Cauchy Problems of Hyperbolic Heat-Conduction Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 232

wave front and rear in the propagation of waves from initial disturbance ψ ( M ) in

some regions (Section 2.8). The sound wave is a typical example of such a wave.
The solution (5.120) of PDS (5.119) differs from that in Eq. (5.123) not only on
the appearance of factor e −

t

2 τ 0 but also on its second term in the right-hand side.

Since

⎛ ⎞

At

⎝ ⎠

d r =

Vo l u m e o f V At ,

dS

− At

S r

the integral of in the second term is, in fact, a volume integral. The thermal

waves from the initial disturbances

ψ ( M )

ψ ( M ) thus have a wave front, but not a wave rear.

t
2τ 0 in Eq. (5.120) causes the temperature to decay as time increases.
This agrees with the physical reality of heat conduction.
The wave front will not appear if there exist global initial disturbances

The factor e −

ψ ( M ) ,

ϕ ( M ) or global source disturbances f ( x , y , z , t ) . The distribution and strength of ini-

tial disturbances will determine

whether the temperature at a given point decays or not and the degree of decaying.

ψ ( M ) , ϕ ( M ) and source disturbances f ( x , y , z , t )