1-3

Sound is a vibration that travels through a medium (such as air, water, or solids) as a pressure wave, typically perceived by the ear. It is caused by the rapid back-and-forth movement of particles in the medium, creating areas of compression and rarefaction.

Longitudinal vs Spherical waves.

A logitudinal wave can be considered a plane wave. For a plane wave the values of the wave in any plane perpendicular to some direction (in this case, perpendicular to the direction of travel) at any moment in time are equal.

At a sufficient distance from the source, we are in the far field and can approximate a spherical wave as a plane wave.

Pressure

Static pressure

(baseline pressure without sound)

Pressure of a wave

(alternating between high (compression) and low (rarefaction) values.)

Speed of propagation

The speed of propagation is the rate at which sound waves travel through a medium, determined by the medium’s properties (e.g., density and elasticity).

c = (K /ρ) 1/2 K being the bulk modulus of the fluid

Particle velocity

Particle velocity describes the motion of particles in the medium as they vibrate back and forth due to a sound wave. It differs from the speed of sound, which refers to the wave's propagation speed.

Energy and intensity

Energy in a sound wave is the energy carried by its oscillations.

Intensity is the power per unit area carried by the wave, measured in watts per square meter (W/m²). It indicates how loud a sound is.

Wavenumber:

• The wavenumber (k) represents the number of wave cycles per unit distance. It is the spatial equivalent of angular frequency in time.

• Formula: k=2π / λ (rad per m)

Spatial Frequency:

• The spatial frequency ξ indicates how many wave cycles occur per unit distance, similar to wavenumber but expressed in cycles instead of radians.

• Formula: f=1 / λ (cycles per meter)

1. Acoustic impedance is a property of a medium that describes the relationship between the pressure of a sound wave and the particle velocity it induces.

2. It determines how much sound energy is transmitted, reflected, or absorbed when sound waves encounter a boundary between two different media.

3. Z=ρc

   
   

When sound waves encounter a boundary between two media with different acoustic impedances, part of the wave is reflected back, and part is transmitted into the second medium. The transmitted wave often changes direction (refraction), and the division of energy between reflection and transmission depends on the impedance mismatch.

1. Refraction occurs when the transmitted wave changes direction at the boundary due to differing sound speeds in the two media.

2. The angle of refraction is determined by Snell’s Law​​

1. The pressure of the reflected and transmitted waves depends on the acoustic impedances of the two media:

   • Reflection coefficient (R) (fraction of wave reflected)

   • Transmission coefficient (T) (fraction of wave transmitted)

2. A larger impedance mismatch results in higher reflection and less transmission.

1. hese relationships assume:

   1. Plane waves: The wavefront is flat.

   2. Perfect boundary: The transition between media is smooth and well-defined.

   3. Fluid/Fluid Interface

2. Deviations (e.g., rough boundaries or high-intensity waves) can alter reflection, refraction, and energy transfer.