On Webster’s horn equation and some generalizations
P. A. Martina)
Department of Mathematical and Computer Sciences, Colorado School of Mines,
Golden, Colorado 80401-1887
~Received 17 October 2003; revised 18 May 2004; accepted 2 June 2004!
Sound waves along a rigid axisymmetric tube with a variable cross-section are considered. The
governing Helmholtz equation is solved using power-series expansions in a stretched radial
coordinate, leading to a hierarchy of one-dimensional ordinary differential equations in the
longitudinal direction. The lowest approximation for axisymmetric motion turns out to be Webster’s
horn equation. Fourth-order differential equations are obtained at the next level of approximation.
Comparisons with existing asymptotic theories for waves in slender tubes are made. © 2004
Acoustical Society of America. @DOI: 10.1121/1.1775272#
PACS numbers: 43.20.Mv @LLT#
Pages: 1381–1388
I. INTRODUCTION
Webster’s horn equation ~1919! gives a one-dimensional
approximation for low-frequency sound waves along a rigid
tube with a variable cross-sectional area A(z). The equation
itself can be written as
S
D
dP
1 d
A~ z !
1k 2 P ~ z ! 50,
A dz
dz
~1!
where k5 v /c, v is the frequency, c is the ~constant! speed
of sound, and z is the coordinate along the tube. Equation ~1!
can be derived by considering a thin layer of the fluid at z,
perpendicular to the z-direction, with the assumption that the
acoustic pressure is constant over this layer; this pressure is
P(z). For a succinct derivation, see ~Pierce, 1989, p. 360!.
Alternative derivations and extensions to other problems
~such as with fluid flow through the tube! are available; see,
for example, ~Reinstra, 2002! and ~Hélie, 2003!. Exact solutions of Eq. ~1! are available for various specific A(z); for
some recent results, see ~Kumar and Sujith, 1997!.
In 1967, Eisner published an excellent review of early
work based on Eq. ~1!. On p. 1127, we read: ‘‘Eq. ~1! is
usually called ‘Webster’s horn equation,’ but we see that
there is little justification for this name. Daniel Bernoulli,
Euler, and Lagrange all derived the equation and did most
interesting work on its solution, more than 150 years before
Webster.’’
We are interested in giving a systematic derivation of
Webster’s equation, and of higher-order variants. We limit
our analysis to axisymmetric tubes, and obtain Webster’s
equation when we seek axisymmetric solutions. We also consider nonaxisymmetric motions. To be specific, we consider
a tube of finite length, and seek the frequencies of free vibration of the compressible fluid within the closed tube.
In 1916 ~three years before Webster’s paper!, Lord Rayleigh published a paper on axisymmetric motions in an axisymmetric tube. He began by writing the general solution of
the axisymmetric Helmholtz equation in cylindrical polar coordinates (r, u ,z) as
a!
Electronic mail: pamartin@mines.edu
J. Acoust. Soc. Am. 116 (3), September 2004
SA
u ~ r, u ,z ! 5J 0 r
d2
dz 2
D
1k 2 u 0 ~ z ! ,
~2!
where u 0 (z)5u(0,u ,z) is the value of u on the axis of the
tube; as the Bessel function J 0 (w) has a power-series expansion in integer powers of w 2 , this provides meaning to the
right-hand side of Eq. ~2! @see Eq. ~26! below#. Rayleigh
then obtained an equation for u 0 (z) by applying the boundary condition on the rigid wall of the tube; as a first approximation, he obtained Eq. ~1! @see Eq. ~8! in ~Rayleigh, 1916!#.
He also discussed some higher-order approximations. We
shall adopt a similar approach, although we do not begin
with an explicit representation such as Eq. ~2!: we shall use a
Frobenius-type power-series expansion for the radial variation of u(r, u ,z).
After formulating the problem in Sec. II, we examine the
special case of circular cylinders in Secs. III and IV. The
exact solution for the eigenfrequencies is recalled in Sec. III.
Then, the approximate method is developed in Sec. IV. It is
based on some observations of Boström ~2000! for the related axisymmetric problems of elastic waves in isotropic
rods. In principle, we can obtain a heirarchy of approximations: We give explicit results for the first two members of
this heirarchy. Apart from the merits of explaining the
method for a simple case, we are also able to give a quantitative comparison with the exact solutions from Sec. III.
In Sec. V, we consider axisymmetric, noncylindrical
tubes. We change variables in the governing Helmholtz equation from r and z to r and z, where r is a scaled version of r
chosen to that the lateral boundary is mapped to r5constant;
this has the effect of complicating the partial differential
equation ~via the chain rule! but has the virtue that the lateral
boundary condition is applied on a coordinate surface. The
complications bring the shape of the boundary into the differential equation @cf. Webster’s equation ~1!# but they can be
dealt with readily because we then use a Frobenius-type expansion in the new ‘‘radial’’ variable r. Again, we obtain a
hierarchy of approximations. The first approximation gives a
second-order ordinary differential equation; it reduces to
Webster’s equation for axisymmetric motions. The second
approximation gives a fourth-order ordinary differential
0001-4966/2004/116(3)/1381/8/$20.00
© 2004 Acoustical Society of America
1381
equation. In order to assess these approximations, we compare with some results of Ting and Miksis ~1983! and Geer
and Keller ~1983!. These authors began with the governing
elliptic boundary-value problem for waves in slender tubes,
and then obtained various asymptotic approximations. The
comparisons are made in Sec. VI. Some concluding remarks
can be found in Sec. VII.
In summary, the approximate method described below
has two virtues. First, the approximations can be improved.
Second, the use of power series means that the basic method
can be applied to much more complicated equations of motion, and to systems of such equations. For example, the
propagation of elastic waves in nonuniform anisotropic rods
can be studied; for some preliminary results in this direction,
see ~Martin, 2004!.
