Design of Waveguide Bars for Transmitting a Pure Shear Horizontal Wave to Monitor High Temperature Components

Guided wave technique could be a possible method for monitoring components working in high temperature above 350 °C. However, this would require the design of an appropriate waveguide bar to transmit the wave, so that its sensing part is not influenced by the high temperature. In the present study, the shape of waveguide bars is designed based on the analysis of wave source characteristics. The critical frequency-width and frequency-thickness products of waveguide bars are analyzed theoretically and numerically to transmit the zeroth shear horizontal wave SH0* in bars. The results show that waveguide bars can cut off all the other wave modes when their frequency-thickness products are smaller than the critical value fd*, and frequency-width products are not smaller than the critical value fw*. Six waveguide bars are designed and fabricated based on the design criteria, and an experiment system is set up to check their work performance. The testing results indicate that the wave signals of the SH0* mode propagate clearly in waveguide bars, and cut off all the other modes when the frequency-thickness products and frequency-width products of the bars meet the design criteria. It is also demonstrated that the dependency of the experimental group velocity of each waveguide bar on frequency is in good agreement with the numerical result. High-temperature experiments also validate the reliability of the designed waveguide bars. Therefore, the critical frequency-thickness product and frequency-width product can be the basis for the design of the waveguide bars.


Introduction
High temperature components are applied comprehensively in aerospace and process industries. Many researchers have suggested that monitoring the structural health of components with permanently installed transducers is a useful way to maintain their safety [1][2][3][4][5][6][7][8][9]. Permanent installation will allow measurements to be more frequent and remove errors introduced by re-installing transducers. The financial consequences of prolonged operational shutdowns have previously aroused strong interest in developing ultrasonic transducers that can operate at elevated temperatures above 350 • C for a long time. Li [10] has developed BiScO3-PbTiO3 (BS-PT) high-temperature transducers. However, lithium niobate decreases in sensitivity over time when left in high temperature. Hou et al. [11] have directly deposited thick piezo films onto high temperature structures. Sinding et al. [12] have sprayed piezoelectric powder on the surfaces of structures. The attachment procedure for these transducers is intricate and time-consuming, requiring several deposition steps and subsequent poling to achieve robust attachment. None of them have achieved ideal results for high temperature longtime usage. the more desirable characteristics than the torsional point sources for the guided wave monitoring system.

Theoretical Analyses
The anti-plane shear line loading source doesn't exist in real life. A waveguide bar of large width-to-thickness ratio (width >> thickness) is the closest practically implementable approximation. The wave excited by the anti-plane shear line loading source is a shear horizontal wave (shorten for SH wave), which can be depicted in Figure 1. The wave propagates in the x1 direction, the particle vibrates in the x3 direction, and there is no out-of-plane particle displacement in the vibration of the SH wave. The explicit solutions for the group velocity in terms of frequency-thickness product fd are constructed by Rose [27]. The group velocity Cg is where d equals to the thickness of the layer, f is frequency, CT is the shear wave speed. n  {0, 1, 2,    }.
Since the material of many high temperature components is made of stainless steel, stainless steel is also chosen as the material of waveguide bars in order to reduce refraction of the wave transmitting from waveguide bars into components. The shear wave speed in a stainless steel layer is CT = 3200 m/s. The group velocity curves for the first six SH modes in a stainless steel layer are plotted in Figure 2.  The explicit solutions for the group velocity in terms of frequency-thickness product fd are constructed by Rose [27]. The group velocity C g is where d equals to the thickness of the layer, f is frequency, C T is the shear wave speed. n ∈ {0, 1, 2, · · · }. Since the material of many high temperature components is made of stainless steel, stainless steel is also chosen as the material of waveguide bars in order to reduce refraction of the wave transmitting from waveguide bars into components. The shear wave speed in a stainless steel layer is C T = 3200 m/s. The group velocity curves for the first six SH modes in a stainless steel layer are plotted in Figure 2. the more desirable characteristics than the torsional point sources for the guided wave monitoring system.

Theoretical Analyses
The anti-plane shear line loading source doesn't exist in real life. A waveguide bar of large width-to-thickness ratio (width >> thickness) is the closest practically implementable approximation. The wave excited by the anti-plane shear line loading source is a shear horizontal wave (shorten for SH wave), which can be depicted in Figure 1. The wave propagates in the x1 direction, the particle vibrates in the x3 direction, and there is no out-of-plane particle displacement in the vibration of the SH wave. The explicit solutions for the group velocity in terms of frequency-thickness product fd are constructed by Rose [27]. The group velocity Cg is where d equals to the thickness of the layer, f is frequency, CT is the shear wave speed. n  {0, 1, 2,    }. Since the material of many high temperature components is made of stainless steel, stainless steel is also chosen as the material of waveguide bars in order to reduce refraction of the wave transmitting from waveguide bars into components. The shear wave speed in a stainless steel layer is CT = 3200 m/s. The group velocity curves for the first six SH modes in a stainless steel layer are plotted in Figure 2.

