notch is plotted as a function of F-. A notch of width 0.10 in on a rod of diameter 0.156 in was used. The theoretical line is also shown. The discrepancy between theoretical and experimental values of I gives the end correction. The second set of measurements, shown in Fig. 4, gives the variation of delay as a function of notch width for two diameters of the notch. It is at once evident that the end correction is a true concept. The end corrections for the two diameters are 0.16 ps and 0.28 ;Ls. From the results shown in Fig. 3 the corrections are 0.15 ;Ls and 0.32 IIS respectively. The agreement is within the limits of accuracy of the observations. Thus, the use of mean values of inertia and compliance to describe the propagation of pulses in corrugated rods is supported by experiment. The discrepancies have been brought together in terms of an end correction. but further theoretical work would be of value to the authors.