III. CIRCULAR TUBE: EXACT SOLUTION
Consider a circular tube with R(z)51 for 0,z,1. We
can write down separated solutions of Eq. ~6!, u(r,z)
5J m (lr)cos mz, where J m is a Bessel function, and l and m
are arbitrary constants satisfying k 2 5l 2 1 m 2 . The boundary
8 and m 5N p , where N>0 and ,>1
conditions give l«5 j m,,
8 )50,
8 is the ,-th zero of J m8 :J m8 ( j m,,
are integers and j m,,
,51, 2,...,. Hence,
8 ! 2 « 22 .
k 2 5 ~ N p ! 2 1 ~ j m,,
~7!
This is the exact solution of the problem. It can be found in
textbooks @for example, p. 221 of ~Kinsler et al., 1982!# and
was given by Lord Rayleigh in the first edition of volume II
of The Theory of Sound, published in 1878 ~Rayleigh, 1945!.
A. Axisymmetric modes „ m Ä0…
II. FORMULATION
Consider a tube of circular cross-section and length L.
Using cylindrical polar coordinates, (r̃, u ,z̃), the interior of
the tube is specified by
0<r̃,aR̃ ~ z̃ ! ,
0< u ,2 p ,
0,z̃,L,
where 0,R̃(z̃)<1, so that 2a is the maximum diameter of
the tube. It will be convenient to define dimensionless variables, using L as our length scale. Thus, we put
r5r̃/L,
z5z̃/L,
R̃ ~ z̃ ! 5R ~ z !
and «5a/L.
0< u ,2 p ,
0,z,1.
Inside the tube, the acoustic potential U(r, u ,z,t) satisfies the wave equation
1 ] 2U ] 2U L 2 ] 2U
]U
1 ]
1 25 2
,
r
1 2
r ]r
]r
r ]u2
]z
c ]t2
S D
~3!
where c is the speed of sound. On the lateral wall of the tube,
the normal derivative of U vanishes
]U
]U
2«R 8 ~ z !
50
]r
]z
on r5«R ~ z ! ,
0,z,1.
~4!
We assume that R(0) and R(1) are both positive, and close
the two ends of the tube with rigid circular discs, giving
] U/ ] z50
at z50
and
at z51.
~5!
We seek free vibrations of the compressible fluid within
the axisymmetric tube. Thus, we put U(r, u ,z,t)
5u(r,z)cos mu cos vt, where m is a non-negative integer
and v is the frequency. The wave equation becomes
]
]u
] 2u
r
1r 2 2 1 ~ k 2 r 2 2m 2 ! u50,
r
]r ]r
]z
S D
~6!
where ~now! k5 v L/c is a dimensionless wave number. We
are interested in determining eigenfrequencies v so that there
is a nontrivial u that satisfies Eq. ~6! and the boundary conditions.
1382
J. Acoust. Soc. Am., Vol. 116, No. 3, September 2004
k5N p for all «.0.
~8!
This set is elementary; it can be found in Sec. 62 of Lamb’s
book ~1960!.
8 .3.832
The next set comes by using j 0,2
k 2 5 ~ N p ! 2 114.68« 22 ,
~9!
N50,1,2,..., .
B. Flexural modes „ m Ä1…
~Later, we shall regard « as a small parameter.! Hence, the
tube becomes
0<r,«R ~ z ! ,
The axisymmetric problem (m50) is unusual, because
we can take l50: both Eq. ~6! and the lateral boundary
condition on r5« are satisfied by u(r,z)5cos mz, so that
one set of eigenfrequencies is given by
We are also interested in nonaxisymmetric motions. As
an example, we consider the case m51 ~‘‘flexural modes’’!.
8 .1.841 and j 1,2
8 .5.331, Eq. ~7! gives the first two
As j 1,1
sets of eigenfrequencies as
k 2 5 ~ N p ! 2 13.39« 22
and k 2 5 ~ N p ! 2 128.42« 22
~10!
for N50,1,2,..., later, we will compare the coefficients 3.39
and 28.42 with those obtained by certain approximate theories.
IV. CIRCULAR TUBE: APPROXIMATE METHOD
If the tube is slender, «[a/L!1, we expect to be able to
derive one-dimensional theories. We shall do this using
power series in r. Note that we do not limit ourselves to
polynomials in r, and so we are not limited, in principle, to
very long waves. Nevertheless, it turns out that the low-order
truncations obtained below work best for longer waves.
We begin, as in the method of Frobenius, by writing
`
u ~ r,z ! 5
(
n50
r 2n1 a u n ~ z ! ,
~11!
where a and u n (z) are to be found. Substitution in Eq. ~6!
gives
`
~ a 2 2m 2 ! u 0 ~ z ! 1
(
n50
r 2n12 $ s n ~ a ! u n11 1u 9n 1k 2 u n % 50,
where s n ( a )5(2n121 a ) 2 2m 2 . Just as in the method of
Frobenius for ordinary differential equations, we require that
P. A. Martin: Webster’s horn equation and some generalizations
every coefficient of r , vanishes. For the first term to vanish,
we obtain a 2 5m 2 . As we want u to be bounded at r50, we
take a 51m, and then we obtain
u 9n ~ z ! 1k 2 u n ~ z ! 524 ~ n11 !~ n1m11 ! u n11 ~ z ! ,
n50,1,2,..., .
Notice that this procedure does not determine u 0 (z).
However, following Boström ~2000!, we note that u 1 ,
u 2 ,..., are all determined by u 0 ; for example, we have
u 1 52
u 90 1k 2 u 0
~12!
4 ~ m11 !
u 2 52
8 ~ m12 !