Calculating the Thickness
According to Figure 2, when n = 0, the wave is in the SH0 mode, the group velocity of which is not frequency-dependent. It is a non-dispersion wave propagating at the shear wave speed C T . All other SH modes (n ∈ {1, 2, · · · }) are dispersive. As the frequency-thickness product fd approaches infinity for any given fixed n, the group velocity of any SH mode approaches that of bulk shear waves C T . In this case, it is different to achieve in actual engineering. Here, we consider that the frequency-thickness product is small enough.
Supposing that the group velocity of SH1 is zero, Equation (1) can be written as In this case, a cut off frequency-thickness product can be calculated according to Equation (2) fd* = 1.6 MHz·mm, When the frequency-thickness product fd is smaller than 1.6 MHz·mm, there is only SH0. Since there is no out-of-plane particle displacement in a SH wave, it is less affected by the presence of surrounding media. Furthermore, the group velocity of the SH0 wave mode isn't frequency-dependent, which can simplify the signal analysis for different acquisition frequencies. The characteristic that the SH0 wave will not convert to other modes when defects exit can reduce the complexity of data processing and improve the ability to identify defects. Therefore, the SH0 mode is preferred in structural health monitoring over all the other SH modes. The SH0 mode is defined as a desired mode, and all the other SH modes are defined as undesired ones in the present study. The frequency-thickness product fd* is designated as the critical value of the frequency-thickness product to cut off the undesired modes.
Useful frequencies for non-destructive inspection normally range from 1 MHz to 5 MHz [25]. When the acquisition frequency of non-destructive monitoring is selected, the thickness of the layer can be calculated according to the critical value of the frequency-thickness product. For example, when the signal frequency is 1 MHz, the thickness d can be calculated as 1.6 mm, according to Equation (3). A geometrical thickness of the layer equal to 1 mm can be selected, taking into account the convenience of material selection. At this condition, the undesired SH modes can be cut off and only SH0 can propagate through the layer with 1 mm thickness, as shown in Figure 3. The first wave packet is excitation signal, and the second is received signal. The received signal is very clear. The presence of other modes that are much weaker than the main signal can be ignored.

Calculating the Thickness
According to Figure 2, when n = 0, the wave is in the SH0 mode, the group velocity of which is not frequency-dependent. It is a non-dispersion wave propagating at the shear wave speed CT. All other SH modes (n  {1, 2,    }) are dispersive. As the frequency-thickness product fd approaches infinity for any given fixed n, the group velocity of any SH mode approaches that of bulk shear waves CT. In this case, it is different to achieve in actual engineering. Here, we consider that the frequency-thickness product is small enough.
Supposing that the group velocity of SH1 is zero, Equation (1) can be written as In this case, a cut off frequency-thickness product can be calculated according to Equation (2) fd* = 1.6 MHz·mm, When the frequency-thickness product fd is smaller than 1.6 MHz·mm, there is only SH0. Since there is no out-of-plane particle displacement in a SH wave, it is less affected by the presence of surrounding media. Furthermore, the group velocity of the SH0 wave mode isn't frequencydependent, which can simplify the signal analysis for different acquisition frequencies. The characteristic that the SH0 wave will not convert to other modes when defects exit can reduce the complexity of data processing and improve the ability to identify defects. Therefore, the SH0 mode is preferred in structural health monitoring over all the other SH modes. The SH0 mode is defined as a desired mode, and all the other SH modes are defined as undesired ones in the present study. The frequency-thickness product fd* is designated as the critical value of the frequency-thickness product to cut off the undesired modes.
Useful frequencies for non-destructive inspection normally range from 1 MHz to 5 MHz [25]. When the acquisition frequency of non-destructive monitoring is selected, the thickness of the layer can be calculated according to the critical value of the frequency-thickness product. For example, when the signal frequency is 1 MHz, the thickness d can be calculated as 1.6 mm, according to Equation (3). A geometrical thickness of the layer equal to 1 mm can be selected, taking into account the convenience of material selection. At this condition, the undesired SH modes can be cut off and only SH0 can propagate through the layer with 1 mm thickness, as shown in Figure 3. The first wave packet is excitation signal, and the second is received signal. The received signal is very clear. The presence of other modes that are much weaker than the main signal can be ignored. Therefore, the critical frequency-thickness product can be considered as the criterion for thickness selection, when the signal frequency is selected.  Therefore, the critical frequency-thickness product can be considered as the criterion for thickness selection, when the signal frequency is selected.

