Acoustic Radiated Power
Acoustic Radiated Power: Theoretical Foundations
What is Acoustic Radiation?
Professor, what is acoustic radiation power?
A vibrating structural surface vibrates the surrounding air to produce sound. It's an index that quantifies the sound power.
$p$: sound pressure on the surface, $v_n$: normal direction vibration velocity of the surface, $*$: complex conjugate. It is the integral of the acoustic intensity over the entire surface.
Radiation Efficiency
Does sound always come out if it's vibrating?
Sometimes vibration does not become sound. That is expressed by the radiation efficiency $\sigma_{rad}$.
$\rho c$: characteristic impedance of air, $S$: radiation area, $\langle v_n^2 \rangle$: surface-averaged squared vibration velocity value.
- $\sigma_{rad} = 1$: Perfect radiator (piston vibration)
- $\sigma_{rad} < 1$: Weak radiation (acoustic short-circuit occurs)
- $\sigma_{rad} > 1$: Rare, but can occur during resonance
What is acoustic short-circuit?
When adjacent regions vibrate in opposite phases, the positive pressure from one is canceled by the negative pressure from the other. This is prominent in the modal vibration of plates and is the reason why radiation efficiency is lower at lower frequencies.
Critical Frequency and Radiation
Radiation efficiency changes significantly with frequency:
- $f < f_c$ (Below coincidence frequency): $\sigma_{rad} \ll 1$. Acoustic short-circuit is dominant.
- $f = f_c$: $\sigma_{rad}$ increases sharply.
- $f > f_c$: $\sigma_{rad} \approx 1$. The entire surface radiates efficiently.
Summary
The concept of radiation efficiency originates from Lord Rayleigh's 1877 paper
The theoretical foundation of radiation efficiency dates back to Lord Rayleigh's 1877 work "The Theory of Sound" Volume 2. He derived the radiation impedance using an infinite flat plate "piston model," but its extension to finite structures had to wait until the 20th century. Wallace (1972) analytically obtained the radiation efficiency of a finite rectangular plate, which became the cornerstone of modern structural acoustics theory.
Computational Methods for Acoustic Radiated Power
Acoustic Radiation Calculation Methods
How do I calculate acoustic radiation power using FEM?
There are mainly three approaches.
1. FEM Coupled Analysis
Structural FEM + Acoustic FEM coupling. The acoustic domain is modeled with finite elements, and a PML (Perfectly Matched Layer) is placed externally.
- Advantage: Can calculate both near-field and far-field.
- Disadvantage: The acoustic domain mesh becomes large.
2. BEM (Boundary Element Method)
Calculates the external sound field using only the structural surface mesh. Can handle infinite domains directly.
$G$: Free-space Green's function. Calculates the sound pressure field using the surface vibration velocity as a boundary condition.
3. Rayleigh Integral (Plane Approximation)
Assumes a vibrating surface mounted on an infinite baffle:
Convenient for quick estimation of radiation power from flat panels. Computational cost is lower than BEM.
Summary
Near-field Acoustic Holography is a masterpiece of inverse problem analysis
Near-field Acoustic Holography (NAH) is an innovative method published by Maynard et al. in JASA in 1985, capable of reconstructing the vibration distribution of a radiating surface from measurements by an array of sound pressure microphones. NAH was introduced as experimental bench equipment by Ford and BMW's NVH departments in the 1990s to non-contact identify contributing noise sources like engine covers and fuel tanks. It is now miniaturized and commercialized as acoustic cameras (e.g., gfai tech's SoundCam).
Acoustic Radiated Power in Practice
Practical Acoustic Radiation Power Analysis
Typical application examples include automotive engine radiation noise, home appliance motor noise, and transformer hum noise.
Analysis Flow
1. Structural Vibration Analysis โ Obtain surface vibration velocity via modal or frequency response analysis.
2. Radiation Power Calculation โ Calculate sound field using BEM or Rayleigh integral.
3. Radiation Efficiency Evaluation โ Plot $\sigma_{rad}$ for each frequency.
4. Sound Pressure Level โ Evaluate sound pressure at observation points in dB(A).
Practical Checklist
Sound Power Level (SWL)
$W_0 = 10^{-12}$ W (reference acoustic power). $L_W = 80$ dB means $W = 10^{-4}$ W (0.1 mW).
Electromagnetic forces dominate the radiated sound of electric motors
Unlike combustion engines, the radiated noise of EV motors is primarily caused by electromagnetic forces due to torque ripple. During the development of Toyota's first-generation Prius (1997), the fact that the motor sound was clearly audible inside the cabin when the engine was off remained unrecognized until late in the design phase. Today, coupled electromagnetic-structural-acoustic analysis using Maxwell stress tensor has become the de facto standard, with combinations like JMAG + ABAQUS + Actran adopted by major domestic suppliers.
Acoustic Radiated Power: Software & Solver Comparison
Tools
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