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·Methods and Instrumentation
Ultrasonic Testing Using a Unipolar PulseY. Udagawa
Imaging Supersonic Laboratories Co.,Ltd. 12-7 Tezukayamanakamachi Nara Japan 6310063
Krautkramer Japan Co.,Ltd. 9-29 Sumida 1-chome Higashiosaka Japan 5780912
Numerical calculation of acoustic field of a transducer was carried on the following equation,
Where P is acoustic pressure at an observing point, f is acoustic pressure of the transmitted pulse, t is time, r is distance from the source to the observing point, v is sound velocity and S means transducer surface. A transducer is plane and circular and the diameter is 20. Linear dimension is normalized by wavelength and time is normalized by period. These are shown in Fig.1.
|Fig 1: Calculation of acoustic field of a circular transducer||Fig 2: Shape of transmitted pulse|
|Fig 3: Near field of a circular plane transducer for Pulse 1,2 and 3|
Calculation was done on the following three pulses.
Pulse 1. A unipolar pulse which is one quota period of sinusoidal wave
Pulse 2. An asymmetric sinusoidal pulse of one and half of waves as a represent of short pulse
Pulse 3. A sinusoidal pulse of four waves as a represent of burst wave
These pulses are shown in Fig.2.
3.2. Result and discussion
Results of calculation are shown in Fig.3. X and Z is shown in Fig.1. Z of 100 is so called transition point from near field to far field. In the figure, the maximum amplitude of pulse is plotted. In (b) and (c), the maximum amplitudes of plus and minus amplitude of pulse are shown.
As shown in Fig.3 (a), near field of Pulse 1, a unipolar pulse, has constant distribution near the transducer surface and its width decreases as Z.
Near field of Pulse 2, asymmetric sinusoidal pulse of 1.5 waves, is shown in Fig.3 (b). The plus amplitude has constant distribution but minus amplitude has a peak at the side of distribution, of which reason will be shown in later.
Fig. 3 (c) shows near field of Pulse 3, sinusoidal pulse of 4 waves. The amplitude fluctuates as well known.
Calculation of eq.(1) also gives the pulse shape and they are shown in Fig.4 for the representative points. In order to understand the calculation result especially for the oscillating pulse, we consider the results on the central axis of transducer.
On the central axis of the transducer, Z axis in Fig.1, eq.(1) can be expressed as a following analytical form, because of compensation of r factors..
Where L0 is length of pulse in distance and T0 is length of pulse in time. Integral limit in eq.(2) is further limited by the source position. The lower limit of t is the shortest travelling time and the upper one is the longest travelling time. If there is no limitation for them and eq.(2) is integrated over domain of T0, it is zero for ordinary oscillating pulse, dipolar pulse. It has practical value only for integral over time domain smaller than T0. That is, eq.(2) is not zero for the first arrival pulse group and the last arrival one. Eq.(2) can be converted for those time domain as following respectively, except constant terms,
Where t starts at the arrival time of first pulse in (2a) and at the arrival time of end pulse in (2b) and the first term of (2b) is usually zero..
For a unipolar pulse, eq.(2) is not zero because of no cancellation of interference. As shown in a left figure of Fig.4 (a) for a unipolar pulse, the pulse succeeds the first integrated pulse continuously, and the total pulse length is elongated in the range of arrival time delay. The pulse length decreases with Z and it becomes as same as the transmitted pulse at Z=100 which is the transition point from the near field to the far field.
|Fig 4a: Pulse 1,A unipolar pulse|
|Fig 4b: Pulse 2, An asymmetric sinusoidal pulse of 1.5 waves|
|Fig 4c: Pulse 3, A sinusoidal pulse of four waves|
|Fig 4: Received pulse shape at point (X,Z) for pulse 1,2 and 3 Abcissa is time in period and ordinate is amplitude of same magnitude as Fig. 2. Start of each pulse is shifted by 0.25|
The left figure of Fig.4 (b) shows received pulse shape of Pulse 2, asymmetric sinusoidal pulse of 1.5 waves, on the central axis. It is clearly shown in the figure for Z=25, that the pulse is composed by two peaks, the arrival peak and the end peak which are given by eq.(2a) and (2b). The end peak looks like as if it starts from the transducer edge. This is a same phenomenon as the edge scattering. Interference of both pulses gives distributions in Fig.3. The left figure of Fig. 4 (c) can be interpreted in the same way. At positions off the central axis, the pulse shape becomes more complex because of r factor. But principle of formation of pulse shape is same.
