Infinite Interior Boundary Value Problem: Waves Propagating in a Cylindrical Pipe

Let us compare waves vibrating in a finite cylindrical cavity ( $0\le z\le L$ ) with waves propagating in an infinite cylinder ( $-\infty <z<\infty$ ).

The wave equation is the same in both cases,

$$\nabla^2\psi -\frac{1}{c^2}~\frac{\partial^2\psi}{\partial t^2} = 0\,.$$ (532)

The boundary conditions along the radial and angular direction (``transverse direction'') are also the same in both cases:

$$\psi (r=0,\theta ,z,t) = \textrm{finite} ~~$$

$$\psi (r=a,\theta ,z,t) = 0 ~\qquad\qquad\qquad\qquad\textrm{(Dirichlet~b.c.)}$$

$$\psi (r,\theta ,z,t) = \psi (r,\theta +2\pi ,z,t) \qquad\textrm{(Periodic~b.c.)}~.$$

These are the familiar two sets of boundary conditions for the two Sturm-Liouville problems on the radial and the angular domain.What do their solutions tell us?

I.) Their eigenfunctions yield the amplitude profile across any transverse cross section ($z=const.$ ) at any time ($t=const.$ ). These cross sectional profiles are determined by the two sets of eigenvalues,

$$m=0,\pm 1,,\cdots ~~\textrm{and}~~k_{mj}: J_m(k_{mj}a)=0; j=1,2,\cdots$$

II.) By virtue of the wave equation (5.32) each of these transverse eigensolutions determines the properties of a wave disturbance

$$\psi=\psi_{mj}(t,z)J_m(k_{mj}r)e^{im\theta}~.$$

propagating along the longitudinal direction. The wave equation tells us that these properties are captured by

$$\frac{1}{c^2}~\frac{\partial^2\psi}{\partial t^2}-\frac{\partial^2\psi}{\partial z^2} +k^2_{mj}\psi =0~.$$ (533)

The mathematical behavior of its solutions is the same as that of a string imbedded in an elastic medium as discussed on page . If one happens to be familiar with its physical properties, one can infer the mathematical properties of wave propagation along the $z$ -direction.

The problem of waves trapped in a cavity is similar to that of waves propagating along a pipe: both are most efficiently attacked in terms of normal modes, which satisfy

$$\frac{\partial \psi}{\partial t}=-i\omega \psi~~~\rightarrow \psi\propto ~~ e^{-i\omega t}~,$$

However, the difference in the boundary conditions on the $z$ -domain demands a different point of view in regard to what is given and what is to be determined. First of all, instead of Dirichlet boundary conditions, the new condition is that for a given frequency $\omega$ the normal modes express waves travelling along the $z$ -direction. This implies that a normal mode satisfies

$$\frac{\partial\psi}{\partial t}\mp i\frac{\omega}{k_z}~\frac{\partial\psi} {\partial z} = 0$$

so that

$$\psi\propto R_{mj}(r)\Theta_m(z) e^{\pm ik_zz}e^{-i\omega t}\,.$$

Second, the fact that this mode satisfies the wave equation,

$$\frac{1}{c^2}~\frac{\partial^2\psi}{\partial t^2} +(k^2_{mj}+k^2_z)\psi =0~,$$

implies that

$$-\frac{\omega^2}{c^2} +k^2_{mj}+k^2_z = 0$$

or

$$k^2_z = \frac{\omega^2}{c^2} - k^2_{mj}\,.$$

Third, and finally, our viewpoint is now necessarily different. Instead of $k_z$ being determined by an eigenvalue problem, we now take $\omega$ to be the given frequency of the wave $\psi$ to be launched into the positive $(+)$ or negative $(-)$ $z$ -direction and ask: for what values of $\omega$ will $k_z$ be real so that $\psi$ expresses a travelling wave

$$\psi\propto R_{mj}(r)\Theta_m(\theta )e^{\pm i\vert k_z\vert z} e^{-i \omega t}$$

and for what values of $\omega$ will $k_z$ be imaginary so that $\psi$ expresses a spatially damped (or antidamped) wave

$$\psi\propto R_{mj}(r)\Theta_m(\theta ) e^{\pm \vert k_z\vert z}e^{-i \omega t}\,.$$

It is evident, that the answer is to be inferred from the dispersion relation

$$k_z = \pm\left(\frac{\omega^2}{c^2} -k^2_{mj}\right)^{1/2}\,.$$

This relation between $k_z$ and $\omega$ depends on the eigenvalues $k_{mj}$ for the amplitude profile in the transverse plane. A wave which decays exponentially along its putative direction of propagation is called an evanescent wave. This happens when the frequency of the launched wave is low enough. It is evident that there is a critical frequency

$$\omega_{critical}= c~k_{mj}$$

at which the wave changes from being a propagating to being an evanescent wave. The eigenvalues $k_{mj}$ are, of course, determined by the given Dirichlet boundary condition. For a hollow cylinder these eigenvalues are the roots of the equation

$$J_m(ka)=0~~.$$

This implies that the smaller the radius of the cylindrical pipe the higher the critical frequency below which no wave can propagate. A wave which meets such a small radius pipe gets simply reflected.

