Describe the production of sound by vibrating sources

3.4 Sound – Production of Sound by Vibrating Sources

1. What is a vibration?

• A rapid back‑and‑forth motion of an object about its equilibrium position.
• The motion repeats at regular intervals and can be described mathematically by

  
  x(t)=A\sin(2\pi ft)

  where x = displacement, A = amplitude, f = frequency (Hz).

2. How a vibrating source produces sound

• The vibrating object repeatedly pushes and pulls on the surrounding medium (air, water or solid).
• This disturbance creates alternating compressions (regions of higher pressure) and rarefactions (lower pressure) which travel away from the source.
• Because the particles of the medium move parallel to the direction of wave travel, sound is a longitudinal wave.
• The wave propagates at the speed of sound in that medium and, when it reaches a detector (e.g., the human ear), the periodic pressure changes are interpreted as sound.

3. Key concepts, formulas and their relevance

4. Typical vibrating sources

5. Example calculations

5.1 Fundamental frequency of a stretched string

For a string of length L, tension T and linear mass density μ, the fundamental (first harmonic) frequency is

f₁ = \frac{1}{2L}\sqrt{\frac{T}{\mu}}

Given L = 0.65 m, T = 80 N, μ = 0.002 kg m⁻¹:

f₁ = \frac{1}{2(0.65)}\sqrt{\frac{80}{0.002}} ≈ 138 Hz

Interpretation: the string will produce a pitch of about 138 Hz – a low‑to‑mid musical note.

5.2 Wavelength of a 500 Hz tone in air

Using λ = v / f with v ≈ 340 m s⁻¹:

λ = 340 / 500 = 0.68 m

The distance between successive compressions is 0.68 m.

5.3 Echo time‑distance method (speed of sound)

Place a flat wall a measured distance d from a loudspeaker. Emit a short pulse and record the time between the emitted pulse and the reflected pulse (the echo) using a stopwatch or a digital timer.

Speed of sound: v = 2d / Δt (the factor 2 accounts for the round‑trip).

Example: d = 5 m, measured echo time Δt = 0.029 s → v = 2×5 / 0.029 ≈ 345 m s⁻¹.

6. Factors that affect the sound produced

• Tension – increasing tension raises the frequency (higher pitch).
• Length – shortening the vibrating length raises the frequency.
• Mass per unit length (μ) – a heavier string lowers the frequency.
• Amplitude – larger amplitude gives a louder sound (intensity ∝ A²).
• Medium – the speed of sound is larger in solids than liquids than gases because v = √(elastic modulus / density). Higher elastic modulus and lower density give a higher speed.
• Temperature (gases) – higher temperature increases the speed of sound (≈ 0.6 m s⁻¹ per °C), slightly raising pitch.

7. Supplementary topics required by the Cambridge syllabus

7.1 Echo

• An echo is the reflection of a sound wave from a surface back to the source.
• For an echo to be heard distinctly, the reflecting surface must be at least 17 m away (since the human ear needs a minimum delay of ~0.05 s).
• Echoes are used in sonar, medical imaging and architectural acoustics.

7.2 Ultrasound and its applications

• Ultrasound = sound with frequency > 20 kHz (outside the audible range).
• Medical imaging (e.g., obstetric scans) – high‑frequency waves give fine resolution but limited penetration.
• Non‑destructive testing (NDT) – detecting cracks or voids in metals.
• Sonar – navigation and ranging for ships and submarines.

7.3 Why sound travels fastest in solids

• In solids the particles are tightly bound, giving a large Young’s modulus (E) and relatively low density (ρ) compared with liquids and gases.
• Since v = √(E/ρ) for solids, v is large. For liquids v = √(B/ρ) with a smaller bulk modulus, and for gases both B and ρ are much smaller, giving the lowest speed.

8. Summary

Sound is generated when a source vibrates, producing longitudinal pressure waves that travel through a medium. The pitch of the sound is set by the vibration frequency, which depends on tension, length, mass, and the elastic properties of the source and the medium. Loudness is controlled by the amplitude of vibration (intensity ∝ A²). The speed of sound varies with the medium because it depends on the ratio of its elastic modulus to its density, leading to the order solid > liquid > gas. Echoes arise from reflection, and ultrasound (> 20 kHz) finds important uses in medicine, industry and sonar.