490
X. Zeng et al.
43.2.2 Dispersion Curve Analysis
According to Silk and Bainton’s theory [7], the guide wave in the cylinder is divided
into three modes: axisymmetric longitudinal mode L(0,m), axisymmetric torsion
mode T (0,m) and axisymmetric bending mode F(n,m) (n,m = 1,2,3…). The modes
reflect the different propagation modes and vibration modes of guided waves in the
cylinder. The phenomenon that phase velocity varies with frequency of wave is called
dispersion [3]. Figures 43.2 and 43.3 show the frequency dispersion curve diagram
of phase speed and group speed for an infinite cylinder with diameter D = 10 mm.
Its material density is 7.932 g/cm
3 , p-wave velocity is 5738.92 m/s, the shear wave
velocity is 3067.6 m/s. The material that used in the calculation is 304 stainless steel,
which is the one made up of the rod waveguide used in our study.
In the dispersion curve shown in Figs. 43.2 and 43.3, at least two or more modes
exist at the same frequency of the wave, which is called multi-mode phenomenon
[8]. As it can be seen from Figs. 43.2 and 43.3, the number of guided wave modes
is the least when the frequency is smaller than 300 kHz. Therefore, in the frequency
region, the signal of AE has the least attenuation when the waves propagate in the
infinite cylinder.
All above analysis were based on infinite cylinder, but in a finite rod, wave motion
becomes even more complicated due to reflection and resonance [9]. Acoustic waves
are partially reflected and transmitted at every interface, the relative amounts of which
depend on the ratio of the relative characteristic impedances of each pair of boundary
materials. This impedance is a function of material density and the speed wave hitting
the interface. If material attenuation is low, these reflections will combine with late
T(0,1)
T(0,3)
L(0,1)
L(0,2)
L(0,4)
F(1,1)
F(1,2)
F(1,3)
F(1,4) F(1,5)
F(1,6)
F(1,7)
0
100 200 300
600 700 800 900 1000
Frequency (kHz)
0
1000
2000
3000
4000
5000
6000
7000
8000
Phase speed (m/s)
Dispersion curves (phase speed)
400 500
Fig. 43.2 The dispersion curves of phase speed for infinitely long rod waveguide
X. Zeng et al.
43.2.2 Dispersion Curve Analysis
According to Silk and Bainton’s theory [7], the guide wave in the cylinder is divided
into three modes: axisymmetric longitudinal mode L(0,m), axisymmetric torsion
mode T (0,m) and axisymmetric bending mode F(n,m) (n,m = 1,2,3…). The modes
reflect the different propagation modes and vibration modes of guided waves in the
cylinder. The phenomenon that phase velocity varies with frequency of wave is called
dispersion [3]. Figures 43.2 and 43.3 show the frequency dispersion curve diagram
of phase speed and group speed for an infinite cylinder with diameter D = 10 mm.
Its material density is 7.932 g/cm
3 , p-wave velocity is 5738.92 m/s, the shear wave
velocity is 3067.6 m/s. The material that used in the calculation is 304 stainless steel,
which is the one made up of the rod waveguide used in our study.
In the dispersion curve shown in Figs. 43.2 and 43.3, at least two or more modes
exist at the same frequency of the wave, which is called multi-mode phenomenon
[8]. As it can be seen from Figs. 43.2 and 43.3, the number of guided wave modes
is the least when the frequency is smaller than 300 kHz. Therefore, in the frequency
region, the signal of AE has the least attenuation when the waves propagate in the
infinite cylinder.
All above analysis were based on infinite cylinder, but in a finite rod, wave motion
becomes even more complicated due to reflection and resonance [9]. Acoustic waves
are partially reflected and transmitted at every interface, the relative amounts of which
depend on the ratio of the relative characteristic impedances of each pair of boundary
materials. This impedance is a function of material density and the speed wave hitting
the interface. If material attenuation is low, these reflections will combine with late
T(0,1)
T(0,3)
L(0,1)
L(0,2)
L(0,4)
F(1,1)
F(1,2)
F(1,3)
F(1,4) F(1,5)
F(1,6)
F(1,7)
0
100 200 300
600 700 800 900 1000
Frequency (kHz)
0
1000
2000
3000
4000
5000
6000
7000
8000
Phase speed (m/s)
Dispersion curves (phase speed)
400 500
Fig. 43.2 The dispersion curves of phase speed for infinitely long rod waveguide