GENERAL EQUATIONS AND ELEMENTARY SOLUTIONS

In a subsonic flow, the main-stream Mach number M — Uja is less

than 1, and the quantity _______

0 = Vl – M2 (1)

is real and positive. Equation 2 of the last section can be written as

1 32ф 2 M 2 32ф 32ф

3? + ~7 dxdt ~ ^ 5×2 = d^

which can be simplified by the following transformation:

where

a = M/iffta)

Equation 2 then becomes the classical wave equation

фтт ~ фїЄ ~f’ Фчч

Introducing polar coordinates

£ = / cos в, fj = r’ sin в

/ = л/£2 + jj2 = ~ Vx2 + /? V

p*a

—■(і)=–■(?)

we have

, . 1 1

фгт — фгУ + -,Ф,’ + – Т^фы

Ґ г

Assuming a solution of the form

ф(/,0,т) = (9)

we see that the equation governing R(r’) is

The brace { } means that either of the two functions enclosed in it may be taken.

The Hankel functions Hna)(cor’) and Яи(2)(а>/) (also called Bessel functions of the third kind) are tabulated in several treatises.* They are

* For example, see Jahnke and Emde, Tables of Functions. Dover Publications.

complex valued for real arguments. We shall not discuss the general properties of the Hankel functions here.* But the following salient properties are needed for the following discussions:

1. The conjugate complex of Hna)(reie) is Hj2)(re~ie) [and not Hna)(re~l6)]. If Hna)(z) is known, tfn(2)(z) can be obtained from the formula

HKVz) = 2Ш – H™(z) (12)

where Jn(z) is the Bessel function of the first kind of order n.

2. For very small arguments (|z| 1),

7/ cz

~ ——- log— (c ~ ev = e0*5772 = 1.781)

7Г 2

H™(z)~- (13)

7TZ

Hnm(z) ~ – і (n — 1)! (-) (и, integer, Ф 0)

7Г Zl

The ~ sign means that, in the Laurent’s series expansion of the Hankel functions, the sum of all the other terms are of order smaller than the first term.

3. For very large argument, the dominant terms of the asymptotic representation of the Hankel functions are (|z| 1):

tf„a)(z)

We can now examine the physical meaning of the solution 11. Con­sider the case n — 0. The solution is

According to Eqs. 12 and 13, it is seen that the absolute values of H0a)(r’) and Я0(2,(/) become infinitely large when r’ -» 0. Hence, both terms of Eq. 15 behave like a source at the origin. To see the properties of the waves corresponding to these two elementary solutions, we examine the solution at large values of cor’. The asymptotic representation for the

* An exhaustive treatment is given by Watson: Bessel Functions, Cambridge Univ. Press. An elegant introduction sufficiently developed for physical applications can be found in Chapter IV of Sommerfeld’s book: Partial Differential Equations, Academic Press (1949).

functions H0a)(a>r’) and #0<2)(cw’) are given by Eqs. 14 by setting n = 0. Hence, asymptotically, at large distance from the source, the acceleration potentials are

ф1 = A(lff0a)(oj/)eim(l+ax) ~ A0 J-2—. e -<-nli)i+iuxxx

—– (16)

<L = A0H0{i)(mr’)eMt+,xx) ~ An J—- eW4)i+»««* eMt-n

N TTU>r

The factor t + / remains constant when t increases if the distance r decreases with suitable speed. Hence, ф1 represents an incoming wave, converging toward the source at the origin. Similarly the factor t — / in ф2 indicates that ф2 represents a wave radiating from the source. If the source at the origin is regarded as the only cause of disturbance in the flow field, we must impose the condition that waves can only radiate out from the source, and, hence, we must discard H0{l)(a>r’) as a physical solution.

The case n = 1 can be similarly examined. The solution that repre­sents waves radiating out from the singularity is given by

The cos в term on the right-hand side is antisymmetrical with respect to the у axis, while the other, sin в, is antisymmetrical with respect to the x axis. Since H^Hcor’) tends to infinity when r —> 0, so the solution

ф(г 0, t) = В sin в Я1(2)(иг’)еіш((+“) (18)

represents a “doublet” at the origin, with axis perpendicular to the x axis, while the other solution involving cos в represents a doublet at the origin with axis perpendicular to the у axis.

If the singularity (a source, a doublet, etc.) is situated at the point (a?0, Уо) instead of at the origin, it is necessary to replace the x and у occurring in the above solutions by (x —■ x0) and (y — y0), respectively. In particular, / should be replaced by

/ = i_ V(x – x()f + (?iy – y, f (19)