MECH7029 Chap.6 Acoustic Analogy, Power Laws and Fan Noise
Acoustic Analogy, Power Laws and Fan Noise
Use the spectrum as a mechanism test
Lighthill's acoustic analogy places nonlinear flow terms on the right-hand side of a familiar wave operator. That rearrangement does not make the source simple, but it separates propagation from generation and reveals how source class changes speed scaling.
Fan spectra then connect the theory to engineering diagnosis: broad humps suggest stochastic turbulence, while blade-passing orders indicate periodic rotating events.
Exact rearrangement precedes approximation
The acoustic analogy rewrites conservation equations as propagation through a simple reference medium driven by equivalent stresses. Exactness belongs to the retained equation.
Approximation enters when source terms are simplified, the turbulent flow is modelled, surface motion is omitted, or radiation boundaries are imposed. Curle's surface-force extension and the moving-surface formulation add physics that a free-flow stress description cannot represent alone.
Ideal monopole, dipole and quadrupole powers carry progressively stronger Mach-number dependence under the course assumptions.
A measured speed sweep can therefore help distinguish mechanisms, but an exponent is not a unique fingerprint when flow structure, geometry or source efficiency also changes.
Fan diagnosis needs several clues
Blade-passing frequency is blade count multiplied by rotational frequency, not by rpm. Lines at that frequency and its harmonics show periodicity locked to blade passage.
They do not by themselves distinguish thickness displacement, rotating loading or rotor-stator interaction. Broadband energy suggests turbulent inflow, wake or edge scattering, while directivity and controlled geometry changes provide further separation. A credible mechanism must predict what happens when blade count, shaft speed, inlet turbulence or an edge treatment is altered.
What this chapter covers
- 01
Lighthill stress tensor
- 02
Equivalent source interpretation
- 03
Mach-number power laws
- 04
Moving-boundary terms
- 05
Broadband fan mechanisms
- 06
Blade-passing frequency and orders
Worked example · free
Track a blade-passing line after a speed change
- 2Convert rotational speed from rpm to hertz.
- 3Multiply by blade count at both speeds.
- 3Interpret the proportional spectral shift without overclaiming.
Key terms
- Acoustic Analogy
- A rearrangement of flow equations into a wave operator driven by equivalent sources.
- Lighthill Stress
- The equivalent quadrupole-type stress tensor in the acoustic analogy.
- Mach Scaling
- A power-law relation between radiated sound and characteristic flow Mach number.
- Blade-Passing Frequency
- Blade count multiplied by rotational frequency.
- Broadband Noise
- Energy spread continuously across a frequency range rather than concentrated at discrete orders.
Acoustic Analogy, Power Laws and Fan Noise FAQ
Does the acoustic analogy require the entire flow to be linear?
No. The propagation operator is written in wave-equation form while nonlinear flow quantities remain in equivalent source terms. Approximations enter when those terms and the surrounding propagation medium are modelled.
Does a blade-passing line prove that loading noise dominates?
It establishes synchronism with blade passage, not a unique source. Thickness, periodic loading and rotor-stator interactions can share the same fundamental order, so use directivity, amplitude scaling and geometry changes as additional tests.
Why are low-Mach power laws useful?
They expose how cancellation makes higher multipoles inefficient as speed falls. They are diagnostic trends under stated similarity assumptions, not universal exponents for every installation or operating condition.
Exam move
Organise revision as an evidence table with columns for spectrum, rotational order, speed scaling, directivity, source region and remaining alternatives. Begin every rotating-machine example by converting rpm to revolutions per second and multiplying by blade count. Mark the fundamental blade-passing line and at least two harmonics, then separate those discrete components from broadband content.
For each candidate mechanism, predict one additional observation: how a line moves with shaft speed, how amplitude changes during a controlled speed sweep, where a directivity null should appear, or how altered inlet turbulence affects the spectrum. Use the acoustic analogy as an audit trail.
Identify what is exact in the rearrangement, what source field has been approximated, what boundary or motion term is retained, and what propagation model carries sound to the receiver. Practise the Mach-power comparison with a doubled speed and convert power ratios with ten times the logarithm. State why the comparison is conditional on similarity and fixed coefficients.
Add one near-field example in which hydrodynamic pressure is not treated as far-field radiation. Finish each diagnosis with an experiment that could disconfirm the preferred explanation. A mechanism that merely fits an existing line is weaker than one that correctly predicts the effect of changing blade count, speed, inflow condition or edge geometry.
Build one final mixed case containing a blade-passing line, a broadband hump and a speed sweep, and require a separate inference from each observation.