5
4
2
u iv
0 12k u 09 1k u 0
32~ m11 !~ m12 !
u n8 ~ 0 ! 5u n8 ~ 1 ! 50,
.
n50,1,2,..., .
~13!
`
~ 2n1m ! « 2n u n ~ z ! 50,
0,z,1.
Eliminating u n in favor of u 0 , we obtain an equation for
u 0 (z)
~18!
At this stage, no approximations have been made. We
obtain various approximations by truncating Eq. ~14!; this is
done next.
A. First approximation
If we discard all terms with powers of « greater than 2 in
Eq. ~14!, we obtain
~15!
where
~16!
From Eq. ~13!, we see that Eq. ~15! is to be solved subject to
u 08 (0)5u 08 (1)50.
If we look for solutions of Eq. ~15! in the form
u 0 ~ z ! 5cos m z,
~17!
(1)
we find that m 2 5Em
(k,«). Then, in order to satisfy the
boundary conditions on the two ends of the tube, we obtain
m 5N p , whence
J. Acoust. Soc. Am., Vol. 116, No. 3, September 2004
~19!
,
32~ m11 !~ m12 !
~ m14 ! « 4
F
m2
~ m12 !~ k« ! 2
.
4 ~ m11 !
~20!
G
The fourth-order Eq. ~18! is to be solved subject to
u 80 ~ 0 ! 5u 0 ~ 0 ! 5u 08 ~ 1 ! 5u 0- ~ 1 ! 50,
~21!
using Eq. ~12! to express u 18 in terms of u 0- and u 08 , and Eq.
~13!.
Substituting Eq. ~17! in Eq. ~18! gives
m 4 2 m 2 Cm ~ k,« ! 1E~m2 ! ~ k,« ! 50,
~22!
and then a routine calculation yields
~23!
say, where n m 5 x m 14(m12) /(m14) and
~ m14 ! « 4
1
~ u iv12k 2 u 09 1k 4 u 0 ! 1¯. ~14!
32~ m11 !~ m12 ! 0
E~m1 ! ~ k,« ! 5k 2 2 @ 4m ~ m11 ! / ~ m12 !# « 22 .
E~m2 ! ~ k,« ! 5k 4 1
~ m14 ! « 2
2
~ m12 ! « 2
~ u 9 1k 2 u 0 !
4 ~ m11 ! 0
u 09 ~ z ! 1E~m1 ! u 0 ~ z ! 50,
8 ~ m12 ! 2
m 2 5k 2 2 n m /« 2 5 m m2 ~ k,« ! ,
05mu 0 1 ~ m12 ! « 2 u 1 1 ~ m14 ! « 4 u 2 1¯
5mu 0 2
If we retain the terms in « 4 in Eq. ~14!, we obtain
Cm ~ k,« ! 52k 2 2
The lateral boundary condition reduces to ] u/ ] r50 on
r5«, and this gives
(
B. Second approximation
where
Then, regardless of the choice of u 0 , the infinite series in Eq.
~11! will give an exact solution of Eq. ~6!, assuming that the
series converges.
The end boundary conditions, Eq. ~5!, give
`
( n50
r 2n u 8n (z)50 at the two ends, whence
n50
This can be compared with the exact solution, Eq. ~7!. For
m50, we recover the exact set of solutions given by Eq. ~8!.
For m51, we can compare with Eq. ~10!; when m51, we
have 4m(m11)/(m12)58/3.2.67 which is in error by
about 20%.
~2!
u iv
0 ~ z ! 1Cm u 9
0 ~ z ! 1Em u 0 ~ z ! 50,
and
u 91 1k 2 u 1
k 2 5 ~ N p ! 2 1 @ 4m ~ m11 ! / ~ m12 !# « 22 .
S
1
x m 56 11 m
4
D
21
A~ m12 !~ 814m24m 2 2m 3 ! .
~24!
Finally, the end boundary conditions give
k 2 5 ~ N p ! 2 1 n m /« 2 .
~25!
1. Axisymmetric modes (mÄ0)
When m50, Eq. ~25! reduces to Eq. ~8! when we take
the minus sign in 6. If we take the plus sign, we obtain k 2
5(N p ) 2 18« 22 , where the coefficient 8 can be compared
with the exact 14.68 in Eq. ~9!.
For this axisymmetric problem, we can compare with
Rayleigh’s approach. As J 0 (w)5121/4w 2 11/64w 4 2¯,
Eq. ~2! gives
u ~ r,z ! 5u 0 ~ z ! 2 41 r 2 ~ u 09 1k 2 u 0 !
1
1
64
4
2
r 4 ~ u iv
0 12k u 09 1k u 0 ! 2¯.
~26!
If we discard the higher-order terms and apply the boundary
condition, ] u/ ] r50 on r5«, we obtain precisely Eq. ~18!
with m50. This is reassuring but not surprising: The representations given by Eqs. ~2! and ~11! are equivalent, although
the latter can be used for nonaxisymmetric problems.
P. A. Martin: Webster’s horn equation and some generalizations
1383
To solve Eq. ~29!, we proceed as in Sec. IV, and write
2. Flexural modes (mÄ1)
`
When m51, we obtain
k 2 5 ~ N p ! 2 1 54 ~ 96 A21! « 22 .
~27!
To obtain the lowest values, we take the minus sign, giving
k 2 5(N p ) 2 13.53« 22 ; the coefficient 3.53 differs from the
exact 3.39 by about 4%. Thus, the fourth-order model gives
good accuracy for the lowest set of eigenfrequencies.
If we take the plus sign in Eq. ~27!, the coefficient multiplying « 22 becomes 10.87, which can be compared with
the exact value of 28.42 given in the second of Eq. ~10!.
