WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

good for low to moderate stiffness contrasts (E0 < 50). Interestingly,
as the material contrast grows to infinity the wave splits up in a forward
and backward travelling wave of equal magnitude.
An external force is generally required to change the material
properties, hence the mechanical energy in the wave is not conserved.
Based on the formulas for the forward and backward travelling
waves in [36] the relative change of mechanical energy
occurring at t ¼ t0 is found as
DEmek
¼ 1
2 ðE0 1Þ; ð6Þ
Emek
E = 1 ρ = 1
E = E0 ρ = 1
t < t 0
t > t 0
t = 0.750 s
t = 0.975 s
Fig. 3. Top: one-dimensional medium with an instant change in the stiffness for
t ¼ t0. Bottom: two snapshots of the wave motion with indication of the
propagation direction.
Amplitude
1
0.8
0.6
0.4
0.2
0
0 1 2 3 4
ln(E0 )
5 6 7 8
Fig. 4. Amplitude of the forward and backward travelling waves versus the stiffness
E0. Comparison between exact (solid lines) and numerical results (discrete
markers).
thus, if the stiffness of the medium is increased (E0 > 1) energy is
supplied to the wave, whereas for E0 < 1 energy is taken out of
the wave. Simulation results show excellent agreement with the
analytical result.
J.S. Jensen / Comput. Methods Appl. Mech. Engrg. 198 (2009) 705–715 707
Fig. 7 shows simulated values for Emek and also Ekin and Epot versus
time for the case of E0 ¼ 3. The instant doubling of the mechanical
energy is noted at the time of instant material property change
– as predicted by (6). Furthermore, oscillations in the kinetic and
potential energy are seen in the short time interval after the material
change for which the forward and backward travelling waves
have a spatial overlap.
Thus, it has been demonstrated that the simple time integration
algorithm produces accurate results in the case of instant material
properties changes, at least in the case of low to moderate stiffness
contrasts.
3.2. Moving material interface
Weekes [35] studied wave propagation in structures with a
moving material interface. Exact results were provided for the
example illustrated in Fig. 5.
Here, the interface between a material with stiffness E ¼ 1 and a
material with stiffness E ¼ E0 moves with constant speed V towards
left. A wave that propagates towards right meets the moving
interface and is partly transmitted and partly reflected at the interface.
It was shown in [35] that the relative amplitudes of the transmitted
wave T and the reflected wave R are independent of the
material interface speed
pffiffiffiffiffi
2
E0 1
T ¼ p ; R ¼ p ; ð7Þ
þ 1 þ 1
ffiffiffiffiffi
E0
ffiffiffiffiffi
E0
which correspond to the known results for transmission and reflection
of waves at an immovable interface.
However, the interface speed V causes a shift in the wavenumber
(or frequency) of the transmitted and reflected waves (Doppler
shift). The relative wavenumber shift of the reflected wave is found
as
Dc
c 0 R
1 þ V
¼ ; ð8Þ
1 V
i.e. the shift depends only on the interface speed. Here, it should be
noted that V is given relative to the wave speed in the material in
which the incoming wave propagates ðc ¼ ffiffiffiffiffiffiffiffi p
E=q ¼ 1Þ. The shift
E = 1 E = E 0
V
t = 0.560 s
t = 0.940 s
Fig. 5. Top: one-dimensional elastic rod with a moving material interface. Bottom:
two snapshots of the wave motion indicating the propagation direction.