Calculating the Width
Dispersion characteristics in flat layers and in waveguide bars are not identical. Strictly speaking, the term 'shear horizontal' doesn't make sense in a waveguide bar other than an infinite plate. Hence, the name SH0* mode is chosen to indicate the zeroth-order SH mode in a waveguide bar. In order to compare the SH0 and SH0* modes, the group velocities of the waveguide bar with 1 mm thickness are simulated by ANSYS/LS-DYNA software (ANSYS 12.0, ANSYS, INC., Canonsburg, PA, USA). The material of the waveguide bar is 316 L steel. The material's characteristics are listed in Table 1. Three waveguide bars are designed, and the geometrical sizes to be analyzed by numerical simulation are inventoried in Table 2. In the process of simulation, excitation signals are sent on one end, and reception signals are caught on the same end of bars. The reception signals are part of excitation signals that travel along the waveguide bar and conduct back from the end of the waveguide bars. The group velocities of waveforms are calculated by the method of Time of Flight [28]. The group velocities' curves versus the frequency-width product of the reception signals are calculated and shown in Figure 4. It is found that the dispersion behavior in stainless steel waveguide bars is a function of the product of the frequency of the signal and the width of the waveguide bars. All SH0* curves of waveguide bars with different widths are coincident. At frequency, well above the cut-off fw* = 15 MHz·mm, the group velocity of SH0* modes asymptotically approaches the bulk shear velocity of the SH0 mode. The waves also propagate clearly with advantageous non-dispersion, which can be noticed from Figure 3 in Section 3.2.

Calculating the Width
Dispersion characteristics in flat layers and in waveguide bars are not identical. Strictly speaking, the term 'shear horizontal' doesn't make sense in a waveguide bar other than an infinite plate. Hence, the name SH0* mode is chosen to indicate the zeroth-order SH mode in a waveguide bar. In order to compare the SH0 and SH0* modes, the group velocities of the waveguide bar with 1 mm thickness are simulated by ANSYS/LS-DYNA software (ANSYS 12.0, ANSYS, INC., Canonsburg, PA, USA). The material of the waveguide bar is 316 L steel. The material's characteristics are listed in Table 1. Three waveguide bars are designed, and the geometrical sizes to be analyzed by numerical simulation are inventoried in Table 2. In the process of simulation, excitation signals are sent on one end, and reception signals are caught on the same end of bars. The reception signals are part of excitation signals that travel along the waveguide bar and conduct back from the end of the waveguide bars. The group velocities of waveforms are calculated by the method of Time of Flight [28]. The group velocities' curves versus the frequency-width product of the reception signals are calculated and shown in Figure 4. It is found that the dispersion behavior in stainless steel waveguide bars is a function of the product of the frequency of the signal and the width of the waveguide bars. All SH0* curves of waveguide bars with different widths are coincident. At frequency, well above the cut-off fw* = 15 MHz·mm, the group velocity of SH0* modes asymptotically approaches the bulk shear velocity of the SH0 mode. The waves also propagate clearly with advantageous non-dispersion, which can be noticed from Figure 3 in Section 2.2.   Therefore, the critical frequency-width product can be the criterion for width design to acquire the ideal wave mode. Therefore, the critical frequency-width product can be the criterion for width design to acquire the ideal wave mode.

Frequency Dependence
The curves of the group velocity dispersion versus frequency for the SH mode of 1-mm thick steel bars of different widths (30, 25 and 20 mm) are plotted in Figure 5. It is found that the group velocity dispersion has a cut-off frequency that depends on the width of the bars. At frequencies well above the cut-off, the group velocity asymptotically approaches the bulk shear velocity C T . Based on the dispersion characteristics of the SH wave in waveguide bars, it can be found that the signal frequency and the geometric width of waveguide bars are strongly interrelated at low frequency-width products. Therefore, the signal frequencies in the application of non-destructive testing should be high enough to make sure that the frequency-width product isn't lower than the critical value. In this condition, the waves propagating through the waveguide bars can be guaranteed to be in the SH0* mode.

Frequency Dependence
The curves of the group velocity dispersion versus frequency for the SH mode of 1-mm thick steel bars of different widths (30, 25 and 20 mm) are plotted in Figure 5. It is found that the group velocity dispersion has a cut-off frequency that depends on the width of the bars. At frequencies well above the cut-off, the group velocity asymptotically approaches the bulk shear velocity CT. Based on the dispersion characteristics of the SH wave in waveguide bars, it can be found that the signal frequency and the geometric width of waveguide bars are strongly interrelated at low frequencywidth products. Therefore, the signal frequencies in the application of non-destructive testing should be high enough to make sure that the frequency-width product isn't lower than the critical value. In this condition, the waves propagating through the waveguide bars can be guaranteed to be in the SH0* mode.

