Wave Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 110

Fig. 2.9 u value and intersection of S at with Ω

d and D be the minimum and the maximum distance of M 0 from Ω , respectively

(Fig. 2.10). For a fixed point M 0 , we have, as t and v are varied

⎧
⎨

d
a ,

= 0 , for t < Effect of initial disturbances

has not propagated to point M 0 yet,

for d

D
a ,

u ( M 0 , t )=

= 0 ,

a <

t < Effect of initial disturbances has arrived

at point M 0 ,

⎩

D
a ,

= 0 , for t > Effect of initial disturbances has passed.

For a sufficiently short or long time, S at and

0.
Take initial disturbances as the wave source. The effect of the wave source always
appears on a spherical surface of the sphere of origin M 0 . Such a type of waves
is thus called a spherical wave . Also, the effect of the wave source for point M 0
will disappear after a sufficiently long time. For example, the propagation of sound
waves has this property.
Take the center of

Ω have no common part so u ( M , t )=

Ω as the origin O . For any specified time instant t 0 such that

0, consider the region between the spherical surface S at 0 − R and the sur- at 0 − R >

Fig. 2.10 Minimum and maximum distances of M 0 from Ω