I
,,-
14
~C_JULI
THE VTLOCITY POTENTIAL OF AN
HYPERBOLIC HORN
by
JOHN EDWIN FREEHAFER
B.S.,Lehigh University
1931
M.S.,Lehigh University
1933
Submitted in partial fulfillment of the
requirements for the degree of
DOCTOR OF PHILOSOPHY
from the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
1937
Signature of Author..
v..............
Department of Physics, May 13, 1937
Signature of Professor
in Charge of Research
...
Signature of Chairman ojDepartment.
Committee on Graduate Students....
/
_.,-_.--
\
~C1_
__
--1C
TABLE OF CONTENTS
Page
I
II
III
INTRODUCTION
V
VI
.
.
.
.
.
.
.
.
.
THE CHARACTERISTIC VALUES .
.
.
.
.
.
1
.
.
7
THE FIRST CHARACTERISTIC VALUE AIND
FUNCTION .
IV
.
.
.
.
.
.
.
.
NORMALIZATION OF THE FIRST
CHARACTERISTIC FUNCTION.
THE SECOND AND HIGHER
CHARACTERISTIC FUNCTIONS
THE RADIAL FUNCTIONS .
BIOGRAPHICAL NOTE
215592
.
.
.
.
a * .
•
0
•
0
S
0
20
**
.0
0
.
0
0
0.
0
0
24
14
.
.
32
..
0
..
0
..
0
..
0
..
0
..
0
40
I_ ~-
--1-
ABSTRACT
The wave equation is separable in the oblate spheroidal coordinate system, and hence it is possible to reduce the
problem of finding the velocity potential within a horn
bounded by an hyperboloid of one sheet to a study of the
ordinary differential equations which result from the separation. It turns out that both the radial and angular functions can be obtained if we have adequate knowledge of the
solutions of the one ordinary differential equation
(-_1) W"- 2(.Q.+I) 7 W' + (b -C2)W
where a is an integer, c
=0
is prescribed by the frequency
of the wave and b is chosen so that W(1)= 1 and W (z.)- O,
zo being the cosine of one half the angular opening of the
horn. In the case of a velocity potential which is symmetrical about the axis of the horn, a is zero and this case is
the one which is investigated.
For a given value of cZ, b is limited to a set of
values b, , the so-called characteristic values. The b,'s
are functions of c , and a method is given for expressing
the b's in a series of powers of c. Simultaneously one obtains expressions for the characteristic functions. For the
range of values of c2 in which we are interested, the b's
are approximately linear in c . Curves giving the first
MWF-M_
three characteristic functions over an appropriate range of
values of c
for 15"and 300horns have been obtained experi-
mentally by means of the differential analyzer, the characteristic values having been computed from the theory. These
curves show that the first characteristic function is the
one most dependent upon c*
.
A given characteristic function
shows less dependency upon c , the smaller the angular
opening of the horn.
In order to obtain the radial functions it is necessary to represent solutions of the differential equation,
given above, over the entire imaginary axis. The exact representation is rendered difficult by the singular points
of the equation at
1. Accordingly the W-K-B approximation
is derived in such a way that it is possible to get a second
order approximation and hence to establish the validity of
using this method of representing the radial functions. Two
cases occur according to whether b is positive or negative.
In the case of positive b, there is a region in the vicinity
of the diaphragm within which the disturbance is not propagated as a wave. It is also shown that the phase velocity
at points near the throat is a function of frequency; so
that the air behaves like a dispersive medium. Finally
a picture is obtained of the velocity potential of the
elementary waves, which combine to form a given wave within
the horn, which in the case of the first elementary wave
shows a marked departure from the conditions which obtain
within a conical horn.
NW-PNMi
i
-- ·
THE VELOCITY POTENTIAL OF AN
HYPERBOLIC HORN
I Introduction
Since the wave equation in the oblate spheroidal coordinate system is separable, it is possible to calculate the velocity potential of a horn formed by rotating an hyperbola about
an axis perpendicular to the line joining its foci.
Consider points lying in a plane through the f-axis making an angle qý with the f-axis and let r be the distance of a
point from the 7-axis. The coordinates jA and 8 are defined by
the equations
By eliminating
r= -~coshA
sine
SSIl@A
COS6
e, we obtain
O
+
showing that all points for which~Ais a constant lie on an
ellipse whose foci are at r=+-oi
and whose major and minor
axes are
cos
and
respectively. By eliminating .A,
S'cl
we obtain
I
S-
and hence, all points for which A•is a constant lie on an hyperbola whose asymptotes make an angle 6 with the ~-axis and
whose foci are coincident with those of the ellipse. See Fig.1
The Cartesian coordinates f, r and 7
dal coordinates are
of a point whose spheroi-
9,,
and 97 are therefore given by
-
S
Q
sin 6E) cos
csi51"o
2
sinC
The boundary surface of the horn is obtained by putting 6= 0o
where e8is one half the angular opening of the horn. The
radius of the throat is(,/X)SiyeO
Proceeding in the usual manner, we assume that the
velocity potential V(heCP,t)is given by
v= y (,,e.cp)ez
-1-
i
----
-·
111*1111--
where V is the frequency of the wave. The differential equation for 4) (obtained by putting V into the wave equation) is
where v is the velocity of the wave. Let K= Z2
where
N =
wavelength.
Equation (1) in oblate spheroidal coordinates is
COA
oA_
I
Put
-
A-
2J
-
'S/in 9 6E) 0
cosVIk sibn e
()
U
Y = H () 0(e)
+
)
4Q
Equation (la) separates in a straightforward way, yielding
the three ordinary differential equations
__
(cios
_
dt
+
+
a and (32 are the separation constants.
and hence a must be an integer.
Fig. 1
-2-
(3)
fis single-valued
__
~ii~ii
---
Now let us consider equation (3). We shall first
change the dependent variable, defining W by the equation
0 = W sinae
Equation (3) becomes
-p )W=o
a•--•Coso
S+6(n a cose
We now change independent variables by putting
e= Case
The equation for W is
or
(4--)
where
-2(4L+I)t +(b-e'
U
b . -.ftk- ..
