Hopf equation In Lecture 1 we considered propagation of sound in a compressible gas with the use of equation,

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1 Lecture 4 In preceding lectures we discussed dispersion effects defined as a nontrivial dependence of phase velocity of harmonic waves on their wave number. Due to dispersion effects, envelope of a wave pulse propagates with the group velocity. It also changes its form during propagation. In the lowest approimation, evolution of the envelop function is governed by the parabolic equation. Oscillations can also arise at sharp edges of the pulse. All these effects were considered in framework of linear theory. Now we shall study consequences of nonlinear effects neglecting dispersion. Hopf equation Hopf equation In Lecture we considered propagation of sound in a compressible gas with the use of equation, ρ t + (ρv) = 0, v t + (v )v = (/ρ) p. () where the pressure p and the density ρ were connected by a material state equation It is known from kinetic theory of gases that in the ideal gas p = p(ρ). (2) p = NkT = (ρ/m)kt. (3) Hence in the limit m the pressure vanishes and we get the equation of state p = 0. (4) of heavy dust particles. Correspondingly, the gas dynamics equations become independent of each other: the velocity field is governed by v t + (v )v = 0, (5) and when v(r, t) is known, the density can be found from the continuity equation ρ t + (ρv) = 0. (6) In case of one-dimensional flow, when all variables depend on one space coordinate only, equation (5) takes the form This equation is called the Hopf equation. v t + vv = 0. (7)

2 2 Solution Solution of Hopf equation As we know from Lecture, if the coefficient before v in Eq. (7) were constant, v t + cv = 0, then we would have the solution in the form v = v 0 ( ct), c const, where v 0 () is the initial distribution of the flow velocity at t = 0. Therefore, it seems natural to check whether the solution of equation is given implicitly by the epression v t + vv = 0. (8) v = v 0 ( vt). (9) Indeed, its differentiation gives v t = v 0v v 0v t t, v = v 0 v 0v t, where v 0() = dv 0 /d, and, hence, v t = vv 0 + v 0 t, v = v 0 + v 0 (0) t. Substitution of Eqs. (0) into Eq. (8) shows at once that Eq. (9) is the solution satisfying the initial condition v = v 0 () at t = 0. We have found the general solution of Eq. (8) in the form vt = w(v), () where w(v) is an arbitrary function. If we put here t = 0, we find that w(v) is an inverse function of the initial distribution v 0 (), and we denote this function as (v). Then the solution (9) of the Hopf equation can be written in the form = (v) + vt, which means that the velocity v is constant along the straight lines = + v 0 ( )t, v = v 0 ( ), (2) in the plane (, t), where is the value of at which the initial distribution has the value v of the initial velocity, i.e., it is the solution of the equation v = v 0 ( ). Let the initial distribution be given by the function v 0 ( ) = /( + 2 ). 2

3 _ u 0() _ Formula = + v 0 ( )t, v = v 0 ( ), gives the map of the initial points into the corresponding points at the moment of time t. = + v ()t b t = t b t > t b b t < t _ - At t = 0 this is the identical map =, but with growth of t its deviation from = increases proportionally to the values of v 0 ( ). We see that with growth of t the region of the plot ( ) corresponding to points with v 0 < 0 flattens, and at some moment t = t b the plot acquires a point with a horizontal tangent line. At greater times t > t b there arises a region < < +, where each value of corresponds to three values of. This means that after t = t b the faster particles begin to overtake the slower ones. Three values, 2, 3 corresponding to one value of give the three values of the initial velocities v 0 ( i ), i =, 2, 3, of particles which come to the point at the moment t. Hence, the plot of v() is three-valued in the region < < +. 3

4 v() t = 0 t = t b t > t b b t 2 t _ (t) M + (t) t b _ b 3 The boundaries of the multi-valued region can easily be found by noticing that at these points the plot of ( ) has a horizontal tangent line, / = + v 0( )t = 0. (3) Hence, Eqs. (2) and (3) define the functions ± (t) in a parametric form, ± = v 0 ( )/v 0( ), t = /v 0( ), (4) where plays the role of the parameter. The wave-breaking point b corresponds to the infleion point of the curve ( ), where 2 / 2 = 0, so the corresponding value of is determined by the equation v 0 ( b ) = 0. (5) If we find this value of b and substitute it into Eqs. (4), then we obtain the point ( b, t b ) in the plane (, t), where the boundary curves + (t) and (t) merge together. 4

5 It is useful to look at this figure from another point of view. At each value of Eq. (2) determines the straight line in the plane (, t) the characteristic of Eq. (8). Variation of gives the family of straight lines whose envelope is determined by the known from elementary differential geometry equations that is, just by Eqs. (4). t ( ) = 0, / = 0, Simple wave Simple wave Now we generalize the above consideration by relaing the condition p = 0. Namely, we look for the solution of the system for the one-dimensional flow ρ t + (ρv) = 0, v t + vv = (/ρ) p (6) under supposition that there is a relationship between the flow velocity v and the gas density ρ: v = v(ρ), ρ = ρ(v). Under this supposition, we may rewrite Eqs. (6) as ρ t + [d(ρv)/dρ] ρ = 0, (7) v t + [ v + (c 2 /ρ)(dρ/dv) ] v = 0. (8) Noticing that we obtain from Eq. (7) that ( ) t ρ t /ρ = ( / t) ρ, ρ = d(ρv) dρ = v + ρ dv dρ, 5

