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Sound

Ch. 8Std 11

Easy Overview

When you pluck a guitar string, it vibrates, pushes the air around it, and those ripples in air pressure travel to your ear, where your brain interprets them as sound. But how do those vibrations travel? Why do some sounds hit a high note and others a low note? Why does a siren sound different when it is coming toward you versus moving away? And how does a hall like the Sydney Opera House make music sound so good? This chapter is the physics of sound - mechanical waves that travel through matter. You will learn about longitudinal and transverse waves, the wave equation, speed of sound in different media, the Doppler effect, beats, standing waves in pipes and strings, and acoustics. Sound is not just about music - it is about how we communicate, how bats navigate in the dark, how ultrasound scans create images, and how a glass can shatter from a singer's high note. By the end, you will never hear a sound the same way again.

Nature of Sound Waves

Sound is a mechanical wave - it needs a medium to travel through. Sound cannot travel through a vacuum. That is why there is no sound in space. Sound waves in air are longitudinal - air molecules vibrate parallel to the direction of wave propagation. Regions of compression (high pressure) and rarefaction (low pressure) travel through the air. A sound wave is characterized by its frequency (pitch), amplitude (loudness), and wavelength. The human hearing range is 20 Hz to 20,000 Hz. Below 20 Hz is infrasound. Above 20 kHz is ultrasound (used in medical imaging and bat echolocation). The speed of sound in air at 0 C is about 331 m/s, increasing by about 0.6 m/s for every 1 C rise.

Transverse and Longitudinal Waves

In transverse waves, particles vibrate perpendicular to wave travel - like a wave on a string. Light is a transverse wave. In longitudinal waves, particles vibrate parallel to wave travel - like sound in air. Transverse waves can be polarized (vibrations filtered to one plane), but longitudinal waves cannot. Sound in solids can be both longitudinal and transverse - solids can support shear stresses. Transverse sound waves in solids are generally slower than longitudinal. Earthquakes produce both types: P-waves (longitudinal, faster) and S-waves (transverse, slower). The time difference between their arrival helps locate the epicenter.

Speed of Sound in Different Media

Speed of sound depends on the medium's elasticity and density. In solids: v = sqrt(Y/rho) (Young's modulus Y, density rho). In fluids: v = sqrt(B/rho) (bulk modulus B). In gases: v = sqrt(gamma P/rho) = sqrt(gamma RT/M), where gamma = C_P/C_V (about 1.4 for air). Sound travels fastest in solids (steel: ~5000 m/s), slower in liquids (water: ~1480 m/s), slowest in gases (air: ~343 m/s at 20 C). Particles in solids are more tightly packed, transmitting disturbances faster. Temperature increases speed in gases. Humidity also affects it - moist air has lower density, so sound travels slightly faster in humid air.

Reflection of Sound - Echo and Reverberation

Sound reflects off surfaces. An echo is a distinct reflected sound arriving after the original sound stops. For a distinct echo, the reflecting surface must be at least 17 m away (the ear can distinguish two sounds 0.1 s apart). Echoes are used in sonar to detect objects underwater. Reverberation is the persistence of sound after the source stops, caused by multiple overlapping reflections. Too much reverberation makes sound muddy. Too little makes sound dead. Reverberation time is the time for sound intensity to drop by 60 dB. Optimal time: about 1-2 s for concert halls, about 0.5-1 s for speech.

Doppler Effect

The Doppler effect is the change in frequency of a wave due to relative motion between source and observer. When a source moves toward you, waves get compressed - you hear a higher frequency. When it moves away, waves get stretched - you hear a lower frequency. That is why an ambulance siren sounds higher as it approaches and lower as it moves away. The formula: f' = f(v +/- v_o)/(v -/+ v_s), where v is speed of sound, v_o is observer velocity, v_s is source velocity. Numerator: + if observer moves toward source. Denominator: - if source moves toward observer. Applications: radar speed guns, Doppler ultrasound (measuring blood flow), astronomy (redshift of galaxies).

Beats

Beats occur when two sound waves of slightly different frequencies interfere. The amplitude varies periodically - you hear a loud-soft-loud wobbling sound. Beat frequency = |f_1 - f_2|. If one tuning fork is 256 Hz and another is 260 Hz, you hear 4 beats per second. Beats are used by musicians to tune instruments - when two strings are perfectly in tune, beats disappear. The time between successive maxima is 1/f_beat. Beats result from superposition - at some instants waves add constructively (loud), at others destructively (soft). Above about 10-15 beats per second, the ear cannot follow individual pulses and hears a continuous rough sound.

Standing Waves in Strings

A string fixed at both ends vibrates in specific patterns called standing waves. Boundary conditions: displacement zero at both ends (nodes). Possible wavelengths: lambda_n = 2L/n, where n = 1, 2, 3, ... Frequencies: f_n = nv/(2L) = n f_1, where f_1 = v/(2L) is the fundamental. For a stretched string, wave speed v = sqrt(T/mu), where T is tension and mu is mass per unit length. So f_n = (n/(2L)) sqrt(T/mu). Higher tension = higher pitch. Shorter string = higher pitch. Lighter string = higher pitch. When you press a guitar string against a fret, you shorten the vibrating length, increasing frequency. First three modes: fundamental (n=1), first overtone (n=2), second overtone (n=3).