W 0-o LI)
C
Putting in the value of Iwe find
-sin
Radiation of shorter wavelength than the radius of
the throat is not appreciably diffracted; accordingly, the
maximum numerical value for c' in which we are interested
is given by
(Sin, 0
For a 15. horn
-_00 <
< 0
For a 30W horn
- 160 <
0O
Let us now return to equation (4), and change the
dependent variable to F and the independent variable to
x whereos
M= Fo'sh >A
X SIVAe
Equation (4) then becomes
or using the notation introduced previously
+%.x+-
F
(b+c
)F
0
(+(b)
It is worth noting that by putting x= iz, equation
(6) is reduced to equation (5). With the equations written
in the form of (5) and (6) both x and z are real and 04x
while z,< z<1
Now we shall seek V, a complex solution of the
wave equation whose real part is the real velocity poteneA=o
tial within the region bounded by the surfaces e-e.
V
must'satisfy
Then
R.
and "= co. We shall call this region
the following conditions:
(1) It must be finite and single-valued in R.
(2) Its normal derivative must vanish over the surface
8= eo
Its normal derivative at.=o must be a function
L(e. p.t) prescribed by the behavior of the diaphragm.
(4) For large values offA, it must represent waves
traveling out from the origin.
*. If it is to repLet us consider the function
resent conditions at the diaphragm which make sense physically, it must be a periodic function of p with period 2n .
We shall assume that it is also periodic in t with period zu
Now condition (2) requires thatA&(/c (=Q We can satisfy this
requirement. y choosing b so that for a given pair of values
of a and ct, (0/de=O. From this fact and the differential
equation for the O's, it can be shown in the standard way
that the O'sare orthogonal. That is,
(3)
J
sl
i O
=
0=
when b and s are unequal.
0
According to the Sturm-Louisville theory the set of O'sis
also complete, so that we are able to write.
Q. b,.
In the notation used above, a b or c written as a
subscript is intended to represent the ordinal number decreased by one of the particular member of the set of permissible values which the parameter is assuming at the time.
We use bn to represent the(n+ st characteristic value because the (n+ st characteristic function has n zeros.
-4-
qii-~-I
.·IW ·._ ~_
Thus o.4, means that a is zero, b is equal to the fifth
characteristic value, and c has a value corresponding to ti
second harmonic. Then by making use in the usual way of the
orthogonal properties of the functions involved, it may be
shown that
Yk
-i)
C
0
The function V is then expressed in the form
V = R,
abe
When r
e
MZ abc(ue
Wet
:o we must have by condition (3) that
Thus to satisfy the boundary condition at the diaphragm we
put
For the simple case in which the diaphragm is assumed to move like a piston, the velocity at the diaphragm
is independent of position and
L= Vo•
where Vo is the maximum velocity. For this case
0
Thus B is zero unless a = 0O and u=-We . We shall drop the sat
scripts a and c, B. being the coefficient when b= b., a= 0,
and c has only the one value obtained from the frequency at
which the piston is being driven. Hence
e.
B,
=
V
sine e,ade
fO
B n6
-5-
s,ede
.•~
..
Or in terms of W and 7
BA =
V
.
W, (cose) de
W,
Il
(cose) de
and the velocity potential is given by
v e .w.l
Wt) F.
n=0
-6-
T
II THE CHARACTERISTIC VALUES
We now seek integrals of equation (5) which are consistent with conditions (~) and (2) stated above. Equation (5)
has a regular singular point at +1. One integral of the equation is analytic at +1 and the other independent solutions
have logarithmic singularities at that point, The condition
of finiteness requires that we choose the analytic integral.
The requirement that the normal derivative of the velocity
potential shall vanish at the boundary requires in turn that
LBI =o
or in terms of W and 7- that
- .)lk•I
-
.W-o2
=__
(7)
These two conditions upon W limit the permissible
values of b for a given value of cL to a discrete set, the
so-called characteristic values. When b is a characteristic
value, the integral W which is finite at +1 is a characteristic function. It is convenient , though not essential, to
choose W(1)= 1. Let us represent the set of b's by b., b, ,
b2, where b,< b,, . We will label the corresponding functions
Wo, WI, Wz, etc. It is well known from the Sturm-Louisville
theory' that in the region from i°to 1, Wo has no zeros,
W,, one zero etc. There is an interesting geometrical interpretation of equation (7). w(*)/w~gis evidently the distance
between the point Z.and the intercept on the z-axis of the
, then when a= 1,
tangent drawn to the curve at- .. Let D=_:i
the tangents drawn to the characteristic ~functions at a.
all pass through the point P which lies on the axis at a
When a = 2 all of the tangents
distance D to the left of 4•a
pass through P'which is only one half as far away from e.
as P etc. When a=0 the tangents are parallel to the ?-axis.
These relationships are indicated in Fig. 2
See Ince,
Ordinary Differential Equations Chapter X
See Ince, Ordinary Differential Equations Chapter X
-7-
C.=
2
Fig.2
and
When a= O, the velocity potential is independent of<r
horn.
the
of
axis
the
about
symmetry
cylindrical
shows
hence
We shall confine our attention to this case.
Equation (5) reduces to
0
C••0___(-e
(Q--•) lhkW _
(W
and the characteristic functions W have zero derivatives
at L,. Putting z =l in equation (5) and recalling that
W(1) = 1, we obtain
(=
W'(1)
u-
Now b-cialmust be positive at least over a portion of the
range from ?- to 1; otherwise the functions are monotonic
and could not have zero derivatives at Z. . Remembering
that c' is negative we conclude that
b- Cz > o
It also follows that all of the characteristic functions
have positive slopes at +1. Hence the first, second,
fourth, sixth etc. characteristic functions have minima
at 7, and the odd-numbered characteristic functions except the first have maxima. The ever-numbered characteristic functions are all positive at 7oand the odd-numbered ones are all negative. These relationships are clear
from Fig. 3
-8-
~L
FA~~A.1·I
w."e
Wo
CIIA~
~·LC·A~
M)to) "0
Wn(-EC)
If we put
-=Z. in 5a
and solve for b, we
obtain
Fig. 3
)
"(u..____
. -0-s ) w
W (a.)
b =-
Now (1- 40)>O and hence for the first characteristic function b < cOZ, . For all other characteristic functions
b > c 2z, . Thus we may summarize
c2 < b. < c 2 Z
C ,-lrb
n
o
be is thus confined to the space between the lines b = cz
and b= cfz
t2
n
, bn (n
o ) must lie above the line b -- c
We are interested in finding the b 's as a function
It turns out that if the b 's together with the
of c
characteristic functions are known for some value of c t ,
say k, the characteristic values and functions may be obtained for values of cL in the neighborhood of k. Since the
method applies for all values of a provided that the boundary condition at m, is that the derivative of the characteristic function is zero, it may be of interest to carry
through the discussion retaining E even though the results
obtained for a different from zero do not apply to the
horn problem.
