Infinite beams (i.e. we don't account for boundary conditions) made of a homogeneous and isotropic material are considered here.
For 3 different models, the dispersion relations, i.e. the expressions of the phase velocity $c_{\phi}$ and group velocity $c_g$ as functions of the wavenumber $k$ are presented.

The overview of these models will lead us to discuss points which are usually not considered or even ignored in continuum mechanics books (maybe to the exception of Isaac Elishakoff's "Handbook On Timoshenko-Ehrenfest Beam And Uflyand-Mindlin Plate Theories" but the current price is prohibitive).

The equations of motion are not fully derived. Please refer to some books/lectures in continuum vibrations for complete derivations (20 years ago I wrote a PDF document for master students on the topic, it's in French, it's ugly, there are probably some errors but it's free).

# Dispersion relations

When adding two waves with close frequencies (and wavenumbers), $\textcolor{#aaccff}{\blacksquare}$ and $\textcolor{#003380}{\blacksquare}$, the resulting wave exhibits beats $\textcolor{#0066ff}{\blacksquare}$. The red bullet ($\textcolor{#ff0000}{\bullet}$) (on a node of the beat envelope) is propagating with the group velocity. The yellow bullet ($\textcolor{#ffd42a}{\bullet}$) (starting at the same position than the red one) is propagating with the phase velocity. Here the phase velocity is faster than the group one (and motions are in the same direction).

By definition, the phase velocity is the ratio between the pulsation $\omega=2\pi f$ (where $f$ is the frequency) and the wavenumber : $c_{\phi} = \frac{\omega}{k}$ and the group velocity is defined as : $c_g = \displaystyle\frac{d\omega}{dk} = c_{\phi} - \lambda\displaystyle\frac{dc_{\phi}}{d\lambda}$ In this page, all quantities are expressed in the international system of units, i.e. $\omega$ is expressed in s$^{-1}$ (sometimes reported as rad.s$^{-1}$ to emphasis on the $2\pi$ term), $f$ is given in Hertz or s$^{-1}$, $k=2\pi / \lambda$ in m$^{-1}$ and $\lambda$ the wavelength in $m$.

### Euler-Bernoulli beams

Euler-Bernoulli model for beam bending leads to an homogeneous forth-order differential equation with a characteristic equation which can be written as : $k^4 - \omega^2 \frac{\rho A}{EI} = 0$ or $\omega^2 = k^4 \displaystyle\frac{EI}{\rho A}$ Thus, 4 values for the wavenumber are possible from the characteristic equation. Among those, 2 are related to evanescent waves (in opposite directions) and the 2 remaining values are related to propagating waves (in the opposite directions). The 2 propagative solutions are of the form $\exp(\pm jkx)$ with : $k = \sqrt{\omega}\displaystyle\left(\frac{\rho A}{EI}\right)^{1/4}$ from which we can express the phase ($c_{\phi}$) and group ($c_{g}$) velocities : $c_{\phi} = \frac{\omega}{k} = k\displaystyle\sqrt{\frac{EI}{\rho A}} = \sqrt{\omega}\displaystyle\left(\frac{EI}{\rho A}\right)^{1/4}$ $c_{g} = \frac{d\omega}{dk} = 2k\displaystyle\sqrt{\frac{EI}{\rho A}} = 2 \sqrt{\displaystyle\omega}\displaystyle\left(\frac{EI}{\rho A}\right)^{1/4} = 2 c_{\phi}$
One major drawback of Euler-Bernoulli beam model is the unrealistic prediction of infinite group and phase velocities at high frequencies (when the wavenumber $k$ tends to infinity).

### Rayleigh beams

Jacques Antoine Charles Bresse and then, independently, John William Strutt (3rd baron Rayleigh) added the rotary inertia to Euler-Bernoulli model. Adding the rotary inertia to Euler-Bernoulli model (which already accounts for the transversal inertia), Jacques Antoine Charles Bresse and then, independently, John William Strutt (3rd baron Rayleigh) have obtained a beam model with bounded group and phase velocities for high frequencies.

The characteristic equation for Rayleigh beams writes : $k^4 - k^2 \omega^2 \frac{\rho}{E} - \omega^2 \frac{\rho A}{EI} = 0$ or $\omega^2 = k^4 \displaystyle\frac{EI}{\rho A + k^2 \rho I}$ Again, 4 values for the wavenumber are possible. Among those, 2 are related to evanescent waves (in opposite directions) and the 2 remaining values are related to propagating waves (in the opposite directions). The 2 propagative solutions are of the form $\exp(\pm jkx)$ with : $k = \sqrt{\omega}\displaystyle\left(\frac{EI}{\rho A + k^2 \rho I}\right)^{1/4}$ $c_{\phi} = \frac{\omega}{k} = k\displaystyle\sqrt{\frac{EI}{\rho A + k^2 \rho I}}$ $c_{g} = \frac{d\omega}{dk} = k\displaystyle\sqrt{\frac{EI}{\rho A+\rho I k^2}} = 2 c_{\phi} - \displaystyle\frac{\rho}{E}c_{\phi}^3$

### Timoshenko-Ehrenfest beams

Stephen Timoshenko, together with Paul Ehrenfest, later added the shear deformation making use of a shear coefficient that will be discussed in a section below.
Note that it is not clear (at least to the author) if Timoshenko & Ehrenfest developed their model based on the work by Rayleigh (which they were aware of) or from the work by Jacques Antoine Charles Bresse (who developped, years before Rayleigh, a beam model accounting for both rotary inertia and shear deformation but without any correction that will lead to the introduction of the shear coefficient).

