Sound Waves

Overview

Sound is a longitudinal mechanical wave that propagates through a medium by compression and rarefaction of the medium's particles.

Nature of Sound

Characteristics

• Longitudinal wave (particle motion parallel to wave direction)
• Requires a medium (cannot travel in vacuum)
• Produced by vibrating objects
• Detected by the ear or microphones

Speed of Sound

In air (at temperature $T$ in Kelvin):

$$v = 331 \sqrt{\frac{T}{273}} \text{ m/s}$$

At 20°C (293 K):

$$v \approx 343 \text{ m/s}$$

General formula:

$$v = \sqrt{\frac{B}{\rho}}$$

Where $B$ = bulk modulus, $\rho$ = density

Medium	Speed (m/s)
Air (20°C)	343
Water	1480
Steel	5960

Properties of Sound

Frequency and Pitch

• Higher frequency → higher pitch
• Human hearing: 20 Hz to 20,000 Hz
• Infrasound: < 20 Hz
• Ultrasound: > 20,000 Hz

Amplitude and Loudness

• Larger amplitude → louder sound
• Related to intensity and pressure variations

Intensity and Decibels

Sound Intensity

$$I = \frac{P}{A}$$

Unit: W/m²

Intensity Level (Decibels)

$$\beta = 10 \log_{10}\left(\frac{I}{I_0}\right)$$

Where $I_0 = 10^{-12}$ W/m² (threshold of hearing)

Sound	Intensity (W/m²)	Level (dB)
Threshold of hearing	$10^{-12}$	0
Whisper	$10^{-10}$	20
Normal conversation	$10^{-6}$	60
Rock concert	$10^{-1}$	110
Pain threshold	1	120

Properties of Decibels

• 10 dB increase = 10× intensity
• 20 dB increase = 100× intensity
• Doubling intensity adds 3 dB

Resonance and Standing Waves

Open Pipe (Open at Both Ends)

Antinodes at both ends

Wavelengths:

$$\lambda_n = \frac{2L}{n} \quad (n = 1, 2, 3, ...)$$

Frequencies:

$$f_n = \frac{nv}{2L} = nf_1$$

All harmonics present

Closed Pipe (Closed at One End)

Node at closed end, antinode at open end

Wavelengths:

$$\lambda_n = \frac{4L}{n} \quad (n = 1, 3, 5, ...)$$

Frequencies:

$$f_n = \frac{nv}{4L} \quad (n = 1, 3, 5, ...)$$

Only odd harmonics present

Beats

When two waves of slightly different frequencies interfere:

$$f_{\text{beat}} = \lvert f_1 - f_2 \rvert$$

• Heard as periodic variation in loudness
• Used for tuning instruments

Interference of Sound

Constructive Interference

• Path difference = $n\lambda$
• Loud sound

Destructive Interference

• Path difference = $(n + \frac{1}{2})\lambda$
• Quiet or no sound

Examples

Example 1: Speed of Sound

Find speed of sound at 30°C.

$$v = 331\sqrt{\frac{303}{273}} = 331 \times 1.054 = 349 \text{ m/s}$$

Example 2: Wavelength of Sound

A 440 Hz tuning fork in air at 20°C. Find wavelength.

$$\lambda = \frac{v}{f} = \frac{343}{440} = 0.78 \text{ m}$$

Example 3: Open Pipe Harmonics

An open pipe is 0.5 m long. Find fundamental and first two harmonics.

$$f_1 = \frac{v}{2L} = \frac{343}{2 \times 0.5} = 343 \text{ Hz}$$ $$f_2 = 2f_1 = 686 \text{ Hz}$$ $$f_3 = 3f_1 = 1029 \text{ Hz}$$

Example 4: Closed Pipe

A closed pipe produces fundamental at 256 Hz. Find length.

$$f_1 = \frac{v}{4L}$$ $$L = \frac{v}{4f_1} = \frac{343}{4 \times 256} = 0.335 \text{ m}$$

Next harmonic (3rd):

$$f_3 = 3f_1 = 768 \text{ Hz}$$

Example 5: Decibel Calculation

Sound intensity is $10^{-5}$ W/m². Find decibel level.

$$\beta = 10 \log\left(\frac{10^{-5}}{10^{-12}}\right) = 10 \log(10^7) = 70 \text{ dB}$$

Example 6: Beats

Two tuning forks: 440 Hz and 443 Hz. Find beat frequency.

$$f_{\text{beat}} = \lvert 443 - 440 \rvert = 3 \text{ Hz}$$

(3 beats per second heard)

Example 7: Combined Sound Levels

Two sounds each at 60 dB. Find combined level.

Two equal intensities: $I_{\text{total}} = 2I$

$$\Delta\beta = 10 \log(2) \approx 3 \text{ dB}$$

Combined level ≈ 63 dB

Applications

• Musical instruments: Standing waves in strings and pipes
• Medical ultrasound: High-frequency sound for imaging
• Sonar: Using sound reflection for detection
• Noise cancellation: Destructive interference