Sound waves and acoustics

Sound as a mechanical wave

Sound is a mechanical wave, meaning it requires a material medium. In air, sound travels as a longitudinal pressure wave. Air molecules oscillate back and forth, producing traveling regions of compression and rarefaction.

A compression is a region of higher pressure and density. A rarefaction is a region of lower pressure and density. The disturbance travels, but individual air molecules only oscillate around equilibrium positions.

Sound speed

The speed of sound depends on the medium. In a fluid, a useful expression is

ho}}$$ where $B$ is bulk modulus and $ ho$ is density. A medium with greater stiffness transmits sound faster; a medium with greater inertia transmits sound more slowly. In air near room temperature, $$vapprox 343 m/s$$ Sound travels faster in water and faster still in many solids. ## Frequency and pitch Pitch is the perception of frequency. A higher-frequency sound is heard as a higher pitch. Human hearing typically ranges from about $$20 Hz$$ to $$20,000 Hz$$ though the upper limit often decreases with age. Frequencies above human hearing are called ultrasound. Frequencies below human hearing are called infrasound. ## Amplitude and loudness Loudness is related to sound intensity and amplitude, but perception is not perfectly linear. A larger pressure amplitude generally corresponds to a louder sound. A simple pressure wave can be written $$Delta P(x,t)=Delta P_{max}cos(kx-omega t)$$ The pressure amplitude $Delta P_{max}$ is related to intensity. For sinusoidal sound waves, intensity is proportional to the square of pressure amplitude. ## Timbre Timbre is the quality of a sound that lets us distinguish instruments or voices with the same pitch and loudness. Timbre depends on harmonic content, attack, decay, and resonance characteristics. A flute and violin playing the same note have the same fundamental frequency, but their waveforms and spectra differ. Fourier analysis helps explain timbre by decomposing sound into harmonic components. ## Acoustics of rooms Acoustics is the study of sound behavior in spaces. Sound reflects from walls, floors, ceilings, and objects. Reflections can create echoes, reverberation, standing waves, and interference patterns. Reverberation is the persistence of sound due to many reflections. Some reverberation makes music sound rich, but too much can reduce speech clarity. Acoustic design balances reflection, absorption, diffusion, and room geometry. ## Resonance in sound Sound systems often involve resonance. A guitar body resonates with string vibrations. An organ pipe resonates at standing wave frequencies. The human vocal tract shapes sound by resonating at certain frequencies called formants. Resonance can amplify selected frequencies and shape the character of sound. ## Hearing The ear converts pressure waves into nerve signals. Sound vibrates the eardrum, tiny bones transmit the motion, and fluid waves in the cochlea stimulate hair cells. Different locations in the cochlea respond most strongly to different frequencies. This biological frequency analysis is one reason humans can distinguish pitch and timbre. ## The big idea Sound is a longitudinal mechanical wave involving pressure, density, and particle motion. Its speed depends on medium properties, frequency relates to pitch, amplitude and intensity relate to loudness, and harmonic content shapes timbre. Acoustics applies these ideas to rooms, instruments, speech, and hearing.