Wave Equation with Radiation

8.1.1 Basic Energy Balance

Multiplying (8.1) by ˙u and integrating by parts over a space period, we obtain ∫

∫ ′ ∫ ′ ... (¨ u ˙u + ˙u u ) dx − γ u ˙u dx =

f ˙u dx, which we can write

(8.2) where

E=a−r ˙

E(t) ≡ 2 ( ˙u(x, t) +u (x, t) ) dx

is the internal energy viewed as heat energy, and ∫

f (x, t) ˙u(x, t) dx, r(t) = − γ u (x, t) ˙u(x, t)dx, (8.4) is the absorbed and radiated energy, respectively, with their difference a − r

a(t) =

driving changes of internal energy E.

43 Assuming time periodicity and integrating in time over a time period, we

8.1. A BASIC RADIATION MODEL

have integrating by parts in time,

(8.5) showing the dissipative nature of the radiation term.

R≡ 2 r(t) dt = γ¨ u(x, t) dxdt ≥ 0

If the incoming wave is an emitted wave f = −γ U of amplitude U , then ∫ ∫

2 2 1 A−R≡

(f ˙u − γ ¨u )dxdt = γ( ¨ U¨ u − ¨u ) dx ≤ (¯ R in −¯ R), (8.6)

2 with R in = ∫∫γ¨ U 2 dxdt the incoming radiation energy, and R the outgoing.

We conclude that if E(t) is increasing, then ¯ R≤¯ R in , that is, in order for energy to be stored as internal/heat energy, it is required that the incoming radiation energy is bigger than the outgoing.

Of course, this is what is expected from conservation of energy. It can also be viewed as a 2nd Law of Radiation stating that radiative heat transfer is possible only from warmer to cooler. We shall see this basic law expressed differently more precisely below.

Figure 8.1: Standing waves in a vibrating rope.

44 CHAPTER 8. WAVE EQUATION WITH RADIATION