The_wave_equation_ii.pdf

The Wave Equation II Osama K Mahmood [email protected] 7 June 2016

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Causality

The effect of an initial velocity ψ will be that the wave will spread out at a speed ≤ c. This means that it can only effect the wave in the shaded region of Figure 1. This is known as the Principle of Causality.

In other words, the value of u at the point (x, t) is affected by the value of φ at x ± t and the value of ψ in the interval [x − ct, x + ct]. We call the interval (x − ct, x + ct) as the interval of dependence on t = 0. The shaded area in triangle ∆ in Figure 2 is referred to as the Domain of Dependence or the Past History of the point (x, t).

The Domain of Dependence is bounded by the characteristic lines that pass through (x, t).

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2

Energy

Imagine an infinite string with constants ρ and T . Then ρutt = T uxx for −∞ < x < +∞. We know that the Kinetic Energy is given by: Z +∞ 1 1 2 KE = mv = ρ u2t dx (1) 2 2 −∞ The Potential Energy is given by the following formula: Z 1 +∞ T u2x dx PE = 2 −∞

(2)

Combining (2) with (3) we get that: We assume that φ = ψ = 0 for |x|> R and therefore according to the Principle of Causality, u will be also zero on |x|> R + ct. Differentiating (1) before some other substitutions gives: dKE 1 = ρ dt 2 Z =ρ

Z

+∞

−∞ +∞

d 2 (u )dx dt t

ut utt dx

−∞ Z +∞

=T

uxx ut dx (ρutt = T uxx ) −∞ Z +∞

= −T

(3)

utx ux + [ux ut ]+∞ −∞ dx

−∞

=−

1 d 2 dt

+∞

Z

T u2x dx (

−∞

1 d 2 u = utx ux ) 2 dt x

Combining (1) with (2) gives: 1 E= 2

Z

+∞

(ρu2t + T u2x ) dx

(4)

−∞

Getting the derivative of equation (4) yields: d [KE + P E] = 0 dt

(5)

d d As we can note that dt KE = − dt P E. Hence Energy is a constant independent of t. This is the Law of Conservation of Energy.

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3

Example

Solution

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