Question

A quantity \(u\) satisfies the wave equation $$ \nabla^{2} u-\frac{1}{c^{2}} \frac{\partial^{2} u}{\partial t^{2}}=0 $$ inside a hollow cylindrical pipe of radius \(a\), and \(u=0\) on the walls of the pipe. If at the end \(z=0, u=u_{0} e^{-i \omega_{01}}\), waves will be sent down the pipe with various spatial distributions (modes). Find the phase velocity of the fundamental mode as a function of the frequency \(\omega_{0}\) and interpret the result for small \(\omega_{0}\).

Short answer

The phase velocity of the fundamental mode is \(v_p = \frac{\omega}{\sqrt{ \left( \frac{\omega}{c} \right)^2 - \left( \frac{X_{01}}{a} \right)^2 }} \). For small \(\omega_0\), \(v_p \to c\).

Step-by-step solution

- Understand the Problem

The wave equation given is \(abla^{2} u-\frac{1}{c^{2}} \frac{\partial^{2} u}{\partial t^{2}}=0\). The boundary condition is \(u=0\) on the walls of the cylindrical pipe with radius \(a\), and the initial condition at \(z=0\) is \(u=u_{0} e^{-i \omega_{01}}\).

- Express the Wave Equation in Cylindrical Coordinates

In cylindrical coordinates, the wave equation is expressed as: \[abla^{2} u = \frac{\partial^{2} u}{\partial r^{2}} + \frac{1}{r}\frac{\partial u}{\partial r} + \frac{1}{r^{2}}\frac{\partial^{2} u}{\partial \theta^{2}} + \frac{\partial^{2} u}{\partial z^{2}}\].

- Separate Variables

Assume a solution of the form: \(u(r, \theta, z, t) = R(r) \Theta(\theta) Z(z) T(t)\). Substituting this into the wave equation and dividing by \(R\Theta Z T\) gives:\[\frac{1}{R} \left( \frac{d^{2} R}{d r^{2}} + \frac{1}{r} \frac{d R}{d r} \right) +\frac{1}{R r^{2}} \frac{d^{2} \Theta}{d \theta^{2}} + \frac{1}{Z} \frac{d^{2} Z}{d z^{2}} - \frac{1}{c^{2} T} \frac{d^{2} T}{d t^{2}} = 0\].

- Separate Each Term

Each term must be equal to a constant. Let's set: \(\frac{1}{R} \left( \frac{d^{2} R}{d r^{2}} + \frac{1}{r} \frac{d R}{d r} \right) = -k_{r}^{2}\), \(\frac{1}{R r^{2}} \frac{d^{2} \Theta}{d \theta^{2}} = m^{2}\), \(\frac{1}{Z} \frac{d^{2} Z}{d z^{2}} = -\beta^{2}\), and \(-\frac{1}{c^{2} T} \frac{d^{2} T}{d t^{2}} = \omega^{2}\).

- Solve for Radial Component

The radial component separates to: \[\frac{d^{2} R}{d r^{2}} + \frac{1}{r} \frac{d R}{d r} + \left(k_{r}^{2} - \frac{m^{2}}{r^{2}}\right) R = 0\]. This is a Bessel equation whose solution is \(R(r) = J_{m}(k_r r)\). Given the boundary condition \(u(r, \theta, z, t) = 0\) at \(r = a\), it follows that \(J_{m}(k_{r} a) = 0\). The smallest root corresponds to the fundamental mode.

- Interpret Results of Mode Calculation

For the fundamental mode, denote \(k_{r_{01}} = \frac{X_{01}}{a}\) where \(X_{01}\) is the first root of the Bessel function \(J_{0}\). This gives us a specific \(k_{r}\).

- Determine Wave Components

The wave number \(k\) can be expressed as a composite of \(k_{r}\) and \(k_{z}\). For the fundamental mode, \(k_{z} = \beta\), thus:\[k^2 = k_{r}^2 + k_{z}^2 = \left(\frac{X_{01}}{a}\right)^2 + \beta^2\].

