Areas where these arise include acoustics dynamic

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Unformatted text preview: here these arise include acoustics, dynamic elasticity, electromagnetics, and gas dynamics. Here are some examples. Example 10.1.1. The Euler equations for one-dimensional compressible inviscid ows satisfy t + mx =0 (10.1.3a) mt + ( m + p)x = 0 (10.1.3b) et + (e + p) m ]x = 0: (10.1.3c) 2 Here , m, e, and p are, respectively, the uid's density, momentum, internal energy, and pressure. The uid velocity u = m= and the pressure is determined by an equation of state, which, for an ideal uid is m2 ] p = ( ; 1) e ; 2 (10.1.3d) where is a constant. Equations (10.1.3a), (10.1.3b), and (10.1.3c) express the facts that the mass, momentum, and energy of the uid are neither created nor destroyed and are, hence, conserved. We readily see that the system (10.1.3) has the form of (10.1.1) with 23 2 3 23 m 0 4m5 4 m2 = + p 5 4 0 5: u= f (u) = b(x t u) = (10.1.4) e (e + p)m= 0 Example 10.1.2. The de ection of a taut string has the form utt = a2 uxx + q(x) (10.1.5a) 10.1. Conservation Laws 3 u(x,t) T T x=0 x=L Figure 10.1.1: Geometry of the taut string of Example 10.1.2. where a2 = T= with T being the tension and being the linear density of the string (Figure 10.1.1). The lateral loading q(x) applied in the transverse direction could represent the weight of the string. This second-order partial di erential equation can be written as a rst-order system of two equations in a variety of ways. Perhaps the most common approach is to let u1 = ut u2 = aux: (10.1.5b) Physically, u1(x t) is the velocity and u2(x t) is the stress at point x and time t in the string. Di erentiating with respect to t while using (10.1.5a) and (10.1.5b) yields (u1)t = utt = a2 uxx + q(x) = a(u2)x + q(x) (u2)t = auxt = autx = a(u1)x: Thus, the one-dimensional wave equation has the form of (10.1.1) with u = u1 u2 f (u) = ;cu ;cu 1 In the convective form (10.1.2), we have A= 10.1.1 b(x t u) = q(0x) : 2 0 ;a : ;a 0 (10.1.5c) (10.1.5d) Characteristics The behavior of the system (10.1.1) can be determined by diagonalizing the...
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