Dual-Phase-Lagging Heat-Conduction Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 317

and (6.220), Xu andWang (2002) obtain an expression of λ 1 , 2 in terms of ζ m and ω m ,

λ 1 , 2 = ω m 1

− ζ m ± ζ

m

− . (6.221)

Underdamped Oscillation

<

For the case ζ 1, there are two complex conjugates λ 1 and λ 2 , m

λ 1 , 2 = ω m

i 1 − ζ m ± − ζ

m

. (6.222)

Therefore

e − ζ m ω m t a m cos ω m t 1 ω m t 1

m m

Γ m ( t )= − ζ + b m sin − ζ . (6.223)

After the determination of integration constants a m and b m by the initial conditions
[Eq. (6.214)], Xu and Wang (2002) obtain

e − ζ m ω m t ω m t 1 ω m t 1 + ψ m + ζ m ω m φ m

m

ω m 1

m

Γ m ( t )= φ m cos − ζ sin − ζ

m

− ζ

(6.224)

which may be rewritten as

A m e − ζ m ω m t sin Γ m ( t )= ( ω dm t + ϕ dm ) . (6.225)

Here,

ψ m + ζ m ω m φ m
ω dm

2

m

A m = φ + , (6.226)

ω dm = ω m 1 − ζ

m

, (6.227)

tan − 1

φ m ω dm
ψ m + ζ m ω m φ m

ϕ dm = . (6.228)

Therefore, the system is oscillating with frequency ω dm and an exponentially decay-

ing amplitude A m e − ζ m ω m t

[Eq. (6.215)]. Fig. 6.1 typifies the oscillatory pattern, for

ζ m = 0. The wave behavior is still observed in

dual-phase-lagging heat conduction. However, the amplitude decays exponentially
due to the damping of thermal diffusion. This differs enormously from classical heat
conduction.

0 . 1, ω m = 1 . 0, φ m = 1 . 0and ψ m = 0 .

1 forms the condition for thermal oscillation of this kind.
Figure 6.2 illustrates the variation of

ζ m <

Γ m ( t ) with the time t when ψ m is changed

to 1 . 0 from 0. Γ m is observed sometimes to be capable of surpassing φ m . This phe-