CHIN. PHYS. LETT. Vol. 28, No. 7 (2011) 074301
Time-Domain Second-Harmonic Generation of Primary Lamb-Wave Propagation
in an Elastic Plate *
DENG Ming-Xi(邓明晰)1** , XIANG Yan-Xun(项延训)2 , LIU Liang-Bing(刘良兵)1
1
2
Department of Physics, Logistics Engineering University, Chongqing 401331
School of Mechanical and Power Engineering, East China University of Science and Technology, Shanghai 200237
(Received 16 November 2010)
We present an experimental observation of the generation of the time-domain second harmonic by propagation
of the primary Lamb-wave tone-burst. For a case where the phase velocity matching between the primary and
the double frequency Lamb waves is satisfied but the group velocity matching between them is not, our observation clearly shows that the duration of the time-domain second-harmonic tone-burst, as well as its integrated
amplitude, increases with the increasing propagation distance. This experimental result is consistent with the theoretical prediction and demonstrates that group velocity matching is not absolutely necessary for the generation
of the cumulative time-domain second harmonic by primary Lamb-wave propagation.
PACS: 43.25.+y, 43.20.+g
DOI:10.1088/0256-307X/28/7/074301
It is known that the second harmonic signal generated by primary Lamb-wave propagation can include
accurate information about elastic plates and that it
can be taken as an effective method of nondestructive evaluation of plate-like structures.[1−5] However,
due to dispersion and multi-mode characteristics,[6−8]
practical measurement of the cumulative secondharmonic signal generated by the primary Lamb-wave
propagation is very difficult.[1−5,9,10] Some studies
have claimed that besides the phase velocity matching between the primary Lamb wave and the doublefrequency Lamb wave (DFLW),[9] an additional requirement of group velocity matching between them
is strongly suggested for measurement of the cumulative second-harmonic signal generated by propagation
of a primary Lamb-wave tone-burst.[3−5,10] Obviously,
this additional requirement of group velocity matching
makes the corresponding measurement condition become very rigorous because of the dispersive nature of
the Lamb-wave propagation. Recently, a theoretical
study has indicated that the additional requirement
of group velocity matching is not absolutely necessary
for generation of the cumulative second harmonic by
the primary Lamb-wave tone-burst.[11] If so, the corresponding measurement condition will be relaxed. Experimental verification of this theoretical prediction
is of substantial significance for practical applications
such as nondestructive evaluation of plate-like structures using the nonlinear Lamb-wave technique.[1−5]
In this Letter, we present an experimental observation to verify that group velocity matching is not absolutely necessary for the cumulative second-harmonic
generation by Lamb-wave propagation.
The second-harmonic field of the primary Lambwave propagation in an isotropic elastic plate can be
considered as superpositions of the fields of a series
of symmetric DFLWs.[12−15] The elastic plate considered here is a 1.90-mm-thick aluminum sheet (its area
is 50×50 cm2 ). Figure 1 shows the phase velocity dispersion curves for the primary Lamb waves and the
symmetric DFLWs. Clearly, the phase velocity (denoted by 𝑐𝑓𝑃 ) of the primary 𝑆2 or 𝐴2 mode (denoted
by the point 𝑃 ) is equal to that (denoted by 𝑐2𝑓
𝑃 ) of a
specified DFLW mode (the double frequency 𝑆4 mode,
abbreviated as DF-𝑆4 ) at the frequency 𝑓 given by
the vertical dotted line 𝑉 . The DF-𝑆4 mode plays
a dominant role in the second harmonic and its amplitude grows with the propagation distance due to
[14]
𝑐𝑓𝑃 = 𝑐2𝑓
The contribution of the other DFLW
𝑃 .
modes (denoted by the points 𝑃1 , 𝑃2 , 𝑃3 , etc.) to
the second harmonic field of the primary 𝐴2 and/or
𝑆2 mode is negligible because their phase velocities
are quite different from 𝑐𝑓𝑃 . At the same driving frequency 𝑓 , the group velocities (denoted by 𝑐𝑓𝑔 ) of the
primary 𝑆2 and 𝐴2 modes are respectively found to
be 3.750 km/s and 1.770 km/s, and the group velocity
(denoted by 𝑐2𝑓
𝑔 ) of the DF-𝑆4 mode is 1.770 km/s.
Figure 2 illustrates the experimental setup for the
measurement of nonlinear Lamb-wave propagation.
The transmitting transducer 𝑇𝑥 , used for generation
of primary Lamb-wave tone-bursts, consists of a plexiglas wedge and a rectangular 𝑧-cut LiNbO3 wafer
(its size is 20 × 8 × 1.41 mm3 and its longitudinalthickness resonance frequency is around 2.6 MHz).
The oblique angle of the plexiglas wedge, given by
𝜃 = sin−1 (𝑐𝐿 /𝑐𝑃 ), is chosen to be 19.3∘ , where 𝑐𝑃 is
the phase velocity of the primary Lamb wave to be
generated and is determined by the horizontal dotted
line 𝐿 in Fig. 1, and 𝑐𝐿 is the longitudinal wave velocity of the plexiglas (𝑐𝐿 =2.712 km/s). The plexiglas
* Supported
by the National Natural Science Foundation of China under Grant Nos 10974256 and 11004056.
[email protected]
© 2011 Chinese Physical Society and IOP Publishing Ltd
** Email:
074301-1