Exercise 53.1 (AXIALLY SYMMETRIC AMPLITUDES)
The transverse amplitude of an axially symmetric wave propagating in a cylindrical pipe of radius $a$ is determined by the following eigenvalue problem:

$$-{d\over{dr}}~r~{du\over{dr}} = k^2ru~~\qquad~~\qquad~~0\le r\le a$$

$$u(0) = {\rm finite}$$

$$u(a) = 0.$$

The eigenfunctions are $u_m(r)=J_0(rk_m)$ where the boundary condition $J_0(ak_m)=0$ determines the eigenvalues $k^2_m\quad m=1,2,\dots$ .

(a)

Show that $\{J_0(rk_m)\}$ is an orthogonal set of eigenfunctions on $(0,a)$ .

(b)

Using the problem ``How to normalize an eigenfunction'' on page , find the squared norm of $J_0(rk_m)$ .

(c)

Exhibit the set of orthonormalized eigenfunctions.

(d)

FIND the Green's function for the above boundary value problem.

Exercise 53.2 (NORMAL MODES FOR A VIBRATING DRUM)
On a circular disc of radius $a$ FIND an orthonormal set of eigenfunctions for the system defined by the eigenvalue problem

$$-\nabla^2\psi = k^2\psi ~~$$

$${\partial\psi\over{\partial r}}(r=a,\theta) = 0~~\qquad~~\qquad~~ \textrm{a = radius~of~disc}$$

$$\psi(r=0,\theta) = \textrm{finite} \qquad\qquad 0\le\theta\le 2\pi~~.$$

Here $\nabla^2={1\over{r}}{\partial\over{\partial r}}r {\partial\over{\partial r}}+{1\over{r^2}}{\partial^2\over{\partial\theta^2}}$ ,
and EXIBIT these eigenfunctions in their optimally simple form, i.e. without referring to any derivatives.

Exercise 53.3 (CRITICAL FREQUENCIES FOR WAVE PROPAGATION)
Consider a wave disturbance $\psi$ which is governed by the wave equation

$$\left[{\partial^2\over{\partial r^2}}+{1\over{r}}{\partial\over{\... ...r{\partial z^2}}\right]\psi={1\over{c^2}} {\partial^2\psi\over{\partial t^2}}.$$

Let this wave propagate inside an infinitely long cylinder; in other words, it satisfied

$$\displaystyle{\partial\psi\over{\partial z}}=ik_z\psi$$

where $k_z$ is some real number, not equal to zero. Assume that the boundary conditions satisfied by $\psi$ is

$$\psi(r=a) = 0\quad {\rm with}~a =~~{\rm radius~of~cylinder}$$

$$\psi(r=0) =~~{\rm finite}$$

(a)Find the ``cut off'' frequency, i.e. that frequency $\omega=\omega_{\rm critical}$ below which no propagation in the infinite cylinder is possible.

(b)Note that this frequency depends on the angular integer $m$ and the radial integer $j$ . For fixed $j$ , give an argument which supports the result that smaller $m$ means smaller critical frequency.

(c)What is the smallest critical frequency, $\omega_{\rm critical}$ , in terms of $a$ and $c$ to an accuracy of $2\%$ or better?

Exercise 53.4 (PIE-SHAPED DRUM)
Consider the circular sector

$$S~\colon~0 \le r\le a$$

$$\le \theta\le\alpha$$ 0

(a)

Exhibit the set of those normalized eigenfunctions for this sector which satisfy

$$(\nabla^2+k^2)\psi =$$ 0

$$\psi = 0~~{\rm on~the~boundary~of}~~S$$

(b)

Compare the set of normal modes of a circular drum with the set of normal modes in Part A when $\alpha=2\pi$

Exercise 53.5 (VIBRATING MEMBRANES)
Consider

(a)

a circular membrane of radius $\underline a$

(b)

a square membrane

(c)

a rectangular membrane which is twice as long as it is wide.

Assume the two membranes

(i)

have the same area.

(ii)

obey the same wave equation $$\bigtriangledown^2 \psi={1\over{c^2}} {\partial^2\psi\over{\partial t^2}}$$

(iii)

Have the same boundary conditions $\psi=0$ at their boundaries

(A)

TABULATE

(i)

the $3$ lowest frequencies for each of the two membranes

(ii)

all the concomitant normal modes.

(B)

For each of the normal modes of the circular membrane DRAW a picture of the nodes,
i.e. the locus of points where $\psi=0$ . LABEL each of the normal mode pictures.

(C)

Do the same for the other membrane. (Caution: Watch out for degeneracies!)

Roots. $\lambda_j$ is the $j$ th root of the Bessel Functions $J_m(\lambda_j)$ :

=5truein    #    &&   #    &&&&&    &$m=0$ & $m=1$ & $m=2$ & $m=3$ & $m=4$ &&&&& 6 $j=1$ & 2.405 & 3.832 & 5.136 & 6.380 & 7.586 6 $j=2$ & 5.520 & 7.016 & 8.417 & 9.760 &11.064 6 $j=3$ & 8.654 & 10.173 & 11.620 &13.017 &14.373 6 $j=4$ & 11.792 & 13.324 & 14.796 &16.223 &16.223 6

Lecture 45