We consider small departures from an unperturbed right-cylindrical shape of an incompressible fluid. Therefore a normal mode can be expressed uniquely in terms of the deformed surface. Hence we may assume that the deformed interface is described by

$r={R}_{0}+\epsilon (t){R}_{1}+\cdots $

(28)

with

${R}_{1}=exp(i(kz+m\phi )).$

(29)

Here

${R}_{1}$ is the elevation of the surface wave measured from the unperturbed position,

*k* (real number) is the longitudinal wave number,

*m* (integer) is the transverse wave number. The amplitude

$\epsilon (t)$ of the perturbation is given by

$\epsilon (t)={\epsilon}_{0}exp(\sigma t),$

(30)

where ${\epsilon}_{0}$ (=*ε* at $t=0$) is the initial amplitude and *σ* is the temporal amplification. If *σ* ($=i\omega $, $i=\sqrt{-1}$) is imaginary, then $\omega /2\pi $ is the oscillation frequency of the propagating wave in the fluid.

As the initial streaming state is perturbed, every physical quantity

$Q(r,\phi ,z;t)$ may be expanded as

$Q(r,\phi ,z,t)={Q}_{0}(r)+{Q}_{1}(r,\phi ,z,t).$

(31)

Here *Q* stands for *P*, *u*, *V*, ${V}^{ex}$, *H*, ${H}^{ex}$, and ${N}_{s}$ while ${Q}_{0}$ indicates the unperturbed quantity and ${Q}_{1}$ is a small increment of *Q* due to disturbances.

In view of the expansion (31), the basic equations of motion (3)-(13) in the perturbation state give

where equations (

33) and (

34) have been used to obtain equation (

35). Based on the linear perturbation technique, the linearized quantity

${Q}_{1}(r,\phi ,z;t)$ may be expressed as

${Q}_{1}(r,\phi ,z;t)={\epsilon}_{0}{q}_{1}(r)exp(\sigma t+i(kz+m\phi )).$

(41)

By means of the expansion (41), equations (

36) and (

40) give the second-order ordinary differential equation

$\left(\frac{1}{r}\right)\left(\frac{d}{dr}\right)\left(r\left(\frac{d{\varphi}_{1}(r)}{dr}\right)\right)-(\left(\frac{{m}^{2}}{{r}^{2}}\right)+{k}^{2}){\varphi}_{1}(r)=0,$

(42)

where

${\varphi}_{1}(r)$ stands for

${V}_{1}(r)$ and

${V}_{1}^{ex}(r)$. The solution of equation (

42) is given in terms of the ordinary Bessel functions of imaginary argument. For the problem under consideration, apart from the singular solution, the solutions of equations (

36) and (

40) are finally given by

Here ${I}_{m}(kr)$ and ${K}_{m}(kr)$ are the modified Bessel functions of the first and second kind of order *m*, while *A* and *B* are constants of integration to be determined.

Using the space-time dependence (41) for equation (

32), we get

$(\sigma +imW+ikU)\underline{{u}_{1}}-(i\mu k/4\pi \rho ){H}_{0}\underline{{H}_{1}}=-\mathrm{\nabla}{\mathrm{\Pi}}_{1}$

(45)

with

${\mathrm{\Pi}}_{1}=\left(\frac{{P}_{1}}{\rho}\right)-{V}_{1}+(\mu /4\pi \rho )({\underline{H}}_{0}\cdot {\underline{H}}_{1}).$

(46)

Also, equation (

35) yields

${\underline{H}}_{1}=\left(\frac{ik{H}_{0}}{\sigma +imW+ikU}\right)\underline{{u}_{1}}.$

(47)

By combining equations (

45) and (

47), we get

$\underline{{u}_{1}}=\left(\frac{-(\sigma +imW+ikU)}{({(\sigma +imW+ikU)}^{2}+{\mathrm{\Omega}}_{A}^{2})}\right)\mathrm{\nabla}{\mathrm{\Pi}}_{1},$

(48)

where

${\mathrm{\Omega}}_{A}={\left(\frac{\mu {k}^{2}{H}_{0}^{2}}{4\pi \rho}\right)}^{1/2}$

(49)

is the Alfven wave frequency defined in terms of ${H}_{0}$.

By taking the divergence of both sides of equation (

48) and using equation (

33), we obtain

${\mathrm{\nabla}}^{2}{\mathrm{\Pi}}_{1}=0.$

(50)

Using the space dependence (41) for equation (

50) and following similar steps for the resulting differential equation as has already been done for equations (

36) and (

40), the solution of equation (

50) could be obtained. Therefore, the non-singular solution for

${\mathrm{\Pi}}_{1}(r,\phi ,z;t)$ is given by

${\mathrm{\Pi}}_{1}={C}_{4}{\epsilon}_{0}{I}_{m}(kr)exp(\sigma t+i(kz+m\phi )),$

(51)

where ${C}_{4}$ is a constant of integration to be determined.

The pressure surface

${P}_{1s}$ in the perturbed state due to the capillary force is determined from equation (

37) along with (29) in the form

${P}_{1s}=\left(\frac{-T}{{R}_{0}^{2}}\right)(1-{m}^{2}-{x}^{2})exp(\sigma t+i(kz+m\phi )),$

(52)

where *x* ($=k{R}_{0}$) is the dimensionless longitudinal wavenumber.

Now, equation (

34) means that the magnetic field intensity

${H}_{1}^{ex}$ in the perturbed state may be derived from a scalar function,

${\psi}_{1}^{ex}$ say, such that

${H}_{1}^{ex}=\mathrm{\nabla}{\psi}_{1}^{ex}.$

(53)

By combining equations (

38) and (

53), we get

${\mathrm{\nabla}}^{2}{\psi}_{1}^{ex}=0.$

(54)

Similarly, as it has been done for equation (

50), equation (

54) is solved and its finite solution is given by

${\psi}_{1}^{ex}={C}_{5}{\epsilon}_{0}{K}_{m}(kr)exp(\sigma t+i(kz+m\phi )),$

(55)

where ${C}_{5}$ is a constant of integration to be determined upon applying boundary conditions.