Basics of Acoustics

Wave number

The angular, or circular, wave number 𝑘, often misleadingly abbreviated as wave number, is the modulus of the wave vector. It is inversely proportional to the wavelength λ.

The wave number is the spatial analogue of the angular frequency: application of a Fourier transformation on data in the time domain yields a frequency spectrum, applied on data in the spatial domain (data as a function of position) yields a spectrum as a function of the wave number. Note that the exact definition of the wave number depends on the physics field studied. In mechanics, the angular wavenumber is defined as:

\[ k=\frac{2\pi}{\lambda}=\frac{2\pi f}{\nu_{\phi}}=\frac{\omega}{\nu_{\phi}} \]

where 𝑓 is the frequency (in Hertz), ω is the angular frequency (in rad.s⁻¹) and 𝑣_𝜙​ is the phase velocity of the wave (in m.s⁻¹).

Complex wave numbers

Complex wave numbers are a commonly way to account for dissipations.
For a harmonic time dependence of the form 𝑒^𝑗𝜔𝑡, a pressure wave propagating toward negative 𝑥 writes:

\[ p(x,t)=Ae^{j(\omega t+kx)} \]

If the wavenumber 𝑘 is a complex quantity of the form k=α+jβ with α and β defined as real positive quantities, the pressure wave can be re-written as:

\[ p(x,t) = A e^{j(\omega t + \alpha x)} e^{-\beta x} \]

e^−βx is an exponential attenuation (or decay) with β as the characteristic dimension.

Complex and frequency dependent wave number

The dissipation mechanisms (e.g. visco-inertial effects, thermal effects…) can usually not be considered as constant with the frequency (see e.g. the page about the viscous boundary layer). Thus, the complex wave number is also frequency dependent (as is the velocity 𝑣_𝜙​).
This frequency dependence implies, at least in a given frequency range, a dispersion of the wave following Kramers-Kronig relations.

Helmholtz resonator

The resonance frequency of a Helmholtz resonator is computed as:

\[ f=\frac{1}{2\pi}\sqrt{\gamma\frac{P_{0}}{\rho}\frac{A_{n}}{V_{c}L_{n}}} \]

where:

• γ is the ratio of specific heats (1.4 for air),
• P₀ the atmospheric pressure in N.m⁻² (or Pa),
• ρ the mass density of air at rest (1.2 kg.m⁻³),
• Aₙ the cross section area of the resonator neck,
  • for a circular cross-section of radius rₙ, Aₙ = πrₙ²,
• Lₙ the length of the resonator neck,
• V_c the volume of the resonator cavity
  • for a spherical cavity of radius r_c, V_c = 4/3 × πr_c³.

Helmholtz resonator can be considered as a mass-spring system with a resonance frequency computed as:

\[ f = \frac{1}{2\pi} \sqrt{\frac{k}{m}} \]

where m is a mass and 𝑘 a stiffness.
The mass is composed of the particles inside the neck while the much larger number of particles inside the cavity constitutes the spring stiffness.

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