Longitudinal Waves (Mechanical Waves › Waves in Fluids)

Study of longitudinal waves in fluids, compression, rarefaction, and speed of propagation.

Conceptual context

Longitudinal waves are the fundamental mechanism for transporting mechanical energy through fluids. Unlike transverse waves on strings, where each element moves perpendicular to the direction of propagation, in a longitudinal wave the particles of the medium oscillate along the same direction the disturbance travels. The observable result is an alternating succession of compressions (zones of high pressure and density) and rarefactions (zones of low pressure and density).

This leaf belongs to the mechanical waves block and connects directly with the previous study of propagation speed in strings. Moving from strings to fluids introduces an essential change: the restoring property is no longer linear tension but the volumetric elasticity of the medium, quantified by $B$. Understanding this transition is indispensable for addressing phenomena such as sound, primary seismic waves, and medical ultrasound.

🟢 Essential level

When a piston, tuning fork, or any source vibrates against a fluid, it pushes nearby molecules and creates a region of higher density: a compression. The compressed molecules push their neighbors and retreat, generating a region of lower density: a rarefaction. This cycle repeats and the disturbance travels away from the source without the fluid moving globally.

The speed at which this pattern of compressions and rarefactions advances depends on two properties of the medium. The stiffer the fluid (larger $B$), the faster molecules relay pressure. The denser it is (larger $\rho$), the more inertia they oppose and the slower the disturbance advances. The $f$ is set by the source, not by the medium, while the spatial scale $\lambda$ adjusts as a consequence.

[!TIP] The wave transports energy and momentum, but individual molecules only oscillate around their equilibrium positions.

An everyday example fixes the idea: when you speak, your vocal cords push the air and generate compressions and rarefactions that travel to the listener's ear at about 343 meters per second in air at room temperature. No air molecule travels from your mouth to the listener's ear; what propagates is the pressure pattern.

🔵 Formal level

Basic formulas

The propagation speed of a longitudinal wave in a fluid arises from the balance between elastic restoring force and the inertia of the medium:

• Propagation speed:

Propagation speed in fluids

$B$ represents the bulk modulus: the ratio between the applied pressure increment and the resulting relative volume deformation. High values of $B$ indicate a fluid that is hard to compress, where restoration is vigorous and the disturbance travels fast. $\rho$ contributes the volumetric inertia that slows this advance. The square root reflects that doubling $B$ does not double $v$; the dependence is sublinear.

When a source oscillates at $f$ within a fluid, the spatial separation between consecutive compressions defines the $\lambda$:

• Wavelength:

Wavelength for imposed frequency

If the medium changes $B$ or $\rho$ (for instance, transitioning from air to water), $v$ changes, and with $f$ held constant the spatial scale $\lambda$ readjusts proportionally.

Real student questions

Why doesn't the $f$ change when the wave passes from one fluid to another? Because the source sets the temporal rhythm. At the boundary, each compression that arrives generates a compression in the second medium at the same rate. What adjusts is $\lambda$, not $f$.

Can a longitudinal wave exist in a vacuum? No. A mechanical longitudinal wave needs matter that compresses and expands. Without molecules there is no elastic restoration or inertia; both $B$ and $\rho$ cease to exist.

Does higher density always mean lower speed? Not necessarily. Although $\rho$ appears in the denominator, a dense liquid usually has a much larger $B$ than a light gas, and the net effect can be a higher speed. Water is denser and faster than air because its $B$ is enormously greater.

Does acoustic impedance measure energy loss? No. $Z$ measures the medium's opposition to deformation, not dissipation. Energy loss is associated with viscosity and absorption, which are not part of the ideal model.

Final synthesis

Longitudinal waves in fluids transport energy through successive compressions and rarefactions. The propagation speed reflects the competition between the volumetric stiffness of the medium and its inertia, while acoustic impedance governs how energy is partitioned at interfaces. Mastering these relationships enables the analysis of phenomena ranging from sound propagation in the atmosphere to the design of ultrasonic transducers and the interpretation of medical ultrasound images.