9.5 Sound from sources embedded in flow

A vibrating object moving through fluid (or fluid moving past a stationary object) can emit sound by mechanisms quite different from those of chapter 6. The classical monopole/dipole picture assumed a small source displacing a stationary medium. When the medium is flowing — and especially when it is turbulent — new sound-generation mechanisms appear.

The Aeolian tone

The simplest example: wind blows past a long cylinder. Behind the cylinder, a von Kármán vortex street forms — alternating clockwise and counterclockwise vortices shed at a periodic rate. Each shed vortex pushes momentarily on the cylinder, producing an oscillating side force. That oscillating force is a dipole source (cf. lesson 6.4) and radiates sound at the shedding frequency.

The shedding frequency follows an empirical rule:

f_\text{shed} \;=\; \text{St} \cdot v / d,

where $v$ is the flow velocity, $d$ the cylinder diameter, and St $\approx 0.2$ is the Strouhal number — a roughly constant dimensionless prefactor over a wide range of Reynolds numbers.

The whistling sound of wind through power lines, telephone wires, or chain-link fences is the Aeolian tone. For a 1-cm wire in 10 m/s wind: $f \approx 0.2 \cdot 10 / 0.01 = 200$ Hz. For thinner wires or faster wind, the pitch rises.

Aerodynamic sound generation generally

The Aeolian tone is just one example. A broader theory — developed by James Lighthill in the 1950s — describes how any turbulent flow generates sound. The Lighthill analogy rewrites the Navier–Stokes equations as

\frac{1}{c^2}\, \frac{\partial^2 p'}{\partial t^2} - \nabla^2 p' \;=\; \frac{\partial^2 T_{ij}}{\partial x_i \partial x_j},

where $T_{ij}$ is the Lighthill stress tensor: a known function of the local flow velocity. The left side is the standard wave operator; the right side is a source term describing how the flow generates sound.

The Lighthill source has three components:

A monopole part proportional to mass fluctuations in the flow (rarely dominant for steady turbulence).
A dipole part proportional to fluctuating forces (the Aeolian tone is this).
A quadrupole part proportional to fluctuating Reynolds stresses (the dominant source for free turbulent jets).

For free turbulence (turbulent jets in unbounded fluid), the radiated acoustic power scales as $\rho U^8 D^2 / c^5$ — the Lighthill $U^8$ law. Acoustic power varies as the eighth power of jet velocity. This is why jet engines are loud: doubling the jet velocity multiplies the radiated power by 256.

Compressor noise, propeller noise

A rotating fan or propeller generates sound through a combination of mechanisms:

Thickness noise: the blade displaces air as it moves, monopole-like radiation at the blade-passing frequency and its harmonics.
Loading noise: aerodynamic forces on the blade fluctuate as it passes through a non-uniform inflow; these are dipole sources.
Broadband noise: turbulent boundary-layer fluctuations on the blade surface; quadrupole sources.

Helicopter blade slap, the prominent buzz of small drones, the high-frequency squeal of cooling fans — all are decomposable into combinations of these elementary mechanisms, each with its own scaling with velocity, density, and geometry.

What this opens up

We are at the edge of the Sound book’s scope. The detailed treatment of aeroacoustics — turbulent-flow sound, jet noise, broadband sources — is its own substantial subject, with applications in aircraft design, automobile aerodynamics, HVAC engineering, and biological flight noise.

For our purposes the takeaway is: sound is generated not just by surfaces vibrating in still fluid (chapter 6), but by fluid motion itself. The wave equation’s source term tells us where the sound comes from; everything we have studied about its propagation then applies.

Next lesson: when sound crosses the boundary between still air and a region of flow, it refracts in characteristic ways.