For the sake of ease in writing we shall put Y= 0c
We now seek values of b such that the integral which is finite
at •1 of the equation
) w"- 2 Co+0 .W' +(b-. E) W = O
(5C.)
(-1.W
has a zero derivative at !o
expanded in powersof y- K).
. We assume that b and W can be
Thus
b =I e,(Y- K)
S=o
where the e's are constants and the f's are functions of
-9-
.
--
i
___
Substitution of these expressions into the differential equation yields
5o
cc
s
t- :e nfs • -
=0
Irn--o
5=0
Rewriting the double sum as a single sum and equating the
coefficient of (Y-K)s to zero, we obtain a differential
s
equation for fs
0-"')f- -2(t+I)
z f',+ (e-K
e~-
)f- -
fs =(-
(1)
If W (1) is to be finite then evidently fs(1) must be finite.
Likewise if W'(2.) = 0 for all values of Y, then fs( m) = 0.
We have assumed that we have complete information about the
characteristic values and functions when y=k, so that e.
and fo are known and equation (8) may be used to determine
the e's and f's successively.
It is easily verified that equation (8) may be
written
ettcLl-'
II + Ce.- KZ)(1- e) 0"fs
( 1 ~zy){Cl2-e)f
-
-,
S
eZ s-m
-
A
Note that
O:pj-tL
r*foI
+ (e0-KEL
1;Lf)
=
o
and therefore f, is an integral of the homogeneous equation
corresponding to (8a). fis known and hence we can integrate
If v satisfies the differential equation
(80.
4
(rv') +SV=0
then an integral of the equation
is given by
=
5V4
3",'
-10-
-W
__ __
I
It follows then that
f5
Now fs (
=
6 )
=
1t·c&
o5[
o and fs (1) is finite. Accordingly
S
(t-E)*(~(fef-b~
emfs-jf.fd0
=o
when
-=.or 1. The first condition is met by taking Z, as
the lower limit of integration; the second, by choosing es
so that
I
S
S(-1)GjZl e,)f,, or
e,.
emfs4-]
= O
a
Zoensr
-Z Cmf,,
~lr"c'e),
( iZ)ý
4
(9)
gfl
(10)
Equation (10) does not yield et . For s =1 equation (9) is
and hence
E
e
fo )
z
0
can be computed if we know es-, , es-, ......*
fs-Z ,
......
fe
* Then fs
e
and fr-,
can be determined from the ex-
pression
1k ) CL+1 f2
-11-
I0?
-If
___
·
which by the use of equation (9) becomes
The lower limit of the outer integral is chosen so that
fS(1)= 0 and hence W(1) is independent ofT.
The usefulness of this method, of course, depends
upon a complete knowledge of the characteristic values and
functions at = k. Now if k be taken equal to zero, equation (8) for s= 0 becomes
(I- )ffo"- 2("-"tf.o +eofo=
0
which is the equation for the associated Legendre functions.
f is then Pn, where n must be so chosen that P (Z' 0.
e. is obtained from the relationship
0z)=
e,= n(nUl)
There is a method given by Bholanath Pal of computing the
values of n for which a•P-M(z)= 0 for given values of z and
m. His paper very conveniently furnishes tables from which
the following values are taken
I
Roots of The Equation
Zo
COS
Ph(z,) - 0
Z7%oCOS
15
300
n(n+l)
n
n(nt 1)
n
14.12
213.5
6.83
53.5
26.29
717.5
12.89
179.0
18.93
358.5
38.35
1509.
It is of course possible to obtain values of e0 for all
values of z and m.
jBulletin of the Calcutta Math. Soc. vol. ix (1917-1918),
p.85 and vol x (1918-1919), p. 187
-12-9
I
The method outlined above has the advantage that the
characteristic function is obtained simultaneously with the
characteristic value.
___
___I_
III THE FIRST CHARACTERISTIC VALUE AND FLUNCTION
The first characteristic function is by far the
easiest to treat. This is a consequence of the fact that
fo = 1 and e,= 0 for all values of z. . In the following discussion a is taken equal to zero. For this case
I
I
EafcL
+3
=f:.~
3
From equation (11)
I
0iz
=Je,)
'
Z,-a•J
1
f
2
:a 3a -P)•
: •_ (•-3-e') log C1+-z)-
Ij
r
1 6- _ - ( _3L e l)÷101
b
6-z
r+;+
or
,--
-~
e,
where
3
f
3
is calculated from the expression
which
obtined
is by putting s2 d a inequation (11).
which is
obtained by putting s = 2 and a= 0 in
-14-
equation (11).
1
__1_
-- ~-
The calculation involves exceedingly tedious algebraic manipulation. We shall accordingly state the result and then
verify it.
f2-. 13
924-2.Zz+13
-6
e
92.
4.+)
+(7-3 0)'lo9
t
n-='
( 2.)
This expression is correct if it vanishes at z= 1 and if it
satisfies the differential equation
o
-e Lf
=(-(I- -e,) •,
_•-) • .-zi•) =
_-
where
fl,
-
19
aM
acnd
f,
Let us put
1
Re6
The equation for f. may thus be written
2
~2.By differentiating equation (12),
we obtain
ao
Now
Iog
It
"
h+Z
o
-
--
-15-
~__ _. i_
i.-e
WR
~·
....