Thus, as when m50, the second set of eigenfrequencies is
not approximated well by the fourth-order model.
u~ r,z !5
(
r 2n1 a u n ~ z ! .
n50
~32!
Substitution in Eq. ~29! gives
`
~ a 2 2m 2 ! u 0 ~ z ! 1
(
n50
r 2n12 $ s n ~ a ! u n11 1L n ~ z ; a ! % 50,
where s n ( a )5(2n121 a ) 2 2m 2 , and
L n ~ z; a ! 5R 2 u n9 22RR 8 ~ 2n1 a ! u n8
1 @ k 2 R 2 1 ~ 2n1 a ! 2 R 8 2
1 ~ 2n1 a !~ R 8 2 2RR 9 !# u n .
As before, we take a 51m and then
V. NONCYLINDRICAL TUBES
Once we move away from cylinders, we no longer have
exact solutions. Therefore, an approximate method will be
required. For this reason, we choose to make a simple
change of the independent variables, from (r,z) to ~r,z!, so
that the new geometry is a circular cylinder. The price of this
change is that the new partial differential equation is more
complicated.
Thus, define new variables r and z by
r 5r/R ~ z !
and z 5z,
~28!
so that the tube is mapped onto the circular tube, given by
0<r,«, 0,z,1. ~Later, we will use z in place of z, but it is
clearer to distinguish the two variables at this stage.! The
chain rule gives
]u 1 ]u
,
5
] r R ]r
] 2u
1 ] 2u
]r
R 2 ]r 2
5
2
,
24 ~ m11 ! u 1 5R 2 u 09 22mRR 8 u 08
1 @ k 2 R 2 1m ~ m11 ! R 8 2 2mRR 9 # u 0 ,
S D
1 @ k 2 R 2 1 ~ m12 ! $ ~ m13 ! R 8 2 2RR 9 % # u 1 .
Eliminating u 1 from the last equation, using Eq. ~33!, gives
- 1 g m u 09 1 d m u 08 1« m u 0 ,
32~ m11 !~ m12 ! u 2 5 a m u iv
0 1 b mu 0
~34!
where
« m ~ z ! 5k 4 R 4 12mk 2 R 2 $ ~ m11 ! R 8 2 2RR 9 % 2mR 3 R iv
]u
]
r
.
]r
]r
1m ~ m11 ! $ R 2 ~ 4R 8 R - 13R 9 2 !
2 ~ m12 ! R 8 2 @ 6RR 9 2 ~ m13 ! R 8 2 # % .
Hence, Eq. ~6! becomes
]
]u
]u
] 2u
r
1 r 3 ~ R 8 2 2RR 9 ! 1 r 2 R 2 2
]r
]r
]r
]z
S D
When Eq. ~32! is substituted in the lateral boundary condition Eq. ~30!, together with a 5m, we obtain
`
] 2u
1 @~ k r R ! 2 2m 2 # u50,
22 r 3 RR 8
]r]z
~29!
05mu 0 ~ z ! 1
]u
]u
2«RR 8 50
]r
]z
on r 5«,
0, z ,1,
~30!
and the end boundary conditions, Eq. ~5!, become
]u
R8 ]u
2r
50
]z
R ]r
1384
and
« 2n12 $ ~ 2n121m ! u n11 ~ z !
at z 51.
J. Acoust. Soc. Am., Vol. 116, No. 3, September 2004
~31!
~35!
Similarly, the end boundary conditions Eq. ~31! give
`
F
R8
G
( r 2n u n8~ z ! 2 R ~ 2n1m ! u n~ z ! 50
n50
z50
at z 50
(
n50
1 ~ 2n1m ! R 8 2 u n 2RR 8 u n8 % .
the lateral boundary condition, Eq. ~4!, becomes
~ 11« 2 R 8 2 !
b m ~ z ! 524mR 3 R 8 ,
2 ~ m11 ! R 8 @ 3RR 9 2 ~ m12 ! R 8 2 # % ,
S D S D
~ 11 r 2 R 8 2 ! r
~33!
28 ~ m12 ! u 2 5R 2 u 19 22 ~ m12 ! RR 8 u 18
d m ~ z ! 524mR $ k 2 R 2 R 8 1R 2 R -
]u R8 8
R 8 ] 2u
r
2r
5
22
2
2
R
]r]z
]r R
]z
]z
2
In particular, we find that
g m ~ z ! 52R 2 $ k 2 R 2 13m @~ m11 ! R 8 2 2RR 9 # % ,
] 2u
R8
1r
R
n50,1,2,...,.
a m ~ z ! 5R 4 ,
]u ]u
R8 ]u
5 2r
,
] z ]z
R ]r
] 2u
L n ~ z;m ! 524 ~ n11 !~ n1m11 ! u n11 ~ z ! ,
and
at z51,
at
~36!
which immediately gives
P. A. Martin: Webster’s horn equation and some generalizations
R ~ z ! u n8 ~ z ! 2R 8 ~ z !~ 2n1m ! u n ~ z ! 50
and
2. Flexural modes (mÄ1)
at z50
~37!
at z51,
for n50,1,2,..., .
Below, we shall use the following shorthand notation:
S1 ~ z ! 5R 8 /R,
S2 5R 9 /R,
S3 5R - /R,
S4 5R iv/R.
~38!
A. First approximation
If we retain only the terms up to « 2 in Eq. ~35!, we find
that
05mu 0 « 22 1 ~ m12 ! u 1 1mR 8 2 u 0 2RR 8 u 80 .
Upon using Eq. ~33!, this gives
u 90 ~ z ! 1D ~m1 ! ~ z ! u 80 ~ z ! 1E ~m1 ! ~ z ! u 0 ~ z ! 50,
~39!
where
When m51, Eq. ~39! reduces to
F
subject to Ru 08 2R 8 u 0 50 at z50 and at z51.
