Wave Mechanics & Acoustics for CAE Engineers
Wave Equation, Acoustic Impedance, NVH & Structural-Acoustic Simulation
Acoustics and wave mechanics govern everything from the roar of a jet engine to the barely perceptible hum inside a premium sedan. In modern CAE practice, noise and vibration analysis — collectively called NVH — is a major design driver, often receiving as much engineering attention as structural strength. Understanding the physics from first principles enables engineers to diagnose noise problems, design quieter systems, and trust their simulation results.
1. The Acoustic Wave Equation
Where does the acoustic wave equation come from? Is it a completely separate thing from the Navier-Stokes equations, or derived from them?
It's derived from them — it's a linearized, inviscid, irrotational simplification of the full Navier-Stokes equations. You start with the continuity and momentum equations, assume small perturbations around a quiescent mean state (no mean flow), neglect viscosity and nonlinear terms, and you end up with the classical wave equation for acoustic pressure. That linearization is the key step: acoustics is the physics of small-amplitude waves. Once amplitudes get large — near a jet engine nozzle, for example — you need nonlinear acoustics, and the simple wave equation breaks down.
1.1 Derivation from Fluid Mechanics
Decompose pressure and density into mean and perturbation: $p = p_0 + p'$, $\rho = \rho_0 + \rho'$, velocity $\mathbf{v} = \mathbf{v}'$ (zero mean flow). Linearize the continuity and Euler momentum equations:
Using the isentropic relation $p' = c^2 \rho'$ and taking the divergence of the momentum equation combined with the time derivative of continuity:
This is the classical linear acoustic wave equation — a second-order hyperbolic PDE. The same form also applies to displacement potential $\phi$ where $\mathbf{v}' = \nabla\phi$.
1.2 Helmholtz Equation (Frequency Domain)
For time-harmonic fields $p'(\mathbf{x},t) = \hat{p}(\mathbf{x})e^{i\omega t}$, the wave equation reduces to the Helmholtz equation:
where $k$ is the wavenumber. The Helmholtz equation is an elliptic PDE in the spatial domain, much easier to solve numerically than the time-domain wave equation for narrow-frequency-band problems. Most commercial acoustic FEM solvers (ACTRAN, Comsol Acoustics, Nastran SOL 108) work in the frequency domain.
1.3 Wave Equation with Source
When volume sources are present (e.g., a vibrating boundary acting as a monopole source):
where $q(\mathbf{x},t)$ is the mass source rate per unit volume. The Green's function solution in free space is:
2. Sound Speed & Acoustic Pressure
Why does sound travel faster in water than in air, even though water is much denser? I'd expect heavier media to slow it down.
Great intuition — but density alone isn't the whole story. Speed of sound depends on the ratio of the restoring force (bulk modulus $K$) to the inertia (density $\rho$): $c = \sqrt{K/\rho}$. Water has a bulk modulus around 2.2 GPa — it's extremely stiff, barely compressible. Air has a bulk modulus of only 142 kPa. Even though water is 800 times denser, it's roughly 15,000 times stiffer, so the ratio $K/\rho$ is about 20 times larger in water. Steel is even more extreme — sound travels at ~5000 m/s because steel is incredibly stiff.
2.1 Speed of Sound
For an ideal gas, the isentropic (adiabatic) bulk modulus $K_s = \gamma P$. For air at 20°C: $c = \sqrt{1.4 \times 287 \times 293} = 343$ m/s.
| Medium | $c$ [m/s] | $\rho_0$ [kg/m³] | $Z = \rho_0 c$ [Pa·s/m] |
|---|---|---|---|
| Air (20°C, 1 atm) | 343 | 1.204 | 413 |
| Water (20°C) | 1482 | 998 | 1.48 × 10⁶ |
| Steel | 5960 | 7850 | 4.68 × 10⁷ |
| Aluminum | 6420 | 2700 | 1.73 × 10⁷ |
| Concrete | 3100 | 2300 | 7.1 × 10⁶ |
| Human tissue (soft) | 1540 | 1060 | 1.63 × 10⁶ |
2.2 Sound Pressure Level (SPL)
Sound pressure level uses a logarithmic scale referenced to the threshold of human hearing $p_{\text{ref}} = 20\,\mu$Pa:
Key reference points: 0 dB = threshold of hearing, 65 dB = normal conversation, 85 dB = prolonged exposure causes hearing damage, 130 dB = pain threshold, 194 dB = theoretical maximum for undistorted sound in air.