3.3. Section Conclusion
A received pulse of a unipolar pulse in the near field has shape of elongated unipolar pulse. The peak amplitude is given by integral of emitted pulse and the width is given by the delay of pulse emitted from the transducer edge. The peak amplitude distributes uniformly and it is useful for inspection of constant sensitivity.
4.1 Numerical simulation
Calculation was done on the following assumption.
Attenuation is caused by the grain boundary reflection. Reflection direction is isotropic and the reflection points distribute uniformly. Only the primary reflection is included in the calculation. The secondary and the tertiary reflections and so on are neglected. Linear dimension is normalized by wavelength. The sound pressure is calculated in the far field. Integration is given by the following equation.
Where m is attenuation coefficient and V is volume in which reflection occurs. Distance R1 and R2 are shown in Fig.5. If the reflection induces phase change, sign of eq.(3) is inverted.
4.2. Result and discussion
Fig.5 shows pulse shapes of a received pulse for a unipolar pulse of Fig.2 observed at Z=100. The transducer is a circular one of radius 10. Ordinate is the same magnitude as Fig.2. In Fig.5 a directly transmitted pulse and a superposed pulse of scattered pulses are shown. Legth of the superposed reflected pulse is very long compared to the emitted pulse.
|Fig 5: Integration of reflected pulse||(a)m=0.03/wavelength||(b)m=0.1/wavelength|
|Fig 6:Pulse shape of a directly transmitted pulse and a superposed pulse of reflected pulses by grain boundaries for a unipolar pulse emitted from a circular transducer of radius 10 at Z=100.|
In Fig.6, shape of received pulse at Z=100 for a sinusoidal pulse of 1.5 waves, Pulse 2 in Fig.2, emitted from the same transducer is shown. The length of superposed pulse of reflected pulse is same as the length of transmitted pulse. The same result is obtained for Pulse 3.
In comparison of Fig.5 and Fig. 6, The most significant difference is length of pulse. Reflected unipolar pulses compose a long pulse, but reflected dipolar pulses can not form a long pulse because of interference. Pulse energy becomes large as pulse length. Then we can detect the intense unipolar pulse by using an amplifier, which has an appropriate low frequency band.
Above-mentioned result is right qualitatively although it is not correct quantitatively because of rough assumption. If there is phase change at reflection and the secondary reflections occur, results become more complex.
|Fig 7: Pulse shape of a directly transmitted pulse and a superposed pulse of reflected pulses by grain boundaries for a sinusoidal pulse of 1.5 waves emitted from a circular transducer of radius 10 at Z=100. µ is 0.1/wavelength.|
4.3. Section Conclusion
A unipolar pulse has low frequency component as well known in d function and it is sometimes thought that it can penetrate the coarse grain materials. However the present calculation shows that the superposition of scattered pulses having various delay times gives long pulse length and large energy to the transparent pulse. This result suggests that the transmission of a unipolar pulse and receiving it by an amplifier of low frequency band is effective for ultrasonic testing of coarse grain materials.
|(a) Transmitted pulse||(b) Received pulse after passing through concrete of 250mm t.||
(c) Received pulse after passing through cork of 25mm t.
||Fig 8: Pulse shapes of (a) transmitted pulse, and pulse after passing through (b) concrete and (c) cork.
As shown in Fig.8, pulse length of transmitted pulse is about 2 micro seconds and received pulse lengths after passing through concrete and cork are a few tens of micro seconds. This means that many delayed pulses through materials superpose and make a long pulse.
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