B. Second approximation
If we retain the terms up to « 4 in Eq. ~35!, we obtain an
equation containing u 0 , u 80 , u 1 , u 18 , and u 2 . If we eliminate
u 1 , u 81 , and u 2 , using Eqs. ~33! and ~34!, we obtain a fourthorder equation for u 0 (z),
~2!
~2!
~2!
~2!
u iv
0 1B m ~ z ! u 0- 1C m ~ z ! u 09 1D m ~ z ! u 08 1E m ~ z ! u 0 50,
~43!
where
B ~m2 ! 54S1 ~ 422m2m 2 ! / ~ m14 ! ,
D ~m1 ! ~ z ! 52S1 ~ 22m 2 ! / ~ m12 ! ,
E ~m1 ! ~ z ! 5E~m1 ! ~ k,«R ! 1
m ~ m11 !~ m22 ! 2
S1 2mS2 ,
m12
(1)
are defined by Eq. ~16!. From Eq. ~37!, Eq. ~39! is to
and Em
be solved subject to Ru 80 2mR 8 u 0 50 at z50 and at z51.
Equation ~39! can be transformed so as to eliminate the
first-derivative term. Thus, put
u 0 ~ z ! 5R g U 0 ~ z !
with
g 5 ~ m 2 22 ! / ~ m12 ! .
C ~m2 ! 5Cm ~ k,«R ! 26m $ S2 1 ~ 42m2m 2 ! S21 ~ m14 ! % ,
~44!
4mS1
~2!
2
D m 5Dm ~ k,«R ! 24mS3 1
$ ~ m11 !~ 42m ! S21
m14
13m ~ m13 ! S2 % ,
E ~m2 ! 5E~m2 ! ~ k,«R ! 2mS4 13m ~ m11 ! S22
~40!
22mk 2
Then, we find that U 0 (z) solves
U 90 ~ z ! 1 @ k 2 2K ~ z !# U 0 ~ z ! 50,
~41!
H
J
m
2m
2 ~ m11 !
S21 1S2 1 2 2 ,
m12
m12
« R
subject to (m12)U 80 22(m11)S1 U 0 50 at z50 and at z
51. Notice that Eq. ~41! has oscillatory solutions when k 2
.K but exponential solutions when k 2 ,K.
Equation ~39! and its associated boundary conditions
can also be written as a regular Sturm-Liouville problem.
Thus,
~ p ~ z ! u 08 ! 8 1 @ q ~ z ! 1lw ~ z !# u 0 ~ z ! 50,
q ~ z ! 5mR 22 g
H F
m ~ m11 !~ m12 ! 2
~ m 2m24 ! S41
m14
(2)
Em
8m ~ m12 !
2
~ m14 ! « R
When m50, Eq. ~39! reduces to
~ Au 08 ! 8 1k Au 0 50,
Dm ~ k,« ! 5
H
4S1
~ 422m2m 2 ! k 2
m14
4
«2
J
~ m12 !~ m 2 22 ! .
Ru 08 2mR 8 u 0 50
~42!
where A(z)5 p a 2 @ R(z) # 2 is the area of the ~circular! crosssection at z; Eq. ~42! is recognized as Webster’s horn equation Eq. ~1!. The appropriate boundary conditions are
u 08 (0)5u 08 (1)50.
J. Acoust. Soc. Am., Vol. 116, No. 3, September 2004
$ ~ m11 !~ m22 ! S21 2 ~ m12 ! S2 % ,
~45!
Note that Eq. ~43! reduces to Eq. ~18! when R(z)[1.
Equation ~43! is to be supplemented with boundary conditions. From Eq. ~37!, these are
1. Axisymmetric modes (mÄ0)
2
2
are defined by Eqs. ~19! and ~20!, respectively,
1
G J
J
1
where p5w5R 22 g , l5k 2 , g is defined by Eq. ~40!, and
4
m11
~ m22 ! S21 2 2 2 2S2 .
m12
« R
42m2m 2 2
4m 2 ~ m13 !
S1 1S2 1
S1 S3
m14
m14
6m 2 ~ m11 !~ m12 ! 2
S1 S2
m14
2
Cm and
and
H
2
where
K~ z !5
G
2
8
2
u 90 ~ z ! 1 S1 u 08 ~ z ! 1 k 2 2 S21 2S2 2 2 2 u 0 ~ z ! 50,
3
3
3« R
and Ru 18 2 ~ m12 ! R 8 u 1 50
~46!
at z50 and at z51. Eliminating u 1 from the second of Eq.
~46!, using Eq. ~33!, and then using the first of Eq. ~46!, we
obtain
u 0- 23mS1 u 09 1 $ 2 ~ m 2 21 ! S21 13S2 % u 08 2mS3 u 0 50;
~47!
P. A. Martin: Webster’s horn equation and some generalizations
1385
both this condition and u 08 5mS1 u 0 are to be imposed at z
50 and at z51. Notice that the boundary conditions do not
involve k 2 .
A. Wavelength comparable to L
For axisymmetric motion, we obtained the fourth-order
Eq. ~48!, which we write here as
2
~ R 2 u 08 ! 8 1k 2 R 2 u 0 5 81 « 2 R 3 ~ Ru iv
0 14R 8 u 0- 12k Ru 09
1. Axisymmetric modes (mÄ0)
14k 2 R 8 u 08 1k 4 Ru 0 ! .
When m50, Eq. ~43! reduces to
u iv
01
S
D
S
In this equation, k and u 0 (z) are unknown; they are to be
determined subject to the four boundary conditions, Eq. ~21!.
Let us look for solutions of the form
D
4R 8 2
4
4
4R 8
u - 12 k 2 2 2 2 u 09 1
k 2 2 2 u 08
R 0
R
« R
« R
S
1 k 42
8k 2
2
« R
2
D
~48!
u 0 50.