3. Acoustic Impedance & Reflection
Specific acoustic impedance is $Z = \rho_0 c$ [Pa·s/m = rayl]. At an interface between media 1 and 2 (normal incidence), the pressure reflection coefficient is:
The intensity transmission coefficient is:
In an ultrasound scan, why is gel applied between the transducer and the skin? Seems like an odd requirement.
Exactly the impedance mismatch issue. Air has $Z \approx 413$ Pa·s/m, human tissue has $Z \approx 1.63 \times 10^6$ Pa·s/m. Plug those into the formula: the transmission coefficient is $4 \times 413 \times 1.63\times10^6 / (413 + 1.63\times10^6)^2 \approx 0.001$ — less than 0.1% of the sound energy gets through! The gel has impedance close to tissue, eliminating the air gap and allowing nearly perfect transmission. This impedance matching principle is also used in acoustic transducer design, building insulation design, and sonar systems.
3.1 Oblique Incidence and Snell's Law
For waves incident at angle $\theta_i$ to the interface normal, Snell's law applies:
When $c_2 > c_1$, a critical angle $\theta_c = \arcsin(c_1/c_2)$ exists beyond which total internal reflection occurs — no energy is transmitted. This phenomenon is used in acoustic waveguides and explains certain room acoustics anomalies.
4. Standing Waves & Resonance
Our car cabin has a noticeable booming around 50 Hz. The acoustic simulation shows a standing wave. How do I figure out if this is a cavity mode or something coming from the structure?
Start with a quick back-of-envelope: the first longitudinal acoustic mode of a rectangular cavity is approximately $f = c/(2L)$ where $L$ is the longest dimension. A typical car cabin is about 2 meters long, giving $f \approx 343/(2\times2) = 86$ Hz. Your 50 Hz boom is below that, which suggests it's likely a coupled structural-acoustic mode — the body-in-white (BIW) panels are participating. Do a vibro-acoustic analysis: first check which structural modes fall near 50 Hz, then look at whether those mode shapes pump air efficiently into the cavity. If you see a highly-coupled mode where the floor panel moves in phase with the cavity pressure, that's your culprit. Stiffening or adding damping to that panel is the fix.
4.1 1D Standing Waves in a Duct
For a duct of length $L$ with rigid walls at both ends (velocity nodes), resonance frequencies are:
Pressure mode shapes: $\hat{p}_n(x) = A\cos(n\pi x/L)$. Maxima at walls (pressure antinodes), zero at center for $n=1$.
For a tube open at one end (quarter-wave resonator), only odd harmonics exist:
4.2 3D Acoustic Cavity Modes
For a rectangular cavity with dimensions $L_x \times L_y \times L_z$, the natural frequencies are:
where $m, n, l = 0, 1, 2, \ldots$ (not all zero simultaneously). The $(0,0,0)$ mode doesn't exist — that would be uniform pressure change with no wave structure.
5. Structural Acoustics & Vibro-Acoustics
When my structure vibrates, some modes are "good radiators" of sound and others are not. What physically determines whether a structural mode radiates efficiently?
The key concept is radiation efficiency — or more precisely, whether the structural wavenumber matches the acoustic wavenumber. Think of it this way: a structural mode has a certain spatial pattern of displacement. If adjacent regions of the plate are moving in opposite directions (out of phase), the sound they radiate tends to cancel. This is efficient cancellation — the mode "shorts out" the acoustic radiation. But if the structural wavelength is long compared to the acoustic wavelength at that frequency (i.e., the structure is supersonic), the radiation can't cancel and the mode radiates powerfully. This is the concept of the critical frequency or coincidence frequency — below it, radiation is inefficient; above it, a panel can become a very efficient radiator.
5.1 Radiation Efficiency
For a baffled panel, the radiation efficiency $\sigma$ characterizes how well a structural mode radiates compared to a piston of the same area:
where $W_{\text{rad}}$ is radiated acoustic power and $\langle v^2 \rangle$ is the mean-square surface velocity. $\sigma \ll 1$ for most structural modes below the critical frequency.
5.2 Coincidence Frequency
For a thin plate (thickness $h$), the coincidence (critical) frequency is:
For a 2 mm steel panel: $f_c \approx 2500$ Hz — above this, the panel radiates efficiently. For a 50 mm concrete wall: $f_c \approx 120$ Hz — below this, sound insulation is poor. This is why concrete walls transmit bass frequencies so easily.