Structural Design Criteria
Based on the above-mentioned analysis, the critical frequency-thickness product and frequencywidth product can be the design criteria of the geometrical structure of the waveguide bar for a given frequency in the normally used non-destructive frequency range (1 MHz, 5 MHz). When the frequency-thickness products are smaller than the critical value fd*, and frequency-width products are not smaller than the critical value fw*, waveguide bars can cut off the undesired wave modes. For the waveguide bars designed by these criteria, the SH0* mode of ultrasonic waves can propagate through with advantageous non-dispersion. It is an ideal condition to detect the deterioration process of components.

Experimental System
In order to verify the reliability of the design criteria regarding the frequency-thickness product and the frequency-width product, and thus design the geometrical structure of the waveguide bar, an experimental system is set up by consulting references [29,30]. The diagram and the picture are shown in Figure 6. The experimental system includes an ultrasonic testing system RITEC-SNAP 5000 (Ritec, INC., Warwick, RI, USA), an oscilloscope, an attenuator, an amplifier, a duplexer, a waveguide bar, and a transducer. The transducer is the Olympus V153-RM (Olympus NDT, INC., Waltham, MA, USA), which has a central frequency of 1 MHz and can excite shear horizontal waves. The duplexer is the RDX-6 (Ritec, INC., Warwick, RI, USA), which is a transformer arrangement that delivers highpower pulses to a transducer, while returned signals from the same transducer are transferred to a receiver. It can also filter signals at the same time. Ten cycle tone bursts modulated with a Hanning window are generated at 1 MHz using the testing system, and recorded at a sampling rate of 50 MHz. Moreover, six different waveguide bars are fabricated, and the pictures are shown in Figure 7. The

Structural Design Criteria
Based on the above-mentioned analysis, the critical frequency-thickness product and frequency-width product can be the design criteria of the geometrical structure of the waveguide bar for a given frequency in the normally used non-destructive frequency range (1 MHz, 5 MHz). When the frequency-thickness products are smaller than the critical value fd*, and frequency-width products are not smaller than the critical value fw*, waveguide bars can cut off the undesired wave modes. For the waveguide bars designed by these criteria, the SH0* mode of ultrasonic waves can propagate through with advantageous non-dispersion. It is an ideal condition to detect the deterioration process of components.

Experimental System
In order to verify the reliability of the design criteria regarding the frequency-thickness product and the frequency-width product, and thus design the geometrical structure of the waveguide bar, an experimental system is set up by consulting references [29,30]. The diagram and the picture are shown in Figure 6. The experimental system includes an ultrasonic testing system RITEC-SNAP 5000 (Ritec, INC., Warwick, RI, USA), an oscilloscope, an attenuator, an amplifier, a duplexer, a waveguide bar, and a transducer. The transducer is the Olympus V153-RM (Olympus NDT, INC., Waltham, MA, USA), which has a central frequency of 1 MHz and can excite shear horizontal waves. The duplexer is the RDX-6 (Ritec, INC., Warwick, RI, USA), which is a transformer arrangement that delivers high-power pulses to a transducer, while returned signals from the same transducer are transferred to a receiver. It can also filter signals at the same time. Ten cycle tone bursts modulated with a Hanning window are generated at 1 MHz using the testing system, and recorded at a sampling rate of 50 MHz. Moreover, six different waveguide bars are fabricated, and the pictures are shown in Figure 7. The geometric structures of the bars are listed in Table 3. A purpose-made installation tool is designed to fix the transducer on one end of the waveguide bar. In the process of testing, the transducer works as exciter and receiver. The installation picture of transducers is evinced in Figure 8. The SWC shear wave couplant is applied between the transducer and the end of the waveguide bar in experiments.  Table 3. A purpose-made installation tool is designed to fix the transducer on one end of the waveguide bar. In the process of testing, the transducer works as exciter and receiver. The installation picture of transducers is evinced in Figure 8. The SWC shear wave couplant is applied between the transducer and the end of the waveguide bar in experiments.
(a) (b)      geometric structures of the bars are listed in Table 3. A purpose-made installation tool is designed to fix the transducer on one end of the waveguide bar. In the process of testing, the transducer works as exciter and receiver. The installation picture of transducers is evinced in Figure 8. The SWC shear wave couplant is applied between the transducer and the end of the waveguide bar in experiments.
(a) (b)      Table 3. A purpose-made installation tool is designed to fix the transducer on one end of the waveguide bar. In the process of testing, the transducer works as exciter and receiver. The installation picture of transducers is evinced in Figure 8. The SWC shear wave couplant is applied between the transducer and the end of the waveguide bar in experiments.

