
545Lagrangian and Hamiltonian Formulations for Continuous Systems
© 2010 Taylor & Francis Group, LLC
Equation 16.33 becomes
Q
tkxtkx
x
=
−+ −−+
≅
2
22
2
1
co
os
cos(
ωω ω
ω
ΔΔ
Δ tkxtkx−−Δ)cos( ).ω
(16.34)
This represents an oscillation of the original frequency ω but with a modulated amplitude as
shown in Figure 16.5. A given segment of the wave system, such as AB, can be regarded as a “wave
packet” and moves with a velocity V
g
(not determined yet). Because Δk ≪ k, this segment contains a
large number of oscillations of the primary wave that moves with the phase velocity V. The velocity
V
g
with which the modulated amplitude propagates is called the g ...