Flexural Waves in Beams

Dispersive propagation of bending waves in elastic beams according to Euler-Bernoulli theory.

Conceptual context

When a metal bar or a train rail is struck, the disturbance does not propagate as a simple wave at constant speed. Bending waves — also called transverse curvature waves — are inherently dispersive: their propagation speed depends on frequency. High-frequency components travel faster than low-frequency ones, causing an initially compact pulse to broaden progressively as it traverses the beam.

This phenomenon explains why a blow on a rail sounds different depending on distance from the impact point: the listener perceives the high-frequency (high-speed) components first, followed by the low-frequency (low-speed) ones. Understanding flexural dispersion is fundamental in structural acoustics, musical instrument design, machinery vibration analysis and building noise control.

This leaf presents classical Euler-Bernoulli theory, which describes the propagation of bending waves in slender beams, and establishes the relationships between frequency, wavenumber, phase and group velocities, and the geometric and material parameters of the cross-section.

🟢 Essential level

A bending wave is a transverse disturbance that travels along a beam, producing local curvature. Unlike longitudinal waves (compression-tension), here the displacement is perpendicular to the beam axis.

The most important property of flexural waves is their dispersion: the speed at which they propagate depends on frequency. High-frequency components travel faster. This contrasts with waves on an ideal string, which all travel at the same speed regardless of frequency.

The propagation speed of bending depends on three factors:

• The bending stiffness of the beam, combining the material's Young's modulus ($E$) with the cross-section shape (second moment of area $I$). A stiffer beam propagates waves faster.
• The mass per unit length ($\rho A$). A heavier beam propagates waves more slowly.
• The frequency of the wave ($\omega$). Higher frequency means higher speed.

Two relevant velocities exist:

• Phase velocity ($v_f$): the speed at which a crest-and-trough pattern moves.
• Group velocity ($v_g$): the speed at which energy travels. In beams, group velocity is exactly twice the phase velocity.

[!NOTE] Bending waves are always slower than longitudinal waves in the same material. At 1 kHz in a 5 mm steel bar, flexural phase velocity is about 217 m/s, compared to 5100 m/s for the longitudinal wave.

🔵 Formal level

Basic formulas

The Euler-Bernoulli dispersion relation establishes the connection between angular frequency $\omega$ and wavenumber $k$:

Flexural dispersion relation (Euler-Bernoulli)

This quadratic relation ($\omega$ proportional to the square of the wavenumber) is the source of all the dispersion. Phase velocity is defined as the ratio between angular frequency and wavenumber:

Flexural wave phase velocity (Euler-Bernoulli)

The square-root dependence on $\omega$ confirms that higher-frequency waves propagate faster. Group velocity, defined as the derivative of $\omega$ with respect to $k$, turns out to be exactly twice the phase velocity:

Flexural wave group velocity

The 2:1 ratio between group and phase velocity is a direct consequence of quadratic dispersion and constitutes a diagnostic signature of the Euler-Bernoulli model.

Real student questions

Why does flexural phase velocity depend on frequency while longitudinal does not? Because the restoring force in bending involves the fourth spatial derivative of displacement, not the second. The fourth-order derivative introduces a wavenumber dependence that produces dispersion. In longitudinal waves, the restoring force only depends on the second derivative and there is no dispersion.

Can flexural phase velocity exceed longitudinal velocity? Not in physical reality. The Euler-Bernoulli model predicts a phase velocity that grows without bound with frequency, but this is an artifact of the simplified model. At high frequencies, shear deformation limits the actual velocity to the shear wave speed (about 3100 m/s in steel).

Why is group velocity greater than phase velocity? Because the curve of $\omega$ versus wavenumber is convex. The tangent (group velocity) has a greater slope than the secant from the origin (phase velocity), typical of bending-wave dispersion.

What is the practical use of knowing group velocity? To calculate how long a perturbation's energy takes to traverse the beam. If a blow occurs 10 meters away and group velocity at the dominant frequency is 500 m/s, the perturbation arrives in 20 milliseconds. Phase velocity does not provide this information because it does not transport energy.

Final synthesis

Bending waves in beams are inherently dispersive: their propagation speed grows with the square root of frequency. Group velocity is exactly twice the phase velocity in the Euler-Bernoulli model, reflecting quadratic dispersion. The control parameters are bending stiffness EI and linear mass $\rho A$, and the model is valid as long as wavelength is much greater than the section thickness.