6.3.1.1. Field Generated in a Three-Dimensional Infinite Space

The problem can be written as:

(6.104)

The equivalent integral equation is, according to equation (6.60)

(6.105)

This solution is called "delayed potential" and its interpretation is straightforward.

Specific example: the punctual source is moving at the constant velocity . In accordance with equation (3.29), it is a source of volume velocity equal to . The solution for the pressure field p = - ? ₀ ? ?/ ?t can then be written, according to equation (6.105), as

(6.106)

The change of variable leads directly to the following result (Figure 6.5):

(6.107)

Figure 6.5: Moving punctual source (constant velocity )

6.3.1.2. Initial Values Problems

At any given point in an infinite medium, without any source, the values of the field ? ₀ and its derivative with respect to the time v ₀ = ? ? ₀/ ?t ₀ are assumed known at t = t _i. The integral equation (6.60) becomes

(6.108)

This result is applied to three different problems: one-, two- and three-dimensions.

6.3.1.2.1. One Dimensional Initial Values Problems: Infinite String in Vacuo

The Solution (6.108) then Becomes

(6.109)

The substitution of the Green's function (3.54) and its derivative (since 1-U(-u) = U(u))

into equation (6.109) and writing t _i = 0 leads to

(6.110)

It is the well-known D'Alembertian solution of the initial value problem at one-dimension. One can...