Introduction - Heat Conduction: Mathematical Models and Analytical Solutions - page 13

This theorem can be shown by reduction to absurdity and can be extended to
cases of multiple dimensions. Using this approach, we also use the Gauss formula
S · =

∇ ·

,

a n d S a d v

Ω

i ∂
∂

j ∂
∂

k ∂
∂

where ∇ =

x + y + z ,

a = P ( x , y , z ) i + Q ( x , y , z ) j + R ( x , y , z ) k , ( x , y , z ) ∈ Ω ,

Ω is the integration domain, S stands for the external surface of Ω and n is the

outward unit normal of S .

Approach 3.

This approach develops the desired equations from some known equations through
differential and integral operations. A typical example is that of developing the equa-
tion of electromagnetic waves from the Maxwell system of differential equations
through differential operations.

1.2.3 Wave Equations

We develop wave equations and introduce some relevant concepts by considering
the equation satisfied by the small transverse displacements of a vibration string.
Consider a homogenous and perfectly flexible thin string of length L fixed at
the end points in Fig. 1.1 where u

is the displacement of the string at point x
and time t . For a small transverse vibration such that u x ≈

( x , t )

0, d s = Δ x , α ≈ 0and

β ≈ 0 (Fig. 1.1), we determine what equation governs the motion of the string. By

Approach 1, consider a small differential segment d s of the string in Fig. 1.1. Since
α ≈ 0and β ≈ 0,

sin α ≈ tan α = u x ( x , t ) , sin β ≈ tan β = u x ( x + Δ x , t ) .

For a homogenous and perfectly flexible string, both its density ρ and the tension T

are constants. Let F be the external force along the vertical direction per unit

length of the string that is acting on the string, including the gravitational force. By
Newton's second law of motion, for the vertical motion of the string we obtain

( x , t )

−

T sin α + T sin β + F ( ξ , t ) Δ x = T [ u x ( x + Δ x , t ) − u x ( x , t )] + F ( ξ , t ) Δ x

1 1

2 u

x ∂

( ξ 2 , t )

= ρ Δ , ξ 1 , ξ 2 ∈ ( x , x + Δ x ) .

∂ t 2