Standing Waves in Pipes

Pipes produce sound through standing waves of air columns. For a pipe open at both ends: both ends are displacement antinodes. All harmonics present: f_n = nv/(2L), n = 1, 2, 3, ... For a pipe closed at one end: closed end is displacement node, open end is antinode. Only odd harmonics: f_n = nv/(4L), n = 1, 3, 5, ... Closed pipes have a hollower sound because they are missing even harmonics. End correction: the antinode at the open end is slightly outside, so effective length = L + e, where e about 0.6R. Resonance occurs when driving frequency matches a natural frequency - maximum amplitude. That is how blowing across a bottle produces sound.

Forced Vibrations and Resonance

When a vibrating system is driven by an external periodic force, it oscillates at the driving frequency. If driving frequency matches natural frequency, resonance occurs - amplitude becomes very large. This is how an opera singer can shatter a glass. Resonance is why soldiers break step while crossing a bridge - if their marching frequency matches the bridge's natural frequency, the bridge could resonate and collapse. In musical instruments, resonance amplifies sound - the body of a guitar resonates, making the sound louder. The resonance tube experiment uses a vibrating tuning fork above a tube in water - adjusting the water level changes the air column length until resonance occurs (loud sound).

Loudness, Intensity, and Decibels

Intensity I = power/area = (1/2) rho omega^2 A^2 v. Proportional to square of amplitude and frequency. The human ear can detect an enormous range - from 10^-12 W/m^2 (threshold of hearing) to about 1 W/m^2 (threshold of pain) - a factor of 10^12. We use a logarithmic scale: the decibel (dB). Sound level beta = 10 log(I/I_0), where I_0 = 10^-12 W/m^2. Whisper: about 20 dB. Normal conversation: about 60 dB. Rock concert: about 110-120 dB. An increase of 10 dB means a factor of 10 in intensity, which sounds about twice as loud. Prolonged exposure above 85 dB can cause hearing damage.

Quality of Sound - Timbre and Pitch

Pitch depends on frequency - higher frequency = higher pitch. Loudness depends on amplitude - larger amplitude = louder sound. But two instruments playing the same note at the same loudness still sound different. That is timbre (quality) - determined by harmonic content. A violin and a piano both playing middle A (440 Hz) sound different because they produce different combinations of harmonics (overtones). The fundamental frequency gives the pitch, but the relative strengths of higher harmonics create the unique sound signature. A pure tone has only the fundamental. Most musical sounds are complex waves with many harmonics.

Key Points

  • •Sound is a mechanical longitudinal wave - requires a medium. Cannot travel through vacuum.
  • •Speed of sound in air at 0 C: 331 m/s. Increases 0.6 m/s per C rise.
  • •Speed in solids: v = sqrt(Y/rho), liquids: v = sqrt(B/rho), gases: v = sqrt(gamma RT/M).
  • •Human hearing: 20 Hz - 20 kHz. Infrasound < 20 Hz, Ultrasound > 20 kHz.
  • •Echo: reflected sound > 0.1 s after original. Minimum distance: ~17 m.
  • •Doppler effect: f' = f(v +/- v_o)/(v -/+ v_s). Approaching source ? higher frequency.
  • •Beat frequency: f_beat = |f_1 - f_2|. Used for tuning instruments.
  • •String fixed both ends: f_n = nv/(2L). v = sqrt(T/mu).
  • •Open pipe: all harmonics f_n = nv/(2L). Closed pipe: odd harmonics f_n = nv/(4L).
  • •Resonance: driving frequency = natural frequency ? maximum amplitude.
  • •Sound level: beta = 10 log(I/I_0) dB. I_0 = 10^-12 W/m^2. Pain threshold: ~120 dB.
  • •Timbre depends on harmonic content. Same pitch, different instruments ? different timbre.
  • •End correction for pipes: effective length = actual + 0.6R.
  • •Reverberation time: time for intensity to drop 60 dB after source stops.

Practice Questions

  • State Newton's formula for speed of sound in air. What correction did Laplace make?
  • Derive expression for Doppler effect when source moves toward stationary observer.
  • A string of length 1 m, mass 0.5 g, tension 80 N. Find fundamental frequency and first two overtones.
  • What are beats? Prove beat frequency equals difference of frequencies of two waves.
  • An open pipe of length 50 cm produces 340 Hz. Find speed of sound. If closed, what is fundamental frequency?
  • Explain resonance with resonance tube experiment. How is speed of sound determined?
  • A train at 30 m/s sounds its whistle at 500 Hz. Find apparent frequency heard by stationary observer (a) approaching (b) receding. (v = 340 m/s)
  • Define intensity and sound level. A sound has intensity 10^-4 W/m^2. Find dB level. What if intensity doubles?