Wave Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 111

face S R + at 0 .The u

does not generally vanish inside this region and it is zero
outside the region. Therefore, S at 0 − R and S R + at 0 are called the wave rear and the
wave front , respectively (Fig. 2.11). After a time interval

( M , t 0 )

Δ t , the wave front moves

(Fig. 2.12). Thus any point on the S R + at 0 may be viewed as the source
of generating an individual wave. After the time period

to S R + a ( t 0 + Δ t )

t ,the embracing surface
of all these individual waves forms the new wave front S R + a ( t 0 + Δ t )

Δ

. The traveling
of such waves is always characterized by a clear wave rear and a clear wave front.
Such a phenomenon is called the Huygens principle.

Poisson Formula of Two-Dimensional Wave Equations

A similar analysis may be made for the Poisson formula of two-dimensional wave
equations by replacing the

by a circle D and the S at by D at . However, a difference

Ω

Fig. 2.11 Wave rear and wave front

Fig. 2.12 Traveling of waves