. -_ i_
hence
S (4a-6)-
99= - 1
fP ±
-FIb +
6t log L + L-.5
lI 1+1 £+&
03)
I-E'-7~ 3
and
fx
13
+
2. -
-e.3.-3 -+1-.(.-•
8
aol0)9
C'-1 +Q)Io51
- j+e, l(,)
2
- 45z++60 z4-IS
As
-37 +6z-3-6 (-2n2+1_ Z2)
- 1) Z
e
3x~2 1
t 2 log
Hence
- 314+ 4ta-I +
d)
a
2-
109
'j
- )1
We have thus verified that f, as expressed in equation (12)
satisfies the differential equation. f,(l) is obviously zero.
e- is obtained by putting f, (Z ) = 0.
e=
ex__F
I+ +Z. log
a)
(
135
If 1, is expressed in terms
e, becomes
29
99
I9
9
e-
2
+Is
e4
L91
of
of
X--to)- (,O
the expression or
Z.
,
expression
the
92+
_
5
)2 I+-?--9
I M,O
9
1o3 '+.
+70
270
270
270
,+4
9
-16-
for
g9.3-•i27•-14 +4
270
30O+60.3+30a 2
9. 4+9ij0--
t.) 109 ý+o
O+7-3.-2
1-9o
kniMMEMOM
°
X
-13
+
- 9
0
4)
The form in which we have expressed f2 and ez is not
suitable for computation. In order that the series for W and
b may converge with convenient rapidity it is necessary that
fp and e1 be small numbers. As expressed in equations (12)
and (14) f, and ez are the differences of large numbers,
and to compute them requires carrying an inconvenient number
of significant figures. It turns out that e 2 contains (1-z, )
as a factor, which is removed in the following way.
Using the expansion for log -
S+4
2
o2 11
9
1"35
e2,+
, we may write
Z8
Z849 +
njn
+3£72
135
n=4
By expanding and then collecting terms, we arrive at
e = -s0.• s'
+0 z0c ir.-8
540
9
h=4
The numerator of the fraction is factored by synthetic
division
-5+15-3-23 +12+12-8
_1
- 5+10+7-16- 4.8
-5+10 + -76-49 +8 O
U1
- 5 +5+12-4 -8
-5+ 5+1i2- 4-8
- 5+ 0+12+8
-5 + o0n2 + 8
0-
1
c
The numerator must therefore contain (z.-1)3as a factor.
-17-
n-1
~0___
**Iiiia
~
__L
-e
_·
...
·I
·_
Hence
e. Q
0-)3
[
-
.8
(00I+
-
2
(-)
-4
n=4
The series in equation (15) converges rapidly for the values
of z. in which we are interested and equation (15) is entirely satisfactory for computing e2
We now turn our attention to fz * It is desirable to
introduce the following definitions.
A (z) = 9z 4 - 2 2z+ . 13
1080
B (z)
2 log 1+z
+
2
135
(2z -6z + 4) + (7-3z2 ) log l+z
(z) - log 1z +
(1-z)
1
( 2 ) n
2
By putting log lz
-.(1z) 1
,,,( 2 )" n , it is possible to factor
2
(l-z) out of each one of the functions A 1 and U. Thus for
the case of 3
~L'1••2
(vY*
A(1)-
1080 "
I 9Z4-2a1 +88+
--
( )Z(')
"-2)-
:
W
2
--
4
02 [ 9A2I+E-I
In a similar way it may be shown that
2.
= L)124~
2 fl2. (~i~)~1
The expression for C is transformed in the following way.
:-
2-2)n
-18-
"
WAýi
.
From equation (12)
f,= t (L)
-A8
u)+ezlay
2(r)-e,
I
Or if we put
)
f
2
f•=
e
etc-
)
(um~
1
[F1(
Q-t)1
()c(.)]+
f e,lo
9
A, B and C are independent of zo so that when once they are
computed they hold for all horns and all frequencies.
It is not practical to obtain f3 and e3 as explicit
functions of z.
We may summarize the results of this section as
follows.
For the first characteristic value and function:
eo=o
ef,= I
18
where A(s) -
__
_•9"'+_8L-5
G(s),Z
-Lf
z
-9e,
=1
3
~
.-
)-e
C)
-19-
)
ce
~
_I
~LCF~
_~___
-----
i_.
~
~
IV NORMALIZATION OF THE FIRST CHARACTERISTIC FUNCTION
met by
The boundary condition at the diaphragm is
expanding a prescribed function in a series of the characteristic functions. It is therefore necessary to know the value
or
VC\
W
--
This integration may be carried out for the first
characteristic function in the following way.
fr4 + -Y2-v *-. -
0f
W
ff
W2 = fk + 2f.fy.+
f-.. + 2,
W'a=
-
,+f)7a* .
:
ff,
4 + Y,
. .
off,). *
fo is 1, so that for normalization we need the following
integrals
I
,
ofeds
Af-
o
and
-- ""
2 -
-"
Using the expansion for log J+z, and expressing F# in terms
2
of z, , we obtain
,-
I-.-o
[
)'Z* ( -")
•-c•
_'_- _-___)+
v
Ob
b
3
-20-
z
~
+
+
Z••"
4
'
.
.-
* a
,
-
-
OW110
I
· ~I I·-
To obtain
If dz, we evaluate
B(z)dz
A(z)dz,
and
i(z)dz .
It is easily verified by differentiation that
X(z)dz.
- 2(lizo) log L+
27(1-5)-l10(l-z)-45(1-sz)
2
135
S16,200
Using the expansion for the logarithm we obtain in the same
way as before.
+
-3
5,
In a similar manner we find
(-)t = (I
77-(3_6-
o ')o-6)-
- J
and
I
I
}{act
Now
where
= IT+ e••
0
1 I
Tt o
If (9)e(
(
-
()d
-
()
If all quantities are expressed in terms of zo , we have
I:(1Z) 3 27z•2+81z,+22
z~z~;1)
1(z .3z,, -6)
8+"
54
5-40
z____l)2
9
n-1
n=t
8
+5
16,200
(1-
z
n+
(
2
n
,,,
.· I_
~__
--f ~· _C
··
__·
__·
__
__ __
The quantity within the braces contains (1-zo) as a factor
which is removed in the manner illustrated above. It turns
out that
7a 2+ b6S 0o+-57Z
-375
-s-,0so
0=
144
1800
4 +2 3+1)
+" +
n=4
It is readily shown that
Clog
Thus
e2 (,o9
"
'
-
0
'T-q
r-
contains (1-z
I.!_.' a
-
1
o)
-
n-(
22
5
We have shown how to calculate I and e. If~dz is calculated
from thee relation
e
ei
jfdz.