6 and in the same way from Eq. (8) that ( ) t = v + c2 ρ dρ dv. (9) Since ρ is a one-valued function of v, we have ( ) ( = t t and, hence, Thus, we have v ρ ) ρ dv dρ = c2 dρ ρ dv., v dv/dρ = ±c/ρ, (20) and the function v(ρ) (or v(p)) is determined by the relation ρ cdρ v = ± ρ. (2) After substitution of dρ/dv = ±ρ/c into Eq. (8) we obtain v t + (v ± c)v = 0 (22) similar to Hopf eq. v t + vv = 0. By analogy with Hopf equation we can write the solution of (22) = [v ± c(v)]t + (v), (23) where (v) is an arbitrary function, and c(v) can be found with the use of Eq. (2) and the formula c(ρ) = dp/dρ. The straight lines (23) in the plane (, t) are called characteristics of the system (7,8). Eq. (23) can be rewritten in the form v = v 0 [ (v ± c(v))t] (24) similar to Eq. (9), v 0 ( ) being the function inverse to (v); v 0 () defines the initial profile of the wave at t = 0. Points of the profile propagate with velocities V = v ± c. Since V varies from one point to another, the profile changes its form what can lead to the wave-breaking phenomenon. Therefore, the solution (23,24), which is called a simple wave solution, describes evolution up to the wave-breaking point. The time t b of wave breaking is determined by the condition that the profile has an infleion point with a vertical tangent line, ( / v) tb = 0, ( 2 / v 2) t b = 0. (25) 6

7 4 Polytropic gas Polytropic gas Polytropic gas has the equation of state (for air γ = 7/5) Hence the sound velocity c is equal to which substitution into (2) yields v = 2c 0 γ p = p 0 (ρ/ρ 0 ) γ. (26) c = c 0 (ρ/ρ 0 ) γ 2, (27) ( ( ρ ρ 0 ) γ ) 2, (28) and (27) and (28) give and Eq. (22) simplifies to c = c 0 ± 2 (γ )v. (29) v t + ( ±c (γ + )v) v = 0. (30) This equation coincides with Hopf equation. Its solution (23) becomes = ( ±c (γ + )v) t + (v), (3) and according to Eq. (27) the density ρ depends on the velocity v as ρ = ρ 0 ( ± 2 (γ )(v/c 0) ) 2/(γ ). (32) Substitution of Eq. (3) into Eq. (25) gives the conditions for the wave-breaking moment for a polytropic gas: t = 2 γ + (v), (v) = 0. (33) 5 Piston problem Piston problem Let a polytropic gas be located inside a pipe indefinitely long at one side ( ) and closed by a movable piston at the other side ( = 0). At some moment t = 0 the piston starts its motion according to the law = X(t), X(0) = 0. We have to calculate the corresponding flow of gas inside the pipe. We have to satisfy the boundary condition at the piston, v = Ẋ(t) at X(t) = ( c (γ + )v) t + (v), 7

8 which means that the flow velocity of the gas contacting with the piston is always equal to the piston s velocity. These two formulas define the function (v) in a parametric form: ) (τ) = X(τ) (c (γ + )Ẋ(τ) τ, v(τ) = Ẋ(τ). The substitution of the first formula into Eq. (3) yields the solution of the piston problem ] = X(τ) + [c (γ + )Ẋ(τ) (t τ), v = Ẋ(τ). (34) Let us specify these formulas for a simple case of the motion of the piston with constant acceleration, Then Eqs. (34) transform to X(t) = 2 at2, Ẋ(t) = at, a > 0. (35) = 2 aτ 2 + ( c (γ + )aτ) (t τ), v = aτ, and these two formulas give = (/2a)v 2 + ( c (γ + )v) (t v/a). (36) This is a quadratic equation with respect to v, its solution gives v = (/γ) ( c 0 2 (γ + )at) + (/γ) (c0 2 (γ + )at) 2 2γa( c0 t). The sign before the square root is chosen so that v = 0 at the boundary = c 0 t with the gas at rest. The plot of v() acquires a vertical tangent line at = c 0 t b at the moment t b = 2c 0 /[(γ + )a]. (37) v() t = t b t < t b 8

9 The characteristics of the flow are straight lines (36) parameterized by the values of v from the interval at v c 0 t, that is, they stem from the line = at 2 /2 corresponding to the piston s motion. t t b b 6 Problem Problem Problem 4.. Solve Hopf equation u t +uu = 0 for the initial condition u 0 () = /( + 2 ). Find the boundaries of the tree-valued solution and coordinates of the wave breaking point. 7 What to read? What to read? L.D. Landau and E.M. Lifshitz, Fluid mechanics, Section 0 G.B. Whitham, Linear and nonlinear waves, Section 2. A.M. Kamchatnov, Nonlinear periodic waves and their modulations An introductory course, Sections.3.2, Summary Summary of Lecture 4 Prototypic eample of nonlinear wave evolution is given by solution of Hopf equation u t + uu = 0 and simple wave solution of D fluid dynamics equations. 9

10 Typical behavior leads to wave breaking and formation of a non-physical multivalued region in the solution. This means that some other effects should be taken into account for regularizing the singularity. 0