Verification of Thickness
We laid special stress on analyzing the reception signal at one of the normally used frequencies, which is 1 MHz. In order to prove the effect of the thickness of waveguide bars on the purity of the signal, two waveguide bars have been designed and fabricated. The width of the waveguide bars is designed as 15 mm, according to the critical frequency-width product fw*. The thicknesses of the bars are chosen as 1 mm and 4 mm, respectively. When the thickness is 1 mm, the frequency-thickness product fd = 1 MHz·mm, which is smaller than the critical frequency-thickness product fd*. When the thickness is 4 mm, the frequency-thickness product fd = 4 MHz·mm, which is bigger than the critical frequency-thickness product fd*. The picture of the waveguide bars is shown in Figure 9. The experimental waveforms of waveguide bar 1 and waveguide bar 2 are plotted in Figure 10. There is cross-talk at the beginning of the time domain of the waveforms. In the present study, the cross-talk signals are not analyzed; instead, the reception signals back from the ends of the waveguide bars are analyzed. In Figure 10a, there is a main reception signal, and the presence of other modes is much weaker than the main signal, so that the reception signal can be considered as only one mode, and all other modes are cut off. The calculated group velocity 3012 m/s is in good agreement with the theoretical group velocity 3200 m/s of the SH0 wave in the steel layer, so the received signal is in SH0* mode. It is noticed that for the received signals in Figure 10b, waveforms are dispersive. The group velocity of the first one packet is calculated as 3016 m/s, and the group velocity of the second one is calculated as 2740 m/s, which is in good agreement with the theoretical group velocity 2752 m/s of SH1 mode. There are still other modes following after SH1 wave mode.

Verification of Thickness
We laid special stress on analyzing the reception signal at one of the normally used frequencies, which is 1 MHz. In order to prove the effect of the thickness of waveguide bars on the purity of the signal, two waveguide bars have been designed and fabricated. The width of the waveguide bars is designed as 15 mm, according to the critical frequency-width product fw*. The thicknesses of the bars are chosen as 1 mm and 4 mm, respectively. When the thickness is 1 mm, the frequency-thickness product fd = 1 MHz·mm, which is smaller than the critical frequency-thickness product fd*. When the thickness is 4 mm, the frequency-thickness product fd = 4 MHz·mm, which is bigger than the critical frequency-thickness product fd*. The picture of the waveguide bars is shown in Figure 9. The experimental waveforms of waveguide bar 1 and waveguide bar 2 are plotted in Figure 10. There is cross-talk at the beginning of the time domain of the waveforms. In the present study, the crosstalk signals are not analyzed; instead, the reception signals back from the ends of the waveguide bars are analyzed. In Figure 10a, there is a main reception signal, and the presence of other modes is much weaker than the main signal, so that the reception signal can be considered as only one mode, and all other modes are cut off. The calculated group velocity 3012 m/s is in good agreement with the theoretical group velocity 3200 m/s of the SH0 wave in the steel layer, so the received signal is in SH0* mode. It is noticed that for the received signals in Figure 10b, waveforms are dispersive. The group velocity of the first one packet is calculated as 3016 m/s, and the group velocity of the second one is calculated as 2740 m/s, which is in good agreement with the theoretical group velocity 2752 m/s of SH1 mode. There are still other modes following after SH1 wave mode. In other words, when the real frequency-thickness product of the waveguide bar is smaller than the critical value, only SH0* can propagate through. Otherwise, the reception signal will disperse. Therefore, the critical frequency-thickness product fd* = 1.6 MHz·mm can be the design criterion for a given signal frequency.  In other words, when the real frequency-thickness product of the waveguide bar is smaller than the critical value, only SH0* can propagate through. Otherwise, the reception signal will disperse. Therefore, the critical frequency-thickness product fd* = 1.6 MHz·mm can be the design criterion for a given signal frequency.

Verification of Thickness
We laid special stress on analyzing the reception signal at one of the normally used frequencies, which is 1 MHz. In order to prove the effect of the thickness of waveguide bars on the purity of the signal, two waveguide bars have been designed and fabricated. The width of the waveguide bars is designed as 15 mm, according to the critical frequency-width product fw*. The thicknesses of the bars are chosen as 1 mm and 4 mm, respectively. When the thickness is 1 mm, the frequency-thickness product fd = 1 MHz·mm, which is smaller than the critical frequency-thickness product fd*. When the thickness is 4 mm, the frequency-thickness product fd = 4 MHz·mm, which is bigger than the critical frequency-thickness product fd*. The picture of the waveguide bars is shown in Figure 9. The experimental waveforms of waveguide bar 1 and waveguide bar 2 are plotted in Figure 10. There is cross-talk at the beginning of the time domain of the waveforms. In the present study, the crosstalk signals are not analyzed; instead, the reception signals back from the ends of the waveguide bars are analyzed. In Figure 10a, there is a main reception signal, and the presence of other modes is much weaker than the main signal, so that the reception signal can be considered as only one mode, and all other modes are cut off. The calculated group velocity 3012 m/s is in good agreement with the theoretical group velocity 3200 m/s of the SH0 wave in the steel layer, so the received signal is in SH0* mode. It is noticed that for the received signals in Figure 10b, waveforms are dispersive. The group velocity of the first one packet is calculated as 3016 m/s, and the group velocity of the second one is calculated as 2740 m/s, which is in good agreement with the theoretical group velocity 2752 m/s of SH1 mode. There are still other modes following after SH1 wave mode. In other words, when the real frequency-thickness product of the waveguide bar is smaller than the critical value, only SH0* can propagate through. Otherwise, the reception signal will disperse. Therefore, the critical frequency-thickness product fd* = 1.6 MHz·mm can be the design criterion for a given signal frequency.