Finally we must obtain an expression for
k\b
4
7
-%-1
,a 2:
]
-
or a 2
L
) o
+ ,4 102 I+F 2
1+
2
It may be verified by differentiation in respect to zathat
¶' f,3d =
I-n
6
2
36
± 2e,-
Z
_ [-.+
When e"is expressed in terms of z,,
-15
739*-.3 + 199
[=(389'
zo
11:og(2
Z.
'h- M1
-22-
109 1+t
2
the expression becomes
(,
0
24 -
0.5zo 4 90-62.1
The expansion for the logYterm is
1)-3-
ej
d 9f,2 2 e.
(,
- &,(lts.)lo91( I4t·)
zi
2
og +Z
MWM- ___
jl
By using this expansion for log together with that for log
we obtain successively
f,2 dz
(l-o)
[210z+391zN+102z
S
-
.
.
-67z,+24
1620
1--
3
2
36
:
.
.e..(+z).
(7i-2) )
108
ftIdz
n
((1-zo)1 195zoa-330Zo + 43z .2 -56z.-+ 48
.3240
Znl
216
8OZ
4[z
.5Z
-5_ll+22
*2,
3240
I
7.6
dz
2
(l)-so )
,205z2
... .+135z
_ z (Zo
+105z
12960
+ 215z
+
I-1
288
_
za(z.ot-l (7zo-2)
864
V t
IM
L.
192
-) n-S
T ,o)-s-t(Z1
n=S
We are thus able to compute J•wdz and Iwzdz for the range of
variables over which the series converg6. It is evident that
the smaller the angular opening of the horn the better is
the convergence.
-23-
ii
--- "
C·
V THE SECOND AND HIGHER CHARACTERISTIC FUNCTIONS
As we have seen above, the first few e's and f's
first characteristic value and function may be exthe
for
pressed explicitly as functions of ze A different method of
approach must be used for the second and higher values.
Here again the method can be applied when a is different
from zero, the boundary condition at zo being that W'(zo)= O,
and it is interesting to retain the I in the discussion even
though for cases in which a40 the boundary condition is
not appropriate to the horn problem.
Let us suppose, just as we did before, that 2 all of the
characteristic values and functions are known for c =k.
Denote these functions by p.and the corresponding characteristic values by dn * The set of Pn's is orthogonal and complete so that the f's may be expanded in a series of the p's
mA o
where
-
a
The boundary conditions are thus automatically satisfied,
It is also possible to expand ztf, in a series of p's. Thus
2.
2.P
'-
,
where
CISM
q
No
Replacing fS by its series expansion, we obtain
h
to
notao
V" d
If we introduce the notation
wVn
then
M
Eao
'%,--e) p,,,p,,
-I2~1-t')2.
I)
co
s' K v,
-24
-24-
r ....
-
-- --
I
·-·
The differential equation for fs is
Q- i)4s'-
Z(a.i-)-zf• + (,eo- •z)•-- l-Zfs-,
S-w-
In order to obtain ase, we substitute the expansion for fS
the differential equation obtaining
into
@0
m+o
+=
m-o
It must be noted that fo is some one of the p's. If
we are seeking the (r÷ 1 )"* characteristic function then fr Pr
and likewise eo=d,. We must also recall that the p's, being
characteristic functions with characteristic values d when
cL=k, satisfy the equation
-?(L+) 1)-K
(I-wiZ)p
and hence equation (16)
CO
1
)
P C=
EP2,
dm Pm
becomes
CO
Z(C1 r'P.,
Z
m=
ho
=
p,-2 (S .,.,,
rzOO
tY"=O
e )
+=I
By equating coefficients oC pihwe obtain
cxSn-
C-51 ým-I
ct me
Now p=,. fo* It is therefore a solution of the homogeneous
equation corresponding to equation (8); and consequently,
any integral of (8) still satisfies the equation if it be
increased byWpr, where u is an arbitrary constant. Accordingly aSr is completely arbitrary'and the numerator of the
second member of equation (17) must vanish when s =r. This
yields the condition from which es is computed.
s
or
S
G~or
asr is arbitrary in the sense that it is an integration
constant..It is actually chosen so that fs(1)= 0
-25-
IMWim
I
-
P
o
utor
and since fo-Pr
Writing
we have aor= 1.
Cs-a,r =
,Kr,
we obtain
S-i
Sdo
e= I
s-ls,n
Krn-- Z
OL-.ret
Notice that due to the orthogonal properties of the p's
aon= 0 if n cr . From the expression given above for es
o
Hence
eS
cco
where
ion for"z0sm
S-J
Cls-,,
Kn
-Zn-
-tr
(18)
t-z
i~ans n is not taken equal to
terms of the k's is
S- , m
Km~n
ICs",n
. The express(J9)
dr-dCIm
We are thus in a position to determine the a's in turn and
likewise the e's.
To apply this method we make use of the fact that
when k=0 the p's are the associated Legendre functions of
the first kind whose properties are well known.
The discussion is now restricted to the case of a = 0.
As an illustration let us compute the second characteristic
value for a 300 horn. From Pal's investigation we know that
the characteristic functions for c 2:0 are Po , P6
Pf 89 , P~8
1 93
etc. Since we are seeking the second characteristic value
e,= (6.83)(1+ 6.83)= 53.5= dr . Likewise P,=P6s6 3 . Now
e,
= Krr
=
:
S
os3 o
Sos 30.
(0.83
.
There is unfortunately no analytic method for evaluating the
integrals which are involved in the expression for e,.
Accordingly we are forced to use graphical means. Tallqvist'
has published five place tables of P,(cose ) for integral
values of n up to and including 32,e ranging from • to 90"
in steps of one degree. The fractional order Pts can be
iTallqvist : Tafeln der 24 ersten Kugelfunktionen P, (cos e)
and Tafeln der Kugelfunktionen P2 (cos e) bis P3a(cos e )
Societas Scientiarum Fennica Commentationes Physico-Iathematicae vi 3 and Vi 10
-26-
C___
:_~___
-~~--
obtained to three places from the tables by plotting P, (cose)
against n for a given 0 .