From Eq. ~47!, we see that Eq. ~48! is to be solved subject to
Eq. ~21!. We observe that Eq. ~48! involves R and R 8 , but
not higher derivatives of R. Also, Eq. ~48! can be derived
directly by applying Eq. ~4! to Eq. ~26!.
2. Flexural modes (mÄ1)
72
12
C ~12 ! ~ z ! 52k 2 2 S21 26S2 2 ~ «R ! 22 ,
5
5
4
D ~12 ! ~ z ! 5 S1 $ k 2 16S21 112S2 212~ «R ! 22 % 24S3 ,
5
2
E ~12 ! ~ z ! 5k 4 2S4 16S22 2 k 2 $ 2S21 15S2 136~ «R ! 22 %
5
2
1 S1 $ 8S3 218S1 S2 212S31 %
5
1
24
5« 2 R 2
H
2S21 13S2 1
8
« 2R 2
J
.
and u 0- 23S1 u 09 13S2 u 80 2S3 u 0 50
at z50 and at z51.
where k 0 and k 2 are to be found. Thus, we put u 0 (z)
5 v 0 (z)1« 2 v 1 (z)1¯ and Eq. ~50! in Eq. ~49!. This is a
classic singular perturbation, because Eq. ~49! reduces to a
second-order equation when «50, implying that the boundary conditions will have to be modified. Indeed, writing out
the first two terms of the exact boundary condition, Eq. ~36!,
we have
5u 08 ~ z ! 2 41 r 2 R 2 ~ u 09 1k 2 u 0 ! 8 1¯,
at each end of the tube, where we have used Eq. ~33!. We
will not be able to satisfy this condition for all allowable r
with 0<r,«, and so we integrate over each circular end of
the tube, and impose
05u 08 ~ z ! 2 81 « 2 R 2 ~ u 90 1k 2 u 0 ! 8 1¯
and
at z50
~51!
at z51;
~ R 2 v 08 ! 8 1k 20 R 2 v 0 50
VI. COMPARISON WITH ASYMPTOTIC
APPROXIMATIONS
A number of formal asymptotic theories have been developed for waves in slender tubes. In the papers by Ting and
Miksis ~1983! and by Geer and Keller ~1983!, the tubes need
not have circular cross-sections and they need not be
straight. Here, we shall compare the results obtained from
our one-dimensional theory with those obtained by specialising the analysis in ~Ting and Miksis, 1983! and ~Geer and
Keller, 1983! to our axisymmetric geometries.
Two asymptotic regimes are of interest to us. In one, k
5O(1) as «→0, so that the wavelength is comparable to L,
the length of the tube. All solutions of this kind are axisymmetric. In a second regime, k5O(« 21 ) as «→0, so that the
wavelength is comparable to a!L. Both kinds of solution
are seen in the exact solutions for circular tubes ~Sec. III!.
J. Acoust. Soc. Am., Vol. 116, No. 3, September 2004
for 0,z,1,
with
v 08 ~ 0 ! 5 v 08 ~ 1 ! 50.
~52!
This is Webster’s horn equation again, written as a regular
Sturm-Liouville problem: it is an eigenvalue problem for k 20
and v 0 (z); we normalize the solution using
1
E$
0
1386
~50!
this ensures that Eq. ~36! is satisfied in an average sense.
The terms in « 0 from Eqs. ~49! and ~51! give
Then, the differential equation Eq. ~43! ~with m51) is to be
solved subject to
u 80 2S1 u 0 50
k 2 5k 20 1« 2 k 22 1¯,
05u 08 ~ z ! 1 r 2 @ u 81 22 ~ R 8 /R ! u 1 # 1¯
When m51, we obtain
4
B ~12 ! ~ z ! 5 S1 ,
5
~49!
v 0 ~ z ! R ~ z ! % 2 dz51.
~53!
The terms in « 2 give
~ R 2 v 18 ! 8 1k 20 R 2 v 1 52k 22 R 2 v 0 1V,
~54!
where
- 12k 20 R v 09 14k 20 R 8 v 08 1k 40 R v 0 !
V5 18 R 3 ~ R v iv
0 14R 8 v 0
5 41 R 2 ~ R 8 R 9 2RR - !v 08 2 21 R 3 R 9 v 90 ,
~55!
after using Eq. ~52!1 . Similarly, Eq. ~51! gives
v 81 ~ z ! 52 41 RR 8 v 90 52 14 ~ RR 8 v 08 ! 8
and
at z51.
at z50
~56!
Thus v 1 solves a forced version of Webster’s horn equation
with inhomogeneous boundary conditions. As the homogeneous form of Eq. ~54! admits nontrivial solutions, Eq. ~54!
P. A. Martin: Webster’s horn equation and some generalizations
will only have solutions if a consistency condition is satisfied. Thus, we multiply Eq. ~54! by v 0 (z) and Eq. ~52!1 by
v 1 (z), subtract the two, and integrate over 0<z<1. This
gives
F
1
2 R 2 v 0 ~ RR 8 v 08 ! 8
4
G
1
52k 22 1
z50
E
1
4
E
1
0
R 2 ~ RR 8 v 08 ! 8 v 08 dz1
E
0
u 09 /E52« 22 F 2 w 0 1« 21 ~ 2iFw 08 1iF 8 w 0 2F 2 w 1 !
1O ~ 1 ! ,
u 0- /E52« 23 iF 3 w 0 1O ~ « 22 ! ,
V v 0 dz,
1
0
u 80 /E5« 21 iFw 0 1O ~ 1 ! ,
1
where we have used Eqs. ~52!2 , ~53!, and ~56!. In order to
cancel
the
left-hand
side,
write
V5 $ V
11/4@ R 2 (RR 8 v 08 ) 8 # 8 % 21/4@ R 2 (RR 8 v 08 ) 8 # 8 . Then, an integration by parts gives
k 22 5
u 0 /E5w 0 1«w 1 1O ~ « 2 ! ,
~57!
