Class 11 Physics — Chapter 302: Sound

Key concepts: Sound (Class 11 Physics)

Sound is a longitudinal wave — a train of compressions and rarefactions that needs a material medium to travel. This chapter studies how fast it moves and why. Newton's formula v = √(P/ρ) treated the process as isothermal and gave too low a value; Laplace corrected it by taking the changes to be adiabatic, giving v = √(γP/ρ), which agrees with experiment. The speed is raised by higher temperature (v ∝ √T) and by humidity, and lowered in a denser gas, while pressure alone has no effect. Reflection of sound gives echoes, reverberation and the sonar method of measuring distance. Using the principle of superposition, two waves of nearly equal frequency produce beats of frequency |f₁ − f₂|. Standing waves fix the notes of a stretched string (the sonometer and its laws) and of air columns in open and closed organ pipes, studied experimentally with the resonance tube. Finally the Doppler effect explains why the pitch of a moving source rises as it approaches and falls as it recedes.

Sound as a longitudinal wave

Sound propagates as a mechanical longitudinal wave in which particles of the medium oscillate back and forth along the direction of travel, creating alternate regions of compression (high pressure) and rarefaction (low pressure). It cannot travel through a vacuum.

Newton's formula and Laplace's correction

Newton assumed sound travels isothermally, giving v = √(P/ρ) ≈ 280 m/s, about 15% too low. Laplace argued the rapid compressions and rarefactions are adiabatic, so the bulk modulus is γP, giving v = √(γP/ρ) ≈ 332 m/s, which matches measurement.

Factors affecting the speed of sound

In a gas the speed rises with absolute temperature as v ∝ √T (about 0.61 m/s per °C in air) and increases with humidity because moist air is less dense. It is smaller in a denser gas (v ∝ 1/√ρ), while a change of pressure at constant temperature has no effect.

Echo

An echo is the distinct repetition of a sound heard after it reflects from a distant rigid surface. Because the ear separates two sounds only if they are at least about 0.1 s apart, the reflector must be at least roughly 17 m away (at 340 m/s).

Reverberation

Reverberation is the persistence of sound in a closed hall caused by repeated reflections from the walls, ceiling and floor after the source has stopped. Too much of it blurs speech, so auditoriums use soft, absorbing materials to shorten the reverberation time.

Principle of superposition

When two or more waves overlap, the resultant displacement at each point is the vector (algebraic) sum of the individual displacements. This single principle underlies interference, beats and the formation of standing waves in strings and air columns.

Beats

Beats are the periodic rise and fall in loudness heard when two notes of slightly different frequencies superpose. The beat frequency equals the difference of the two frequencies, |f₁ − f₂|; beats are used to tune musical instruments and to find an unknown frequency.

Vibrations of a stretched string

A string fixed at both ends supports standing waves whose fundamental is f = (1/2L)√(T/μ), and it can sound all harmonics f, 2f, 3f… The laws of vibrating strings state f ∝ 1/L, f ∝ √T and f ∝ 1/√μ, and are verified using a sonometer.

Vibrations of air columns

A pipe closed at one end has fundamental f = v/4L and sounds only odd harmonics (f, 3f, 5f…), while a pipe open at both ends has f = v/2L and sounds all harmonics. A small end correction (about 0.3 × diameter per open end) makes the effective length slightly longer.

Resonance and the resonance tube

Resonance occurs when an air column is driven at its natural frequency and vibrates with large amplitude. In the resonance-tube experiment a closed column resonates with a fork when its length equals λ/4, 3λ/4…; the two lengths give λ = 2(L₂ − L₁) and hence the speed of sound.

Doppler effect for sound

The Doppler effect is the apparent change of frequency due to relative motion between source and observer. Approach raises the pitch and recession lowers it, described by f' = f(v ± v_o)/(v ∓ v_s), where the signs are chosen so that approach increases f'.