- Relate Frequency and Phase Velocity

The frequency relation for the wave equation gives: \[\beta = \frac{\omega}{c} = \sqrt{\frac{\omega^2}{c^2} - \left(\frac{X_{01}}{a}\right)^2}.\] Rearrange to find: \[\beta = \sqrt{\left( \frac{\omega}{c} \right)^2 - \left( \frac{X_{01}}{a} \right)^2}.\]

- Compute Phase Velocity

Phase velocity \(v_p\) is given by\[v_p = \frac{\omega}{\beta} = \frac{\omega}{\sqrt{ \left( \frac{\omega}{c} \right)^2 - \left( \frac{X_{01}}{a} \right)^2 }}.\]

- Interpretation for Small \(\omega_0\)

For small \(\omega_0\), the term \(\left( \frac{X_{01}}{a} \right)^2\) becomes significant. Thus the phase velocity: \[v_p \approx \frac{\omega}{\sqrt{- \left( \frac{X_{01}}{a} \right)^2 }} \overset{\omega_0 \to 0}{\to} c.\]

Key concepts

Bessel function

In the context of wave equations in cylindrical coordinates, the Bessel function plays a crucial role. The radial part of the wave equation transforms into a Bessel differential equation: \[ \frac{d^{2} R}{d r^{2}} + \frac{1}{r} \frac{d R}{d r} + \bigg(k_{r}^{2} - \frac{m^{2}}{r^{2}}\bigg) R = 0 \] The solutions to this equation are the Bessel functions of the first kind, \( J_m(k_r r) \). These functions are critical because they naturally incorporate the cylindrical geometry of the problem. The boundary condition \( u = 0 \) at \( r = a \) implies that the Bessel function must vanish at the boundary, i.e., \( J_{m}(k_{r} a) = 0 \). This determines the allowed values for the radial wavenumber \( k_{r} \), leading to a quantized spectrum of radial modes. For the fundamental mode, we typically use the smallest root, yielding \( k_{r_{01}} = \frac{X_{01}}{a} \), where \( X_{01} \) is the first root of \( J_{0} \).

boundary conditions

Boundary conditions are imperative in solving partial differential equations as they provide the necessary constraints for unique solutions. In this problem, the boundary condition is \( u = 0 \) on the walls of the cylindrical pipe. This means that the wave function must vanish at \( r = a \). This boundary condition helps in determining the radial modes via the roots of the Bessel function. Additionally, the initial condition at \( z = 0 \) is provided as \( u = u_0 e^{-i \omega_{0}t} \), implying a time-harmonic solution in the form of traveling waves down the length of the pipe. The presence of these conditions means that the mathematical form of the solution is tightly constrained, enabling us to find specific solutions for the wave function.

phase velocity

Phase velocity is defined as the rate at which a wave phase propagates in space. It's given by the ratio of the wave's angular frequency to its wavenumber. For the fundamental mode, the phase velocity \( v_p \) can be determined as: \[ v_p = \frac{ \omega } { \sqrt{ \left( \frac{ \omega } { c } \right)^2 - \left( \frac{ X_{01} } { a } \right)^2 }} \] This shows how phase velocity depends on both the angular frequency \( \omega \) and the geometry of the system through \( a \) and \( X_{01} \). For small frequencies, the term \( \left( \frac{ X_{01} } { a } \right)^2 \) becomes dominant, and the phase velocity approaches the speed of sound, \( c \). Understanding phase velocity is crucial because it explains how different wave components travel, affecting the wave's overall motion and energy distribution.

wave propagation

Wave propagation in a cylindrical pipe is fundamentally influenced by the system's geometry and the imposed boundary conditions. Waves emanate from the point of excitation at \( z = 0 \) and travel along the length of the cylinder. The solution to the wave equation suggests that these waves exhibit different modes, which are spatial distributions determined by the Bessel and sinusoidal functions. These modes determine how energy is distributed across the pipe's cross-section. The fundamental mode is the simplest and has the least radial variation. As waves propagate, each mode can travel with a different phase velocity, contributing to a complex overall wave pattern. The ability to analyze wave propagation helps in understanding many physical phenomena from acoustics to electromagnetic waves in coaxial cables.

frequency relation

Frequency relation in the context of the wave equation describes how the angular frequency \( \omega \) relates to the wavenumber components. For the cylindrical wave equation, we derived the fundamental relation: \[ \beta = \sqrt{\left( \frac{ \omega } { c } \right)^2 - \left( \frac{ X_{01} } { a } \right)^2 } \] where \( \beta \) is the longitudinal wavenumber. This equation shows the dependency of the longitudinal wavenumber on the frequency and the cylindrical system’s quantized radial component. For very low frequencies, the wavenumber \( \beta \) becomes imaginary, implying evanescent waves which decay exponentially rather than propagate. This frequency relation is crucial in designing systems like coaxial cables and waveguides where controlling wave's properties is essential for efficient transmission.