Acoustics is the science of sound, including its production, transmission, and effects.

This interactive tome represents the collaborative effort of an entire swarm of specialized agents. We have orchestrated experts in mathematics, user interface design, and acoustical engineering to bring you the equations that govern the universe of sound.

Turn the page to begin the journey into the fundamental equations.

The fundamental equation of sound wave propagation in three dimensions. It relates the spatial variation of sound pressure (p) to its temporal variation, governed by the speed of sound (c).

Where ∇² is the Laplacian operator:
∇² = ∂²/∂x² + ∂²/∂y² + ∂²/∂z²

This implies that the acceleration of the particle is proportional to the pressure gradient.

In a plane wave, the sound pressure (p) and particle velocity (u) are everywhere in phase. The ratio of pressure to velocity is the specific acoustic resistance (ρc) of the medium.

Pressure (Solid) and Velocity (Dashed) in Phase

Unlike plane waves, in a spherical wave expanding from a point source, the particle velocity ($u$) leads the pressure ($p$) by an angle that approaches 90° near the source, and 0° far from the source.

Notice the phase shift between P and U near the source.

A point source is a radiator whose dimensions are small compared to the wavelength. It radiates sound equally in all directions, yielding a spherical directional pattern.

A dipole consists of two point sources separated by a distance d, vibrating in opposite phase. It exhibits a bidirectional directivity pattern.

A continuous line source of length L produces a highly directional pattern perpendicular to its axis. It is the foundation of line array theory.

By applying a progressive phase shift (δ) along the length of a line source, the main radiation lobe can be electronically "steered" without physically moving the array.

A circular baffled piston of radius R approximates a typical loudspeaker cone. Its directivity depends on the Bessel function of the first kind ($J_1$).

The fundamental frequency of a stretched string depends on its length (l), tension (T), and mass per unit length (m). It is the basis of all stringed instruments.

Fundamental Frequency: 15.8 Hz

A bar clamped at one end (cantilever) vibrates transversely. The restoring force is due to stiffness, governed by Young's modulus (Q) and the radius of gyration (K).

Animation showing transverse cantilever vibration.

A stretched membrane (like a drumhead or condenser microphone diaphragm) vibrates in 2D modes. Its restoring force is tension.
Tap the membrane below to strike it!

Olson's masterpiece: unifying physics. A series resonant circuit behaves identically across domains. Toggle the domain below to see how terms change while the math remains exactly the same!

The conical horn is the simplest expanding waveguide. Its acoustic impedance behaves similarly to a pulsating sphere, exhibiting poor low-frequency resistance compared to exponential profiles.

Acoustic Resistance (R) & Reactance (X)

The exponential horn is characterized by a "cutoff frequency" (f_c). Below f_c, the throat resistance is zero (pure reactance). Above f_c, it acts as a highly efficient acoustic transformer.

Notice the high-pass filter behavior at f_c.

Unlike the infinite horn, a finite horn has acoustic reflections at the mouth. This creates standing waves (ripples in throat impedance) which cause uneven frequency response.

Without a baffle, the rear wave cancels the front wave at low frequencies (acoustic short circuit). A finite baffle pushes this cancellation to lower frequencies.

A ribbon microphone operates on pressure gradient, giving it a natural Figure-8 (bidirectional) polar pattern, following cos(θ).

Sabine's equation for T60 is the cornerstone of architectural acoustics.

Fletcher-Munson equal-loudness contours show the non-linear frequency response of human hearing.