Longitudinal Waves in Bars

Non-dispersive propagation of compression-tension waves in slender bars and their applications.

Conceptual context

When one end of a metal bar is struck, the disturbance travels along it as a compression-tension wave. Unlike flexural waves, these longitudinal waves are non-dispersive in the slender bar model: all frequencies travel at the same speed. This makes them the simplest case of wave propagation in solids and the natural reference for comparison with dispersive phenomena.

The speed of these waves depends exclusively on two material properties: axial stiffness (Young's modulus $E$) and volumetric density ($\rho$). The cross-section geometry does not influence speed, though it does determine the mechanical impedance and therefore how the wave reflects and transmits at discontinuities.

This leaf covers longitudinal waves in slender bars (transverse dimension much smaller than wavelength). It is the direct complement of the flexural waves leaf: there the fourth spatial derivative produces quadratic dispersion; here the second derivative produces dispersion-free propagation.

🟢 Essential level

A long thin bar can be thought of as a chain of microscopic springs. When one end is compressed, each layer pushes the next and the disturbance advances. The speed at which this disturbance advances is the longitudinal wave speed $c_b$.

The essential point is that this speed depends only on the material:

• Stiff materials (high $E$) transmit fast because the "internal springs" are hard.
• Dense materials (high $\rho$) transmit slower because there is more inertia to move.

The relationship is a square root: $c_b$ grows with the root of stiffness and decreases with the root of density.

Typical values:

• Steel: approximately 5100 m/s
• Aluminum: approximately 5090 m/s
• Copper: approximately 3750 m/s
• Glass: approximately 5200 m/s
• Wood (along grain): 3000–5000 m/s
• Polymers: 1000–2500 m/s

The fundamental property: a compression pulse travels along the bar without deforming. Its shape remains identical regardless of distance traveled (as long as there is no damping or discontinuity). This is a direct consequence of the absence of dispersion.

The mechanical impedance $Z$ is the quantity that controls what happens when the wave encounters a change: another bar, a free end, a clamped end, or a junction. It is calculated as the product of density, cross-sectional area, and wave speed.

🔵 Formal level

Basic formulas

The wave equation for longitudinal displacements u(x,t) in a slender bar is derived from Newton's law on an infinitesimal element combined with Hooke's law:

Longitudinal wave speed in a slender bar

• Net force on an element of length dx is the stress difference across its two faces: dF equals A·d(sigma)/dx·dx
• Stress is sigma equals E·(du/dx) (Hooke)
• Element acceleration is rho·A·dx·(d²u/dt²) (Newton)
• Equating: E·(d²u/dx²) equals rho·(d²u/dt²)

The solution has propagation speed $c_b$ equal to the square root of $E$ divided by $\rho$.

Real student questions

Why doesn't speed depend on cross-sectional area? Because increasing area simultaneously increases both restoring force (more stiff material) and inertia (more mass). Both effects cancel exactly, leaving speed invariant.

Why do steel and aluminum have almost the same speed? Because E/rho for steel (210e9/7850 approximately 2.68e7) and aluminum (70e9/2700 approximately 2.59e7) are very close values. Steel's high stiffness compensates its high density.

Is this the same as the speed of sound? Only in the sense that both are wave speeds. But the speed of sound in air is due to gas compressibility, while in a bar it is due to Young's stiffness. Also, in an infinite solid the speed differs (P-waves) because lateral confinement changes the effective modulus.

What happens when the wave reaches the end of the bar? If the end is free, the wave reflects with stress inversion (compression returns as tension). If clamped, it reflects without inversion. In both cases particle velocity inverts or not according to the boundary condition.

Final synthesis

Longitudinal waves in slender bars are the canonical case of non-dispersive propagation in solids. Their speed depends only on E and rho, is independent of frequency and section geometry. Mechanical impedance (rho·A·$c_b$) controls reflection and transmission at discontinuities. The model is valid as long as wavelength is much larger than transverse dimensions.