530
E. Kamenetskii
where
E n =
gμ 0
4
(β n )
2
.
(19.6)
Here ∇
2
⊥ is the two-dimensional (on the circular cross section of a ferrite-disk
region) Laplace operator, g is a dimensional normalization coefficient (with the
unit of dimension ψ
2 ) for mode n and β n is the propagation constant of mode n
along the disk axis z. The parameter μ n (which is a diagonal component of the
permeability tensor [18]) should be considered as an eigenvalue. Outside a ferrite
μ n = 1. The operator ˆ
G ⊥ is a self-adjoint operator only for negative quantities μ n in a
ferrite. For self-adjointness of operator ˆ
G ⊥ , the membrane function ˜
η n (r, θ) must be
continuous and differentiable with respect to the normal to lateral surface of a ferrite
disk. The homogeneous boundary conditions—the ND boundary conditions—for
the membrane function are:
( ˜
η n ) r =R − − ( ˜
η n ) r =R + = 0
(19.7)
and
μ
∂ ˜
η n
∂r
r =R −
−
∂ ˜
η n
∂r
r =R +
= 0,
(19.8)
where R is the disk radius. MDM oscillations in a ferrite disk are described by real
eigenfunctions: ˜
η −n = ˜
η
∗
n . For modes n and n
, the orthogonality conditions are
expressed as
S c
˜
η n ˜
η
∗
n d S = δ nn ,
(19.9)
where S c is a square of a circular cross section of a ferrite-disk region and δ nn is the
Kronecker delta. The spectral problem gives the energy orthogonality relation for
MDMs:
(E n − E n )
S c
˜
η n ˜
η
∗
n d S = 0.
(19.10)
Since the space of square integrable functions is a Hilbert space with a well-defined
scalar product, we can introduce a basis set. A dimensional amplitude coefficient we
write as A n = c
a n , where c
is a dimensional unit coefficient and a n is a normalized
dimensionless amplitude. The normalized scalar-wave membrane function ˜
η can be
represented as ˜
η =
n
a n ˜
η n . The amplitude is defined as |a n |
2
=
S c
˜
η ˜
η
∗
n d S
2
. The
E. Kamenetskii
where
E n =
gμ 0
4
(β n )
2
.
(19.6)
Here ∇
2
⊥ is the two-dimensional (on the circular cross section of a ferrite-disk
region) Laplace operator, g is a dimensional normalization coefficient (with the
unit of dimension ψ
2 ) for mode n and β n is the propagation constant of mode n
along the disk axis z. The parameter μ n (which is a diagonal component of the
permeability tensor [18]) should be considered as an eigenvalue. Outside a ferrite
μ n = 1. The operator ˆ
G ⊥ is a self-adjoint operator only for negative quantities μ n in a
ferrite. For self-adjointness of operator ˆ
G ⊥ , the membrane function ˜
η n (r, θ) must be
continuous and differentiable with respect to the normal to lateral surface of a ferrite
disk. The homogeneous boundary conditions—the ND boundary conditions—for
the membrane function are:
( ˜
η n ) r =R − − ( ˜
η n ) r =R + = 0
(19.7)
and
μ
∂ ˜
η n
∂r
r =R −
−
∂ ˜
η n
∂r
r =R +
= 0,
(19.8)
where R is the disk radius. MDM oscillations in a ferrite disk are described by real
eigenfunctions: ˜
η −n = ˜
η
∗
n . For modes n and n
, the orthogonality conditions are
expressed as
S c
˜
η n ˜
η
∗
n d S = δ nn ,
(19.9)
where S c is a square of a circular cross section of a ferrite-disk region and δ nn is the
Kronecker delta. The spectral problem gives the energy orthogonality relation for
MDMs:
(E n − E n )
S c
˜
η n ˜
η
∗
n d S = 0.
(19.10)
Since the space of square integrable functions is a Hilbert space with a well-defined
scalar product, we can introduce a basis set. A dimensional amplitude coefficient we
write as A n = c
a n , where c
is a dimensional unit coefficient and a n is a normalized
dimensionless amplitude. The normalized scalar-wave membrane function ˜
η can be
represented as ˜
η =
n
a n ˜
η n . The amplitude is defined as |a n |
2
=
S c
˜
η ˜
η
∗
n d S
2
. The