[Acoustics / Theory] Fundamental Concepts

[Characteristics of fluids]

• Fluids: gases, liquids, etc.
• Lack of constraints to deformation
• Unable to transmit shearing forces
• React against a change of shape only because of inertia
• React against a change in volume with a change of the pressure

[Sound waves]

• Compressional oscillatory disturbances that propagate in a fluid
• Involve molecular of the fluid moving back and forth in the direction of propagation
• Accompanied by changes in the pressure, density and temperature
• Longitudinal waves (AKA transversal waves): particles move back and forth in a direction perpendicular to the direction of propagation

[Sound pressure]

• Difference between the instantaneous value of the total pressure and the static pressure

[Basic assumptions]

• Generally oscillatory undergone by the fluid are extremely small

• For example, at the level of 120dB (very high sound pressure level)

• Fractional pressure variations: 2*10^-4

• Fractional change of the density of the particle: 1.4*10^-4

• Oscillatory change of the temperature < 0.02℃

• Particle velocity: 50 mm/s (1000Hz)

• Particle displacement < 8 ㎛

[Some phenomena of sound waves]

• Reflection by rigid walls

• Absorption

• Scattering

• Refraction

[Mathematical description of sound waves]

• Mass conservation

• Force balance: Local longitudinal force caused by a difference in the local pressure balanced by the inertia of the medium

• Nearly adiabatic (no heat flow)

• Linearization: Neglecting higher order terms

• At very high sound pressure levels (> 140 dB) the linear approximation is no longer valid

[Linearized wave equation]

Linearized partial differential wave equation

in tensor expression:

or

in cartesian coordinate.

where

p: pressure

t: time

c: speed of sound, 

Ks: adiabatic bulk modulus, 

γ: ratio of the specific heat at constant pressure to that at constant volume (≒1.401 for air)

p₀: static pressure (101.3[kPa] for normal ambient conditions)

ρ: equilibrium density of the medium

Equilibrium density of the medium:

where

R: gas constant (≒287[J/(kg·K)])

which shows

Note:

, 

[Linearity]

•  and  are linear operators

• Sinusoidal source will generate a sound field: Pressure at all positions varies sinusoidally

• Linear superposition: Sound waves do not interact with each other

[Boundary conditions]

• Information of reflection / absorbtion / scatter, etc. on surfaces

• Expressed in terms of the particle velocity

• For example,  on a rigid surface: Normal component of the particle velocity is zero on a rigid surface

• Euler equation of motion (Newton's second law of motion for a fluid)