By S. L. Gavrilyuk, N.I. Makarenko, S.V. Sukhinin (auth.)
Starting with the elemental notions and evidence of the mathematical conception of waves illustrated through a number of examples, workouts, and strategies of fixing average difficulties Chapters 1 & 2 exhibit e.g. the right way to realize the hyperbolicity estate, locate features, Riemann invariants and conservation legislation for quasilinear platforms of equations, build and examine suggestions with vulnerable or powerful discontinuities, and the way to enquire equations with dispersion and to build vacationing wave suggestions for versions reducible to nonlinear evolution equations.
Chapter three bargains with floor and inner waves in an incompressible fluid. The potency of mathematical tools is confirmed on a hierarchy of approximate submodels generated from the Euler equations of homogeneous and non-homogeneous fluids.
The self-contained shows of the cloth is complemented via two hundred+ difficulties of alternative point of hassle, quite a few illustrations, and bibliographical recommendations.
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Extra resources for Waves in Continuous Media
Example text
U/ is a convex function. u/. v/ is a convex function of v. , double application of this transform yields the same function e. Hint. u// 1 . 13. Consider the equations of one-dimensional motion of an ideal gas with zero pressure t C . ) Show that this system is not hyperbolic. Find all conservation laws @t P. ; u/ C @x Q. ; u/ D 0 admitted by this system. Find out whether there are laws with a convex function P among these conservation laws. Answer: P. u/; where a and b are arbitrary smooth functions; P is not convex.
For 0 < 6 , where u. / D Q. /= is the vehicle velocity and c. / D Q0 . / is the characteristic velocity. 29. Prove that the propagation velocity of the kinematic shock wave D. Q. / Q. 0 //=. 7 Problems 27 on the other side of the front, satisfies the relation 0 D. /D Z 1 . %/d%: 0 Show that D. / is a monotone decreasing function if Q00 . / < 0. 30. The traffic flow moves with velocity u0 and density 0 along a street where the traffic light turns red at time t D 0 and is kept during the time Tk .
U1 ; u2 / D 1 C 1 u1 C 2 u2 ; 36 1 Hyperbolic Waves where k are the Henry coefficients (0 < u2 > 0 the following assertions hold. a. 1 < 2 ). 26) b. u/ are real and distinct: 0 < 1 < 1 =p < 2 < 2 =p (the strict hyperbolicity of the Langmuir adsorption equations). 55. u/ D 0 by making the change of independent variables D vt x, D x. u/ given by Eq. @ C @ /r D 0 for the characteristic d =d D . 56. 7 Problems 37 Remark The Langmuir adsorption equations are also reduced to a system in Riemann invariants in the general case of an n-component mixture.