Verification of Width
In order to prove the effect of the width of waveguide bars on the purity of the reception signal, two waveguide bars have been designed and fabricated. The thickness of the bars is designed as 1 mm, which is based on the critical frequency-thickness product fd*. The width of the waveguide bars is 7 mm and 15 mm, respectively. When the width is 7 mm, the frequency-width product fw = 7 MHz·mm, which is smaller than the critical frequency-width product fw*. When the width is 15 mm, the frequency-width product fw = 15 MHz·mm, which is equal to the critical frequency-width product fw*. The picture of the waveguide bars is shown in Figure 11. The signals propagated through waveguide bar No. 1, No. 3 and No. 4 are plotted in Figure 12. In Figure 12a,b, only the SH0* mode exists, and all of the other SH modes are cut off. In Figure 12c, waveforms are dispersive, and more than one SH mode exists in the reception signals.

Verification of Width
In order to prove the effect of the width of waveguide bars on the purity of the reception signal, two waveguide bars have been designed and fabricated. The thickness of the bars is designed as 1 mm, which is based on the critical frequency-thickness product fd*. The width of the waveguide bars is 7 mm and 15 mm, respectively. When the width is 7 mm, the frequency-width product fw = 7 MHz·mm, which is smaller than the critical frequency-width product fw*. When the width is 15 mm, the frequency-width product fw = 15 MHz·mm, which is equal to the critical frequency-width product fw*. The picture of the waveguide bars is shown in Figure 11. The signals propagated through waveguide bar No. 1, No. 3 and No. 4 are plotted in Figure 12. In Figure 12a,b, only the SH0* mode exists, and all of the other SH modes are cut off. In Figure 12c, waveforms are dispersive, and more than one SH mode exists in the reception signals. Therefore, when the real frequency-width product of the waveguide bar isn't smaller than the critical value, only the SH0* mode can propagate through. Otherwise, the reception signal will

Verification of Width
In order to prove the effect of the width of waveguide bars on the purity of the reception signal, two waveguide bars have been designed and fabricated. The thickness of the bars is designed as 1 mm, which is based on the critical frequency-thickness product fd*. The width of the waveguide bars is 7 mm and 15 mm, respectively. When the width is 7 mm, the frequency-width product fw = 7 MHz·mm, which is smaller than the critical frequency-width product fw*. When the width is 15 mm, the frequency-width product fw = 15 MHz·mm, which is equal to the critical frequency-width product fw*. The picture of the waveguide bars is shown in Figure 11. The signals propagated through waveguide bar No. 1, No. 3 and No. 4 are plotted in Figure 12. In Figure 12a,b, only the SH0* mode exists, and all of the other SH modes are cut off. In Figure 12c, waveforms are dispersive, and more than one SH mode exists in the reception signals. Therefore, when the real frequency-width product of the waveguide bar isn't smaller than the critical value, only the SH0* mode can propagate through. Otherwise, the reception signal will Therefore, when the real frequency-width product of the waveguide bar isn't smaller than the critical value, only the SH0* mode can propagate through. Otherwise, the reception signal will disperse. Therefore, the critical frequency-width product fw* = 15 MHz·mm can be the design criterion for a given signal frequency.

Verification of the Frequency Dependence
According to the critical frequency-width product fw* = 15 MHz·mm, three steel waveguide bars are chosen. They are No. 4, No. 5 and No. 6. For the signal frequency 1 MHz, the frequency-width products are all bigger than the critical value fw*.
The group velocities of the different wave modes excited in all those three waveguide bars are calculated by time of flight method at different frequencies. The experimental results and numerical results are compared in Figure 13. The experimental data proximately distribute around their corresponding numerical values; that is to say, they are in good agreement with each other. disperse. Therefore, the critical frequency-width product fw* = 15 MHz·mm can be the design criterion for a given signal frequency.