W v 0 dz,
24 4
F w 0 1« 23 F 2 $ F 2 w 1 22i ~ 3F 8 w 0 12Fw 80 ! %
u iv
0 /E5«
1O ~ « 22 ! ,
as «→0.
Let us begin with the second-order equation, Eq. ~39!.
(1)
(1)
(1)
5O(1) and E m
5« 22 Em
( k ,R)1O(1) as
We note that D m
«→0. Then, the terms in « 22 give
where
@ F ~ z !# 2 5E~m1 ! ~ k ,R ! .
2
1
4
W ~ z ! 5V1 @ R ~ RR 8 v 80 ! 8 # 8
The terms in « 21 give
2Fw 80 1 $ F 8 1FD ~m1 ! % w 0 50,
5 14 R 3 R 8 v 0- 1 ~ RR 8 ! 2 v 90 1 21 ~ 3R 2 R 8 R 9 1RR 8 3 !v 80
~58!
and we have used Eq. ~55!. Integrating the first integral in
Eq. ~57! by parts, using Eq. ~52!2 , then gives
k 22 52
1
2
E
1
0
which is a first-order differential equation for w 0 . Rearranging gives
~ w 0 F 1/2! 8
w 0 F 1/2
~ RR 8 v 80 ! 2 dz1K,
where K5 * 10 (W v 0 21/4R 3 R 8 v 90 v 80 )dz. We are going to
show that K50.
Noting the first term in Eq. ~58!, and making use of Eq.
~52!1 , we have
2
21
R 8 v 80 ! 8 #v 0
v0 52 @ k 0 v 08 1 ~ 2R
0v8
0 v 02 v 9
1 ~ k 20 v 0 12R 21 R 8 v 80 !v 08
52R
21
2
R 8 v 80 22R
22
1
0
w0
1
D ~m1 !
R8
F8
52
52q ~m1 ! ,
2F
2
R
~63!
where w 0 , F, and R are all functions of z only, and q
(1)
.
[q m
Next, consider the fourth-order equation, Eq. ~43!. Substituting as before, we find that
B ~m2 ! 5O ~ 1 ! ,
@~ RR 9 2R 8 !v 80
~ RR 8 v 08 ! 2 dz1O ~ « 4 !
w 08
w 20 FR 2q 5constant,
2
C ~m2 ! 5« 22 Cm ~ k ,R ! 1O ~ 1 ! ,
D ~m2 ! 5« 22 Dm ~ k ,R ! 1O ~ 1 !
Hence, K51/2* 10 $ (RR 8 v 08 ) 2 1 @ (RR 8 ) 2 v 80 # 8 v 0 % dz, and this
is seen to vanish after another integration by parts. Thus,
E
5
(1)
5(22m 2 )/(m12). An integration gives
where q m
1RR 8 v 90 #v 0 .
1
k 2 5k 20 2 « 2
2
~62!
and
E ~m2 ! 5« 24 E~m2 ! ~ k ,R ! 1O ~ « 22 !
as «→0. Then, we see that the terms in « 24 give
F 4 2F 2 Cm ~ k ,R ! 1E~m2 ! ~ k ,R ! 50,
as «→0. ~59!
~64!
which should be compared with Eq. ~22!. Thus,
This elegant formula was derived in a different way by Geer
and Keller ~1983!; see their Eq. ~102!.
2
~ k ,R ! ,
@ F ~ z !# 2 5 m m
~65!
where m m is defined by Eq. ~23!.
The terms in « 23 give
B. Wavelength comparable to a
05 $ F 4 2F 2 Cm ~ k ,R ! 1E~m2 ! ~ k ,R ! % w 1
In the differential equations, Eqs. ~39! and ~43!, put k
5 k /« and
u 0 ~ z ! 5E ~ z ! w ~ z !
with
E ~ z ! 5exp
H
i
«
E
z
z0
J
Then, we obtain the following approximations:
J. Acoust. Soc. Am., Vol. 116, No. 3, September 2004
1F 8 Cm ~ k ,R ! 2F 3 B ~m2 ! 26F 2 F 8 % .
F ~ t ! dt ,
~60!
where F(t) and w(z) are to be determined, and z 0 is a constant. Suppose further that
w ~ z ! 5w 0 ~ z ! 1«w 1 ~ z ! 1« 2 w 2 ~ z ! 1¯.
12iFw 08 $ Cm ~ k ,R ! 22F 2 % 1iw 0 $ FDm ~ k ,R !
~61!
The factor multiplying w 1 vanishes, due to Eq. ~64!, leaving
a first-order differential equation for w 0 (z)
w 08
w0
1
F 2 Dm ~ k ,R ! 1FF 8 Cm ~ k ,R ! 2F 4 B ~m2 ! 26F 3 F 8
2F 2 @ Cm ~ k ,R ! 22F 2 #
50.
Rearranging this equation gives
P. A. Martin: Webster’s horn equation and some generalizations
1387
~ w 0 F 1/2! 8
w 0 F 1/2
5
4FF 8 2Dm ~ k ,R ! 1F 2 B ~m2 !
2 @ Cm ~ k ,R ! 22F 2 #
52q ~m2 !
R8
,
R
where
q ~m2 ! 5 @ 4 ~ m12 !~ m 2 22 ! 2m ~ m13 ! n m # / @~ m14 ! x m # .