The followinga integrals were determined graphically.
ios 30
108C[Z=
IC
0Z224
°
2.4 d
S.
d -000930
2 P -•.&Z
00388
coS 300
Cal 30o
P2 3&L_= .02o3
I
Co• 30o
dt - . 02.3
12,89
Accordingly
_~
e,-
.0203 =
.02022
-oTz5
's.
We next determine the a,
c
From (19)
a
dmn-om
aon
G.
-
OL I
.914
n= O
But a on : 0 unless n =r aor =
Therefore:
rr
Kmv"r -
where
-Im KImr
dp~
Z~
p; tCd
0•p
SI
Thus
dI
too P •• .
~. oPp.
CL
3.=8-10'3
(53. X.134)
=r-
53.5-0
-
S?~ooS
= 5.41 x 10
da Pc -2
, .
CoS 300
4
I53
d
CoS 300Pr 89 P63
Cos 30
"'
- 6. 02 XIO
- 9.3 xi0
QX5. 5)(. Oi2-3)
-27-
2
I______ _goMINNOW
__1_--
_
From (18)
e2. 27
in Kr"n
Ko
g p,pod
Vj=
,,KO
o=
k 12 +
+0(,Z
P
Pip
2 P6 a3 P12.
K,~t
C0-3
JCas
Accordingly
3 0
2.U.75
%10o
d
CS 30"* P6.93
to
175
3.88o
R
;t 3
.8
o
n
9.30 x10
I
-4
2.22 x 10'
= 4.18 x 10-
, ') -(6.02glo- 4)(4.18x o-') ..-e =(s.4/x o4)(1-.7Sx
Hence the second characteristic value for 300 is
4
b = 53.5 '+ 0.914cet+7.0o• io'•-C. + ......
7With the use of the following additional
it is possible to obtain the coefficients in the
of the third characteristic value for 300and for
and
for
15f.
.. . . third
....
. . characteristic
. .....
. . . . . . . . . vi values
. . . ..
f r
1
Ie
7.0x 105
given by:
integrals
expansions
the second
A
P .c= 0.00o555
d1o= 0.o0113
0o'S15
COS 30o
,,, .
IX P,
A
=_ 3 xi 0-'
IOSSO
e. p
22
FS.2
14.115J
Ico IS5
-0.00542
dE =0.00308
--
4 2.
2-29
216.29
P
14.12
0. 00302
SP 5.
10o5
COIlSo
The coefficients in the expansions for the first
characteristic values are computed from equation (14).
The expressions for the first three characteristic functions
for 15 ° and 3 0 °are summarized below.
eo= I15"
S So = 30o
bo= 0.96631C2-I.607xIO'¢ -. ..
b,: 2Z1.5+0.977c z + O(C4 do')+.÷ '
t
b,= 717.5 o0.980 C2 0 (C4xl '-)+ ..-
ba= 0.8720 c- 9.0486 x dC4+.. .
5 )
bi.= 179.o + o.919 cz + 0 (C4r10-
The convergence of a series representing a characteristic function depends to a large extent upon the value of e *
-28-
20
b,1 53.5*0.914c2 +7.ox1Coc--.
'
rii~iiii
_
MM
ai
As we have pointed out previously b.h? c' zI for fl 1. Now
all of the curves start out with a slope which is approximately one and if the intercept on the b-axis is not large
the curve must bend soon after leaving the axis to keep
from crossing the line b = c z.1 . Hence if eois small the
series does not converge rapidly over as large an interval
of cL as does a series for which e. is large. The second
characteristic function for 300 is a case in point. The
general features of the characteristic value curves are
shown to an exaggerated extent in Fig. 4
10
Fig; 4
bomust lie between the straight lines b= c "z, and b= c *
The other characteristic function curves start at the axis
with approximately unit slope and approach the line b= c2 z 2
asymptotically. The expansion for b, does not converge
rapidly because of the bend in the curve in the vicinity
of c-=O. Figs. 5 and 6 are included to give a quantitative
picture of the 15*and 30*cases.
We have an interesting check on the values obtained
by the method described above. Suppose that for a fixed
value of c z plots are made, for a series of values of b,
of W'(z) where W is that integral of equation (5a) which
is unity at z= 1. From this set of curves, we can obtain
W:'( z.) for various values of b. W'(z.) is then plotted
against b and the characteristic values are the points where
this curve crosses the axis.
-29-
__II:
QiImiid
i-
~_?___
-·
The problem was originally approached along this line, the
family of curves of W'(z) against z being obtained experimentally by means of the differential analyzer'. A comparison of
the values of b,computed from equation (20) and obtained by
means of the differential analyzer is given in the following
table.
b, for 30
c
_
from 20
by D. A.
- 50
8.5
8.3
-100
- 37.2
- 37.0
-150
-
-200
-127.0
(2.0
"
-126.0
There is considerable error involved in the differential analyzer method, especially in plotting !W'(zý
against b. The agreement is good considering that both
methods involve graphical processes. The accuracy with which
we can know the first characteristic values is much higher
than for the others because the coefficients in the expansion
can be obtained by purely analytic means.
If there were no other means available, the characteristic functions could be computed from the theory. It is more
convenient, however, to calculate the characteristic values
from equations (20) and then to use the differential analyzer to obtain the characteristic functions. The curves given
in Figs. 7 and 8 with the exception of those for c =0 are
the result of the procedure. The c2 : 0 curves were obtained
from tables%. The higher functions show less and less dependency upon c1 , and a given characteristic function depends
less upon c2 the smaller the angular opening of the horn.Thus
in the 30*case, the third characteristic functions over the
range of cl in which we are interested are all contained in the
narrow band indicated in the figure.
IFor a description of this machine see Bush, The Differential Analyzer, A New Machine in Solving Differential Equations, Journal of Franklin Institute Oct.1931.
aloc.cit.
-30-
Qaiýwm_
..
I
.
In the 150case, the individual characteristic functions were
not resolved by the analyzer. Hence only one curve is given.
The second characteristic functions are contained in such
a narrow band that it is not convenient to draw curves for
ca between c 1 0 and ca* -600. Now the area under the c= 0
curves is zero and it follows, therefore, that for the case
of a 15°horn and a diaphragm which behaves like a piston,
the second characteristic function is not extremely important
and the third characteristic function can be altogether neglected. In the 300case, the third characteristic function
is not important.