5.3 Coupled Structural-Acoustic System
The coupled equations for structural displacement $\mathbf{u}$ and acoustic pressure $p$ are:
The coupling matrix $\mathbf{C}$ represents the fluid-structure interaction at the shared interface. The acoustic pressure loads the structure (acoustic force = $p \cdot \mathbf{n} \cdot A$) and the structural velocity drives the fluid (acoustic particle velocity = structural normal velocity at interface).
6. NVH: Noise, Vibration & Harshness
NVH (Noise, Vibration, Harshness) is the automotive and aerospace engineering discipline that addresses the subjective quality of acoustic and vibration responses. "Noise" is airborne sound, "vibration" is structure-borne sensation, and "harshness" refers to high-frequency structural roughness that feels unpleasant rather than sounds problematic.
6.1 Source-Path-Receiver Framework
Every NVH problem decomposes into:
- Source: Engine combustion, road roughness, wind turbulence, electric motor harmonics
- Path: Structure-borne (through mounts, chassis) or airborne (through gaps, panels)
- Receiver: Driver/passenger ear, or a regulatory measurement microphone
CAE NVH work systematically targets each element: source characterization by blocked force measurement, transfer path analysis (TPA) to quantify each path's contribution, and receiver sensitivity analysis.
6.2 Transfer Path Analysis (TPA)
The sound pressure at the receiver due to source $k$ transmitted via path $j$ is:
where $H_j(\omega)$ is the acoustic transfer function (Pa/N) for path $j$ and $F_j$ is the blocked force at that path. TPA allows engineers to rank paths by contribution and target only the dominant ones for treatment.
6.3 Sound Quality Metrics
| Metric | Unit | Description |
|---|---|---|
| SPL (A-weighted) | dB(A) | Overall level weighted by human hearing sensitivity |
| Loudness | sone | Psychoacoustic loudness (Zwicker method) |
| Sharpness | acum | High-frequency content concentration |
| Roughness | asper | Amplitude modulation at 15–300 Hz |
| Tonality | tu | Prominence of discrete tones over background |
7. FEM & BEM for Acoustic Simulation
My colleague says BEM is better than FEM for exterior acoustics problems. Why would BEM be better for exterior problems specifically?
It comes down to the Sommerfeld radiation condition. In exterior acoustics, the sound field extends to infinity — there's no boundary far away that you can use as a wall. With FEM, you have to mesh a finite volume of air and then apply an absorbing boundary condition (like a Perfectly Matched Layer, or PML) at the truncation boundary to prevent reflections. This adds computational cost and the PML itself introduces approximation. BEM works directly with the surface integrals using Green's functions that already satisfy the wave equation everywhere in the unbounded domain — you only need to mesh the structure surface, not the surrounding air. For something like computing the sound radiated from a vibrating car body, BEM is much more efficient. For interior cavities like passenger compartments, FEM wins because BEM struggles with internal problems.
7.1 FEM Formulation for Acoustics
The weak form of the Helmholtz equation with structural coupling boundary condition $\partial p/\partial n = -\rho_0\omega^2 u_n$:
Key FEM rules for acoustic meshes:
- At least 6 linear (or 3 quadratic) elements per acoustic wavelength $\lambda = c/f$
- For air at 1000 Hz: $\lambda = 343/1000 = 0.343$ m → minimum element size ~57 mm (linear)
- Higher frequencies require much finer meshes — acoustic FEM at 10 kHz requires 6 mm elements!
7.2 BEM Formulation
The BEM integral equation (Kirchhoff-Helmholtz integral equation) for the exterior problem:
where $G(\mathbf{x},\mathbf{y}) = e^{-ik|\mathbf{x}-\mathbf{y}|}/(4\pi|\mathbf{x}-\mathbf{y}|)$ is the free-space Green's function and $c(\mathbf{x}) = 1/2$ for smooth boundary points. The BEM assembles a fully-populated (dense) matrix, so iterative solvers and fast multipole methods (FMM) are used for large problems.
7.3 Statistical Energy Analysis (SEA) for High Frequencies
FEM and BEM become impractical above ~2–4 kHz for full vehicle models (mesh size requirements become prohibitive). Statistical Energy Analysis models the system as a set of subsystems exchanging vibrational and acoustic energy:
where $E_i$ is subsystem energy, $n_i$ is modal density, $\eta_i$ is damping loss factor, and $\eta_{ij}$ is coupling loss factor. SEA gives average frequency-band energy levels, not spatial detail, but requires minimal computational cost for high-frequency problems. VA One (ESI), AutoSEA, and ANSYS Mechanical SEA extension are common tools.