Verification of the Frequency Dependence
According to the critical frequency-width product fw* = 15 MHz·mm, three steel waveguide bars are chosen. They are No. 4, No. 5 and No. 6. For the signal frequency 1 MHz, the frequency-width products are all bigger than the critical value fw*.
The group velocities of the different wave modes excited in all those three waveguide bars are calculated by time of flight method at different frequencies. The experimental results and numerical results are compared in Figure 13. The experimental data proximately distribute around their corresponding numerical values; that is to say, they are in good agreement with each other. According to Figure 13, it is also found that the wave signals propagating in the rectangular waveguide bar have a cut-off frequency that depends on the width of the waveguide bars. For the bars with different widths, they have a different cut-off frequency. For a given frequency, the designed width needs to be wider than the calculated width, so that the real frequency-width product isn't lower than the critical value, even when the signal frequency fluctuates in engineering application.

Propagating Characteristics of Designed Waveguide Bars
The frequency-thickness products and frequency-width products of the waveguide bars (No. 3-No. 6) all meet the design standards. The experiments are performed, and reception signals are plotted in Figure 14. Waveforms are very clear with significant non-dispersion. The presence of other modes that are much weaker than the SH0* mode can be ignored.
Therefore, it has been verified that the critical frequency-thickness product fd* and frequencywidth product fw* can be the basis to design the geometrical structures of waveguide bars in order to get pure SH0* mode signal. According to Figure 13, it is also found that the wave signals propagating in the rectangular waveguide bar have a cut-off frequency that depends on the width of the waveguide bars. For the bars with different widths, they have a different cut-off frequency. For a given frequency, the designed width needs to be wider than the calculated width, so that the real frequency-width product isn't lower than the critical value, even when the signal frequency fluctuates in engineering application.

Propagating Characteristics of Designed Waveguide Bars
The frequency-thickness products and frequency-width products of the waveguide bars (No. 3-No. 6) all meet the design standards. The experiments are performed, and reception signals are plotted in Figure 14. Waveforms are very clear with significant non-dispersion. The presence of other modes that are much weaker than the SH0* mode can be ignored.
Therefore, it has been verified that the critical frequency-thickness product fd* and frequency-width product fw* can be the basis to design the geometrical structures of waveguide bars in order to get pure SH0* mode signal.

High Temperature Experimental Validation
Propagating characteristics of designed waveguide bars are then tested at high temperature. In this testing system, a high temperature furnace is applied. The test set-up is shown in Figure 15. The waveguide bar goes through a hole in the insulating layer of the furnace and reaches outside of the furnace. One end of the waveguide bar is bonded inside the furnace, and the other end is coupled to the ultrasonic transducer, which works in room temperature. The waveguide bars designed in the present study can be integrated in ultrasonic testing equipment to monitor high-temperature components. The range temperature of this ultrasonic testing equipment mainly depends on the material of the waveguide bar. The material selected for waveguide bars is 316 L steel, which can be used up to 650 °C. Therefore, the ultrasonic testing equipment is expected to operate up to 650 °C. In the present study, the target temperature of 350 °C is tested. When the furnace is heated to 350 °C, the temperature is holding. The transducer end can be safely held by hand. There is no noticeable increase in temperature. Due to the limitation of high-temperature installation tools, the propagating characteristics of two waveguide bars have been tested, which are No. 1 and No. 4. The received signals are compared with ones from room temperature experiments, as shown in Figure  16. It is noticed that the received signal is very clear, too. The presence of other modes of signals that are much weaker than the main signal can be ignored. These features are in line with those of the room temperature experiment. Signal amplitudes remain strong. There is no drastic change in attenuation. However, differences still exist. The waveforms delay about 4% at higher temperature because of the reduction of group velocity in the waveguide bar at high temperatures.

High Temperature Experimental Validation
Propagating characteristics of designed waveguide bars are then tested at high temperature. In this testing system, a high temperature furnace is applied. The test set-up is shown in Figure 15. The waveguide bar goes through a hole in the insulating layer of the furnace and reaches outside of the furnace. One end of the waveguide bar is bonded inside the furnace, and the other end is coupled to the ultrasonic transducer, which works in room temperature. The waveguide bars designed in the present study can be integrated in ultrasonic testing equipment to monitor high-temperature components. The range temperature of this ultrasonic testing equipment mainly depends on the material of the waveguide bar. The material selected for waveguide bars is 316 L steel, which can be used up to 650 • C. Therefore, the ultrasonic testing equipment is expected to operate up to 650 • C.