Here, we have used F 2 5 k 2 2 n m /R 2 , and Eqs. ~19!, ~23!,
~44!, and ~45!. Hence, an integration gives Eq. ~63! again,
(2)
.
but with q[q m
Let us compare our results ~obtained by solving ordinary
differential equations asymptotically! with those of Ting and
Miksis ~1983! and Geer and Keller ~1983! ~obtained by solving an elliptic boundary-value problem asymptotically!.
First, we obtained approximations for F, given by Eqs. ~62!
and ~65!, whereas the exact formula is
8 /R ~ z !# 2 .
@ F ~ z !# 2 5 k 2 2 @ j m,,
~66!
The errors incurred here are exactly the same as those discussed in Sec. IV for a circular tube ~constant cross-section!:
We cannot expect to do any better for tubes of varying crosssection. Second, the analysis of Ting and Miksis ~1983! and
Geer and Keller ~1983! leads to Eq. ~63! with q51 ~for all
m!. If we take the minus sign in Eq. ~24!, we find that q (2)
0
51 and q (2)
1 51.43; it is not clear why this last number is not
closer to 1, given that the corresponding values of F are
close.
VII. CONCLUDING REMARKS
The method of Sec. V for waves in axisymmetric tubes
leads to eigenvalue problems for ordinary differential equations. The simplest ~first! approximation leads to a regular
Sturm–Liouville problem. This is convenient because efficient software is readily available for solving such problems
numerically ~Pruess and Fulton, 1993!. The next ~second!
approximation is expected to be more accurate, and, indeed,
we have shown this in some asymptotic regimes. The second
approximation leads to an eigenvalue problem for a fourthorder differential equation. However, it does not fall into the
class of regular fourth-order Sturm-Liouville problems; see,
for example, the review by Greenberg and Marletta ~2000!.
In fact, we can say little about the theoretical properties of
the simplest equation, namely Eq. ~48!, which models axisymmetric motions.
Evidently, higher-order approximations could be developed, leading to ordinary differential equations of order 2n
1388
J. Acoust. Soc. Am., Vol. 116, No. 3, September 2004
with n53,4,...; the necessary derivations would be expedited
using software for symbolic manipulations. The method of
Sec. V could also be extended to other cross-sections, using
a scaling r that depends on the angle u as well as on the
longitudinal coordinate z.
Another possibility is to abandon power series in favor
of Neumann series, which are series of Bessel functions of
various orders. This would permit better representation of
u(r,z) for fixed z, but at the expense of additional complication.
Finally, the basic power-series method can be extended
to various elastodynamic problems, generalizing the work of
Boström ~2000! on rods and of Boström, Johansson, and
Olsson ~2001! on plates; for axisymmetric motions in nonuniform anisotropic rods, see ~Martin, 2004!. Indeed, the fact
that the power-series method is relatively insensitive to complications in the governing partial differential equations
means that it may be worth developing further.
Boström, A. ~2000!. ‘‘On wave equations for elastic rods,’’ Z. Angew. Math.
Mech. 80, 245–251.
Boström, A., Johansson, G., and Olsson, P. ~2001!. ‘‘On the rational derivation of a hierarchy of dynamic equations for a homogeneous, isotropic,
elastic plate,’’ Int. J. Solids Struct. 38, 2487–2501.
Eisner, E. ~1967!. ‘‘Complete solutions of the ‘Webster’ horn equation,’’ J.
Acoust. Soc. Am. 41, 1126 –1146.
Geer, J. F., and Keller, J. B. ~1983!. ‘‘Eigenvalues of slender cavities and
waves in slender tubes,’’ J. Acoust. Soc. Am. 74, 1895–1904.
Greenberg, L., and Marletta, M. ~2000!. ‘‘Numerical methods for higher
order Sturm–Liouville problems,’’ J. Comput. Appl. Math. 125, 367–383.
Hélie, T. ~2003!. ‘‘Unidimensional models of acoustic propagation in axisymmetric wave-guides,’’ J. Acoust. Soc. Am. 114, 2633–2647.
Kinsler, L. E., Frey, A. R., Coppens, A. B., and Sanders, J. V. ~1982!.
Fundamentals of Acoustics, 3rd ed. ~Wiley, New York!.
Kumar, B. M., and Sujith, R. I. ~1997!. ‘‘Exact solutions for the longitudinal
vibration of non-uniform rods,’’ J. Sound Vib. 207, 721–729.
Lamb, H. ~1960!. The Dynamical Theory of Sound, 2nd ed. ~Dover, New
York!.
Martin, P. A. ~2004!. ‘‘Waves in wood: axisymmetric waves in slender solids
of revolution,’’ Wave Motion ~to be published!.
Pierce, A. D. ~1989!. Acoustics ~Acoustical Society of America, New York!.
Pruess, S., and Fulton, C. T. ~1993!. ‘‘Mathematical software for SturmLiouville problems,’’ ACM Trans. Math. Softw. 19, 360–376.
Rayleigh, Lord ~1916!. ‘‘On the propagation of sound in narrow tubes of
variable section,’’ Philos. Mag., Series 6 31, 89–96.
Rayleigh, Lord ~1945!. The Theory of Sound ~Dover, New York!.
Reinstra, S. W. ~2002!. ‘‘The Webster equation revisited,’’ paper AIAA
2002-2520, 8th AIAA/CEAS Aeroacoustics Conference, Breckenridge,
Colorado, June 2002.
Ting, L., and Miksis, M. J. ~1983!. ‘‘Wave propagation through a slender
curved tube,’’ J. Acoust. Soc. Am. 74, 631– 639.
Webster, A. G. ~1919!. ‘‘Acoustical impedance, and the theory of horns and
of the phonograph,’’ Proc. Natl. Acad. Sci. U.S.A. 5, 275–282.
P. A. Martin: Webster’s horn equation and some generalizations