The Continuation Problem

Consider the acoustic field generated by some localized noise sources as shown in Figure 14.1. The sound field is governed by the compressible Navier-Stokes equation. Suppose the solution is known. To identify that it is the solution of the Navier-Stokes equations, a subscript NS will be added to the pressure and velocity field, i. e., pNS and vns.

Подпись: p0 d + V P = 0 d. + YP0V ■ v = °.

Now let y be a convex surface enclosing the sources and the near sound field. It will be assumed that y is far enough away from the sources that the disturbances at Y are, for all intents and purposes, linear and inviscid. Under these conditions, the equations of motion outside y are the linearized Euler equations as follows:

Thus, in the region outside y, PNS, vNS and the solution of the linearized Euler equations, denoted by pEuler, vEuler are practically the same, namely pNS ^ PEuler,

vNS ^ vEuler.

Suppose Г is a closed convex surface enclosing y as shown in Figure 14.1. On Г, P = PEuler is known. Now, as a first step toward establishing a way to continue the

solution from surface Г to the far field, consider the following initial boundary value problem in the space outside Г. The governing equations are the linearized Euler equations (14.1) and (14.2). The boundary and initial conditions are as follows:

on Г, p = PEuler (14.3)

At |x| ^ to, p, v behave like outgoing waves (14.4)

At t = 0, p = PEuler (X, °), V = VEuler (X, °) (14.5)

By eliminating v from Eqs. (14.1) and (14.2), the equation for p is the simple wave equation as follows:

d2 p

-2- – «0V2p = (14.6)

Now, it is known that the solution of the simple wave equation (14.6)) satisfying boundary conditions (14.3) and (14.4) and initial conditions (14.5) is unique, but the original solution pNS, vNS which become pEuler, vEuler outside Г is also a solution of Eq. 14.6 and boundary and initial conditions (14.3) to (14.5). Therefore, the two solutions must be equal. In other words, the continuation solution from surface Г to the far field is given by the solution of Eqs. (14.1) to (14.5). Hence, a way to continue a near-field solution to the far field is to solve the initial boundary value problem stated above. How to construct such a solution is the subject of the next few sections of this chapter.

Instead of specifying p = pEuler on Г as boundary condition, the solution of the simple wave equation with dp = apEukI (the normal derivative) specified as boundary condition on Г is also unique. By Eq. (14.1), the normal derivative of p is related to the velocity component in the normal direction as follows:

^ d V „ d Vn

Vp • n=-P0 — ■ n=-P0-df • (14J)

However, if vn (x, t) is known on Г, then is known. Therefore, it is possible to

extend or continue a solution beyond surface Г by solving Eqs. (14.1) and (14.2) with appropriate initial conditions and matching vn to vnEuler on boundary Г. In other words, there are two ways to continue a solution from surface Г to the far field. The first way is to match p on Г. The second way is to match vn on Г.

It is worthwhile to note that the solution to the initial boundary value problem defined by Eqs. (14.1) to (14.5) consists of two parts. One part is associated with the initial conditions and zero boundary values on Г. The other part is the zero initial conditions and nonzero boundary value on Г. In most problems, interest is on the second part of the solution at large time. At large time, the transient solution (the first part) would propagate away. So for a long time solution, it is possible to ignore the initial condition altogether. In the rest of this chapter, attention is focused only on continuing the solution from surface Г without reference to initial conditions.

In the literature, there are currently two favorite methods for continuing a solution to the far field. They are the Kirchhoff method (see, e. g., Lyrintzis (2003)) and the Ffowcs-Williams and Hawkings (1969) integral method. Mathematically, the Kirchhoff method yields a far-field acoustic pressure in terms of an integral

over a closed surface. The integral involves the values of the pressure, its normal derivative, and the time derivative on the closed surface (three quantities altogether). This is well known to physicists and mathematicians. The Ffowcs-Williams and Hawkings method is similarly a surface integral representation. It solves the Lighthill equation instead of the simple wave equation. If the surface under consideration is outside all volume sources (quadrupoles), then the Kirchhoff method and the Ffowcs-Williams and Hawkings method are essentially similar. It is to be emphasized that both are integral representations, not solutions of the governing equations. To evaluate the integrals, the values of the pressure, its normal derivative, and time derivative on the surface must be known. It will, however, be shown later that it is sufficient to continue the pressure field from surface Г to the far field by using only the pressure fluctuations on the surface. Alternatively, it is sufficient to determine the far-field pressure by using the fluctuating pressure gradient or the velocity component normal to the surface. It is not necessary to have three or more sets of information. This, nevertheless, does not mean that the Kirchhoff and the Ffowcs-Williams and Hawkings methods are in error, only that when these methods are used, the prescribed pressure, its normal derivative, and time derivative on surface Г must be accurate. Otherwise, because of the overspecification of the boundary data (Note: in the theory of partial differential equations, when two sets of boundary data are prescribed, whereas one set is sufficient for determining a unique solution, it is referred to as overspecification of boundary data), the computed far – field pressure is likely to incur significant error.