Another feature of the curves which is most clearly
shown by the second characteristic function of the 300case
is that at z- z., the curves are more nearly flat the higher
the value of ct, Furthermore the curves for c2 - 0 for the
functions except the first always lie closest to the axis
S7~t no . These features indicate that
lim
W" (z.)
0
" e'%+ W-
This supports our assertion that the characteristic value
curves except the first approach the line b= caza_ asymptotically. For as we have already seen
b=
z -(1-z)
o~w
W (zTo)
-( zo)
Chcsr~cferis1ic
t//c~
for
/50
HOrn7I
800
600
50o
D00
-400
-00
300
500
zoo
b
ert-•ic Values for 30 ° Horn
100
0
,,D
- 100
Fig 6
-200
_ ____I
1
--
-
1.0
W
0.8
0.6
0.4
0.2
0
-0.2
-0.4
ii-
'
II
~
·
·
-
--
--
--- 1.
------
-----------
_I_
_-----··
ftw;;ý
II_
__
-----~I--LI----
W
).8
).6
).4
).2
0
0.2
0.4
F I G. 8
MEN
VI THE RADIAL FNC2TIONS
The next step is to find a convenient representation
of the functions which express the dependency of the velocity
potential upon ).i, As was previously pointed out this problem
can be reduced to that of finding F along the entire positive
real axis where F satisfies the equation
(1+xl) d2F + 2x dF
dx'
b+cx') F= 0 ......... (6a)
dx
and for large values of x represents a wave traveling out
along the x-axis. The fundamental problem here is that of
analytic continuation. Equation (6a), which is equation (5a)
with z replaced by ix, has singular points at x=ti and x=ms
We can therefore obtain series expansions of two independent
integrals of (6a) which converge for Ixi-l. Let these integrals be represented by H.and H1 . It is likewise possible to
obtain asymptotic series representations of two other independent integrals, say F and Fo which are valid whentcxl
is large. F and Fare of the form.
-
F0
e
i lezi
u(x)
o
e_%i(x)
x
u is expressed as a series of descending powers of ex. It is
evident that F represents a wave traveling outward since the
phase decreases with increasing x. The radial problem would
be completely solved if we could find A and B such that
F=A (b,c2 ) Ho+B(b,
c2 ) H,
See for instance: Maclaurin: On the Solutions of the Equation (Va+kA)• -- 0 in Elliptic Coordinates and their Physical
Applications. Cambridge Philosophical Transactions, vol.xvii
Part I
___
_i~L
dffiiiiI6
For the case in which b is chosen so that there exists
one integral of (5a) which has no singularities in the finite
portion of the complex plane, the connection problem has been
solvediby using integral representations of the solutions.
Thus solutions of (5a) may be written
,'fVW(a) = fc e icet
t)
u~t)dt
where Q(t) itself satisfies (5a) and the path c is such that
the bilinear concomitant
vanishes. It seems unlikely that a suitable path exists when
b is such that there is no solution which is finite at both
+1 and -1. It is evident that the path may not pass through
both +1 and -1
for a has a logarithmic singularity at one
or the other of these points. If u is finite at +1, it is
finite at that point only on one sheet of the Riemann surface so that a contour which starts at +1 and, after passing
around -1, returns to +1 is unsatisfactory. Furthermore,
attempts to find a closed contour on the Riemann surface
which encircles one or both of the singular points have
failed. It is not obvious that there exists a path such that
one or the other of the two quantities (1-t 2 ) and (iczu-dA)
dt
vanishes at each end.
There is another method which although theoretically
possible is impractical. Expansions of solutions of the
differential equation in a two-way Laurent series exist k
which converge in the region outside of a circle of radius
one about the origin. These solutions could be joined at
a common point with the expansions about +1 and the connection would then be complete. The Laurent expansion is of the
form
where the s is determined by a continued fraction equation
which insures the convergence of both the ascending and
descending parts of the expansion. The equation for s must
be solved by a cut and try method which is practically impossible to carry out because s is complex for the values of
b involved.
We know that the radial functions for large values
of x represent waves traveling out along the x-axis. The amplitude as well as the wavelength depend upon x.
t
Stratton: Spheroidal Functions, Proceedings of the National
Academy of Sciences vol.21, 1 pp.51-62
cf. MacLaurin loc. cit
-33-
~------~
Accordingly we are led to seek solutions of the form
Sp(X)eiQt Q. Equation (6a) is of the self-adjoint form
&Lr'j)t
dx
S q=o
Let us assume y= Rewhere P, Q, r, s and x are all real.
Substituting the expression for y into the differential
j
equation we obtain
P+irtP+PQP =0
ir
+sP
ij
PtrQ
r-r
Hence we have two equations
rP-r42P +-
s P0 O
PP=O
rC P t r*Q P +t
The second one may be written
(22)
Hence
A is the integration constant
The first equation may be written
P
rP
r
Let us as a first approximation assume that P is a constant.
Equation (22) yields
Putting this value of Q in (21),
we obtain
(rs)Y4
Thus our first approximation to a solution of (21) is
e
This is the well-known Wentzel-Kramers-Brillouin approximation.
Let us carry the process one step further
p= F
If
S
e
P
_
__
-4
-34-
0-sr
I
Hence the second approximation to q is
4
4+
_
and the third approximation to P is
A
With the aid of this second order approximation, we
can get some estimate of the error involved in using the
W-K-B expression for the radial functions. Let
A
J-
-F
Z
45rr.
......... (24)
Then
P=
[I + a r
(sr.)-
Thus A*L is a measure of the reliability of the W-K'B approximation. For equation (6a)
where X= b/c'
r = (+ xt) and S= -CaAh+X)
Substituting these expressions into the equation (24) we
obtain
a_r_.
5
5(1+_s
)'.Xx+ xOt'* 7-',Zx, )(htx')(a•+ ,)Q
lb c' (t+x )(A+Y )3
This expression is evidently of the order of 1 so that the
approximation becomes better as x increases. x ° We look then
for the greatest error to be committed when x = 0.
S(o)
-8
-a
a
For the first characteristic value of a horn of
reasonable angular opening A is approximately unity. We see
then, that even when cl is as small as fifty, ~1)
is
mod-
erately small compared to 1.The approximation be &es
better
as cl is increased. In the case of characteristic values
other than the first, we must be careful in applying the
W-K-B approximation that N is not too small.
