Comprehensive Study Notes on the Physical Fundamentals and Propagation of Sound

Sound exists simultaneously as a physical phenomenon and as the subjective perception of that physical phenomenon.
From a physical perspective, sound is defined as the compression and relaxation of molecules within an elastic medium.
The process of molecular relaxation during sound propagation is technically termed rarefaction.
An elastic medium is defined as any substance capable of having its molecular composition compressed and expanded.
Media capable of transmitting sound encompass almost all solids, liquids, and gases.
While sound is most commonly experienced traveling through air, it also propagates through liquids such as water and dense solids such as metal and wood.
Sound travels differently depending on the specific medium through which it propagates.

Sound propagation does not involve the physical translocation of molecules across space from one location to another.
Individual molecules vibrate back and forth in place; it is the disturbance created by this vibration that moves through space.
Wave propagation can be understood through the visual metaphor of a field of tall grass:
- When wind blows across a field of tall grass, it initiates a swaying motion at the top of the grass blades.
- Each individual blade of grass sways and causes the adjacent blade to sway, passing the kinetic disturbance along.
- After swaying, each blade relaxes back to its starting position and can repeat the cycle if sufficient energy continues to act upon it.
- Given adequate energy, this disturbance travels across the entire length of the field.
- The individual grass blades do not move across the field; each blade remains stationary at its point of origin, simply vibrating back and forth.

Sounds intuitively decrease in perceived loudness as an observer moves further away from the source.
A theoretical environment used to analyze sound propagation without reflection is known as a free field:
- A free field is a hypothetical, completely open space devoid of all physical boundaries or objects, including floors, ceilings, walls, ground, trees, mountains, or oceans.
- Because there are no obstacles or reflective surfaces in a free field, sound cannot bounce or reflect, causing sound waves to travel strictly in straight lines.
- Sound emitted from a point source in a free field radiates outward in all directions simultaneously, forming an expanding sphere of energy centered on the source.
As the imaginary sphere of sound energy expands, its total energy must cover an increasingly large surface area.
Consequently, sound intensity at any single point on the sphere decreases as the surface area of the expanding sphere increases.
The geometric formula for the surface area of a sphere is: $A = 4\pi r^2$ where $r$ represents the radius of the sphere, which corresponds to the distance from the sound source to the observer.

Sound energy falls off as a function of the square of the distance (radius $r$ ) from the sound source.
Even in real-world environments containing physical boundaries like floors and walls, sound intensity fundamentally decreases as a function of the square of the distance from the source.
The specific mathematical relationship governing sound attenuation with distance is known as the Inverse Square Law: $\text{Intensity} \propto \frac{1}{r^2}$
Numerical proportional relationships governed by the Inverse Square Law:
- Moving to $2\,\text{times}$ the original distance from a sound source reduces the perceived sound intensity to $\frac{1}{4}$ (one-quarter) of its initial value.
- Moving to $3\,\text{times}$ the original distance reduces the perceived sound intensity to $\frac{1}{9}$ (one-ninth) of its initial value.
- Subsequent increases in distance continue to attenuate sound according to the reciprocal of the distance squared ( $\frac{1}{r^2}$ ).

In artistic and philosophical contexts, such as the musical piece The Sinking of the Titanic, concepts exist proposing that sounds never completely disappear, but rather become infinitely fainter over time.
If sound persisted infinitely without decay limits, every sound or spoken word ever produced would continue to exist indefinitely in the environment.
Physically, sounds do not persist infinitely; sound intensity eventually decays below the threshold level of Brownian motion.
Brownian motion is the continuous, random movement of microscopic particles present everywhere at all times.
Brownian motion ceases only when the temperature reaches absolute zero, defined as $-273\,^\circ\text{C}$ (minus $273\,\text{degrees Celsius}$ ).
At $-273\,^\circ\text{C}$ , all molecular motion completely stops.