When compared with Euler-Bernoulli or Rayleigh beam models, Timoshenko-Ehrenfest model allows to predict the larger deflections under static loads of "thick" beams or the lower eigen-frequencies of beams under dynamic behaviors.

The analysis of the free vibrations for such a beam, leads to a coupled system of equations : \begin{aligned} G\kappa A \left( \frac{\partial^2 w}{\partial x^2} + \frac{\partial \phi}{\partial x} \right) - \rho A \frac{\partial^2 w}{\partial t^2} &= 0 \\ EI\frac{\partial^2\phi}{\partial x^2} - G\kappa A \left( \frac{\partial w}{\partial x} + \phi \right) - \rho I \frac{\partial^2 \phi}{\partial t^2} &= 0 \\ \end{aligned} The elimination of the cross-sectional rotation $\phi$ leads to the fourth-order differential equation (both in space and in time) : $EI \frac{\partial^4 w}{\partial x^4} - \rho I \left(1 + \frac{E}{G\kappa} \right)\frac{\partial^4 w}{\partial x^2\partial t^2} + \rho A \frac{\partial^2 w}{\partial x^2} + \frac{\rho^2 I}{G\kappa} \frac{\partial^4 w}{\partial t^4} = 0$ Some authors prefer to eliminate the transverse displacement $w$ instead of $\phi$ and they end up with the same equation, $w$ being replaced with $\phi$ i.e. both variables have the same time and space dependence).

The corresponding characteristic equation is a bi-quadratic equation : $k^4 - k^2 \omega^2 \frac{\rho}{E} \left( 1 + \frac{E}{G\kappa } \right) - \omega^2 \frac{\rho A}{EI} + \omega^4 \frac{\rho^2}{G\kappa E} = 0$ 4 solutions are possible for the wavenumber and the behavior of the beam depends on a "cutoff" pulsation $\omega_c$ defined as the square root of the ratio between the shear stiffness and the rotary inertia : $\omega_c^2 = \frac{\kappa G A}{\rho I}$ (all quantities being positive, there's only one physical solution for $\omega_c$ which is $+\sqrt{\omega_c^2}$).

For $\omega < \omega_c$, 2 wavenumbers describe propagative waves (in opposite direction) and the remaining 2 describe evanescent waves (in opposite direction).
For $\omega > \omega_c$, the 4 wavenumbers describe propagating waves and there is thus 2 different waves propagating at each pulsation above $\omega_c$. The first wave is the flexural one. The second one belongs to what have been called the second spectrum (which can be disregarded under some conditions like long beams and specific boundary conditions).

# About the second moment of area

The second moment of area $I$ is a purely geometric quantity computed here, with respect to the vector space used, as : $I = \iint_{A} z^2dzdy$ Note that $I$ is, sometimes, abusively referred to as an inertia quantity while the density is not involved in $I$.

Below are some values for $I$, expressed in the international system of units as m$^4$. Any resemblance to the buttons of a PS4 console with Nintendo's colors may not be purely coincidental.
$I = \pi (R^2 - r^2) / 4$
$I = BH^3/36$ (not necessarily isosceles triangle but axes should pass through its centroid)
$I = \frac{bH^3+2Bh}{12}$
$I = (BH^3-bh^3) / 12$
$I = \frac{B H^3-(B-b)h^3}{12}$ (for a symmetric beam)
$I = \pi R^4 / 4$
$I = BH^3/12$

A dimensionless shear coefficient $\kappa$ was introduced to account for the fact that the shear stress and the shear strain are not constant over the cross-section of a beam.
There is no unique formula to express this coefficient which depends on the geometry of the cross-section among other parameters. Various expressions have been published, some accounting for the Poisson's ratio $\nu$ of a homogeneous material, some valid in the static regime, some others valid in a dynamic regime. The values reported are usually obtained from comparisons with more general continuum mechanics analyses or from experiments.

Below are some expressions for $\kappa$ found in the literature.

$\displaystyle\frac{6(1+\nu)}{7+6\nu}$ in Cowper 1966 (static regime); $\displaystyle\frac{6(1+\nu)^2}{7+12\nu+4\nu^2}$ in Hutchinson 2001 (dynamic regime).

$2/3$ in Timoshenko 1921; $\displaystyle\frac{10(1+\nu)}{12+11\nu}$ in Cowper 1966 (static regime). Note that this latter expression equals $5/6$ when $\nu=0$. The expression in the dynamic regime proposed by J. R. Hutchinson 2001 is too long to be reported here but it reduces to $\displaystyle\frac{5(1+\nu)}{6+5\nu}$ as one approaches "perfect" plane stress conditions.

Additional expressions, for other shapes or for other conditions, can be found in reviews on the subject by G. R. Cowper 1966, T. Kaneko 1975, J. R. Hutchinson 2001 (and the comments by N. G. Stephen on Hutchinson's work)...

# Script

Obviously, the artwork of this page is inspired from Mario by Nintendo (with all images drawn from scratch and served as svg files). The beam/tube + piranha plant is inspired from the artwork by Manali on Dribbble.