-35-
M
W
FI
PT
PL~-
i.e. that c2 is not in the vicinity of a root of one of the
characteristic values. With this restriction in mind, we
shall write
S+ X
XI)(I+XZ
F =
V4
where k is equal to ci(a real positive number)and X is
assumed to be positive.
Additional evidence for the validity of the WK-B
approximation was furnished experimentally by comparison
with differential analyzer solutions.
In case X is negative X+t 2 has a zero at the point
tx 2 <O0 and when x > x' Atx > 0.
When x < x'
',I'.
x'
t
In the vicinity of x ' the W-K-B approximation breaks
down. It is to be noted that the exact solution which the
expression represents asymptotically has no singuW.-Kx'. In order to get connections between the asympat
larity
totic representation of our function in the two regions we
make use of the Kramers connection formulas.
In the equation
C)(rj + sM=
let r be everywhere positive on the axis of reals and let
S have a zero at x'. We shall call the region on the
x-axis to the left of x', I and to the right, II. Let S <
in I and c>o in II. Then it can be shown that if y is represented asymptotically in II by the expression
y is approximated, but not asymptotically, in region I by
dc
Xi'
the expression
-
AFor a complete discussion of these formulas see eremble,
A Contribution to the B-V+K Method, Phys.Rev.vol.48(1935-6)p.
549
Kemble carries through the proof for r=l. However, there
seems to be no reason why the argument is not equally valid
when r has no roots in the neighborhood of x'and no real roots
-36-
2'
:L---;-
Vis a constant which may have any value which does not
4)
to be much smaller than I sin(-/ 4)l1. The
cause (cos ('connection formula is written
cos(f-n) e
-
cos x r rdx
(IsIr) 4
I[sr]]
By taking Y= 0, we have
[[•JxJ}-d
f' dx
l
-cos
(Isl r)
and by taking
Y= -n/z
6sr]
(Isr ) '4
Henc
Ye
e
-A
a
(1e-~r)
r)"
54
e
.r'
e
Lsr]"'4
we have for region I
Applying this exp:ression to ,,(6a),
I Xta
I
--
1xZ+X1 (i÷xa) •4
and for region II
r-
-x
E
:
iK I
(x'+x) (+x2)
-37-
X o
'
d
"-
. Thus in the pegioi
In I, F has a constant phase
sinhflh '•Ii , t
0
andA=
•-=
for
which
spheroids
the
between
back and
moves
air
the
and
of
position
phase is independent
The di!
piston.
non-rigid
a
of
forth somewhat in the manner
whic)
for
expansion
the
in
turbance corresponding to a term
vicinity
the
in
wave
as
a
b is positive is not propagated
of the diaphragm.
It is interesting to note, that the air near the
throat is a dispersive medium. Let us consider the waves
traveling outward along the axis of the horn. The phase
is given by
It follows then that the phase velocity w is expressed by
w -
C-M.
(A
V
rr
Ici1T
I
/?+X
where v is the velocity which appears in the wave equation
We see therefore that the phase velocity depends upon cz
which in turn depends upon the frequency and hence waves
of different frequency travel with different velocity.
Likewise for a given frequency, the waves corresponding
to the various allowed values of b are propagated with
different velocities.
As we have indicated in the introduction, any
wave inside the horn can be considered to be the resultant of a suitable combination of elementary waves
. Let us examine the velocity potential of the
W, Fn eit
first elementary wave. If the horn does not have too large
an angular opening we may consider A to be unity. The real
velocity potential is therefore given by
V=
)
cos(litKx)
Because of the synnnetry about the axis of the horn
we need to consider conditions in only one of the planes
which passes through the axis. It is convenient to imagine
that the velocity potential at any instant is plotted as
a surface above a section through the axis.
-38-
II
r
Figs. 7 and 8 show that for a given value of x, the velocity
potential is a maximum for z = 1. For a given value of z, the
velocity potential is a damped cosine function of x. Thus
the velocity potential surface has a series of peaks and
depressions along the axis of the horn, the peaks and depressions becoming more and more pronounced the higher the
frequency. A contour map of the surface will show the
general features indicated in Fig. 9. The velocity at any
point is perpendicular to the equipotential lines, so that
there are points at which the velocity is transverse to the
axis. This condition is noteworthy because a similar situation does not exist for the first elementary wave of a
conical horn, the velocity potential of which being independent of the angular displacement from the axis shows no peaks.
Fig. 10 indicates the velocity potential for the second
elementary wave at t= 0.
In this investigation we have considered the two
fundamental problems which are encountered in establishing
a foundation upon which to base a theoretical treatment of
the hyperbolic horn. The first of these problems is the
calculation of the characteristic values in order to obtain the characteristic functions; and the second, is the
representation from the diaphragm to infinity of the
radial functions. A method has been evolved for obtaining
the characteristic values, and the validity of the W-K-B
approximation to the radial functions has been established.
Considered as a boundary value problem, the treatment is
complete. Some of the more obvious physical consequences
have been pointed out, but the details have yet to be
deduced. It seems likely, for instance, that a comparison
between the conical and hyperbolic horns will yield considerable information concerning the effect of the curvature at the throat. It is likewise both desirable and
possible to investigate the case in which the diaphragm
s considered to be a piston driven at a fixed frequency.
I wish to express my indebtedness to Professor
J. A. Stratton for his indispensable guidance and encouragerment.
-39-
Nili
~
q-l-
--· -
- --
Fi9 9
Fig 10
P- Peak
D- Depression
o -0
Potentiql
1---
"
·
BIOGRAPHICAL NOTE
From 1927 to 1931, the author pursued a
course of study at Lehigh University for which
he was awarded the degree of Bachelor of Science
in Engineering Physics. During the next two
years, as a teaching assistant in the IMathematics department at Lehigh, he carried on
work for which he received the Masterts degree
in 1933. In the same year he was awarded a
graduate scholarship and admitted to the Graduate School of the Massachusetts Institute of
Technology. During the years 1936 and 1937 he
held an appointment as a Teaching Fellow in
the Physics department of the Institute. He is
a member of Phi Beta Kappa and Pi Mu Epsilon.
-40-