High Temperature Experimental Validation
Propagating characteristics of designed waveguide bars are then tested at high temperature. In this testing system, a high temperature furnace is applied. The test set-up is shown in Figure 15. The waveguide bar goes through a hole in the insulating layer of the furnace and reaches outside of the furnace. One end of the waveguide bar is bonded inside the furnace, and the other end is coupled to the ultrasonic transducer, which works in room temperature. The waveguide bars designed in the present study can be integrated in ultrasonic testing equipment to monitor high-temperature components. The range temperature of this ultrasonic testing equipment mainly depends on the material of the waveguide bar. The material selected for waveguide bars is 316 L steel, which can be used up to 650 °C. Therefore, the ultrasonic testing equipment is expected to operate up to 650 °C. In the present study, the target temperature of 350 °C is tested. When the furnace is heated to 350 °C, the temperature is holding. The transducer end can be safely held by hand. There is no noticeable increase in temperature. Due to the limitation of high-temperature installation tools, the propagating characteristics of two waveguide bars have been tested, which are No. 1 and No. 4. The received signals are compared with ones from room temperature experiments, as shown in Figure  16. It is noticed that the received signal is very clear, too. The presence of other modes of signals that are much weaker than the main signal can be ignored. These features are in line with those of the room temperature experiment. Signal amplitudes remain strong. There is no drastic change in attenuation. However, differences still exist. The waveforms delay about 4% at higher temperature because of the reduction of group velocity in the waveguide bar at high temperatures. In the present study, the target temperature of 350 • C is tested. When the furnace is heated to 350 • C, the temperature is holding. The transducer end can be safely held by hand. There is no noticeable increase in temperature. Due to the limitation of high-temperature installation tools, the propagating characteristics of two waveguide bars have been tested, which are No. 1 and No. 4. The received signals are compared with ones from room temperature experiments, as shown in Figure 16. It is noticed that the received signal is very clear, too. The presence of other modes of signals that are much weaker than the main signal can be ignored. These features are in line with those of the room temperature experiment. Signal amplitudes remain strong. There is no drastic change in attenuation. However, differences still exist. The waveforms delay about 4% at higher temperature because of the reduction of group velocity in the waveguide bar at high temperatures. Generally, the temperature of high-temperature components fluctuates slightly in a normal working period, so the change of group velocity can be negligible. When the temperature fluctuates obviously, the temperature compensation technology can be utilized. Therefore, the conclusion can be drawn that the waveguide bars designed by the critical frequency-thickness product fd* and frequency-width product fw* can be used to measure high-temperature components by pure SH0* mode signal.

Conclusions
In order to introduce ultrasonic wave technology into high-temperature components monitoring to improve their security, the waveguide bars have been designed to transmit the wave so that the sensing part is not influenced by high temperature. According to wave source characteristics analysis, a large aspect ratio rectangular waveguide bar is designed to approximately load the anti-plane shear line source. In order to get a very clear wave signal with advantageous non-dispersion in the waveguide bar, the transmitting characteristics are analyzed theoretically and numerically. It is noticed that the frequency-thickness product of bars should be smaller than the critical value fd*, and frequency-width product should be not smaller than the critical value fw* to cut off the undesired wave mode. Moreover, some waveguide bars are designed and fabricated based on these design criteria, and experiments are carried out. The experimental dependencies of group velocities on frequencies are in good agreement with numerical simulation results. It is also found from the experimental waveforms that the signals can propagate clearly and non-dispersedly in the waveguide bar when the frequency-thickness products and frequency-width products of the bars meet the design criteria. High temperature experiments are carried out, and the experimental results show that the designed waveguide bars can work quite well. Therefore, the feasibility of the design method is verified.  Generally, the temperature of high-temperature components fluctuates slightly in a normal working period, so the change of group velocity can be negligible. When the temperature fluctuates obviously, the temperature compensation technology can be utilized. Therefore, the conclusion can be drawn that the waveguide bars designed by the critical frequency-thickness product fd* and frequency-width product fw* can be used to measure high-temperature components by pure SH0* mode signal.

Conclusions
In order to introduce ultrasonic wave technology into high-temperature components monitoring to improve their security, the waveguide bars have been designed to transmit the wave so that the sensing part is not influenced by high temperature. According to wave source characteristics analysis, a large aspect ratio rectangular waveguide bar is designed to approximately load the anti-plane shear line source. In order to get a very clear wave signal with advantageous non-dispersion in the waveguide bar, the transmitting characteristics are analyzed theoretically and numerically. It is noticed that the frequency-thickness product of bars should be smaller than the critical value fd*, and frequency-width product should be not smaller than the critical value fw* to cut off the undesired wave mode. Moreover, some waveguide bars are designed and fabricated based on these design criteria, and experiments are carried out. The experimental dependencies of group velocities on frequencies are in good agreement with numerical simulation results. It is also found from the experimental waveforms that the signals can propagate clearly and non-dispersedly in the waveguide bar when the frequency-thickness products and frequency-width products of the bars meet the design criteria. High temperature experiments are carried out, and the experimental results show that the designed waveguide bars can work quite well. Therefore, the feasibility of the design method is verified.