Sound Waves

Misconceptions

Key Concepts

Sound as a Longitudinal Wave

Sound is a mechanical longitudinal wave that propagates through compression and rarefaction of a medium.

Characteristics:

• •Requires a medium (solid, liquid, or gas)
• •Particles oscillate parallel to wave direction
• •Creates alternating high-pressure (compression) and low-pressure (rarefaction) regions
• •Human hearing range: 20 Hz to 20,000 Hz

Speed of Sound

The speed of sound depends on the medium's properties:

In air (approximate):

• •v = 343 m/s at 20C (68F)
• •Temperature dependence: v = 331 + 0.6T (where T is in Celsius)

In general:

• •v = sqrt(B/rho) for fluids (B = bulk modulus, rho = density)
• •v = sqrt(Y/rho) for solids (Y = Young's modulus)

Typical values:

• •Air (20C): 343 m/s
• •Water (25C): 1493 m/s
• •Steel: 5960 m/s
• •Human tissue: ~1540 m/s

Intensity and Decibels

Sound intensity is power per unit area: I = P/A (W/m^2)

Inverse square law: Intensity decreases with distance squared

• •I = P/(4pir^2) for a point source

Decibel scale (logarithmic):

• •beta = 10 * log10(I/I_0)
• •I_0 = 10^(-12) W/m^2 (threshold of hearing)
• •Each +10 dB = 10x intensity
• •Each +3 dB = 2x intensity (approximately)

Reference levels:

• •0 dB: Threshold of hearing
• •60 dB: Normal conversation
• •85 dB: Prolonged exposure may cause damage
• •120 dB: Threshold of pain

Doppler Effect

The apparent change in frequency when source and/or observer are moving.

General formula:

• •f' = f * (v +/- v_o) / (v -/+ v_s)

Convention:

• •Use + in numerator when observer moves toward source
• •Use - in numerator when observer moves away from source
• •Use - in denominator when source moves toward observer
• •Use + in denominator when source moves away from observer

Special cases:

• •Moving observer only: f' = f * (v + v_o) / v (approaching)
• •Moving source only: f' = f * v / (v - v_s) (approaching)

Beat Frequency

When two sound waves of slightly different frequencies interfere:

• •f_beat = |f_1 - f_2|
• •Creates periodic variation in loudness
• •Used for tuning instruments

Standing Waves in Pipes

Open pipe (open at both ends):

• •Antinodes at both ends
• •Harmonics: f_n = n * v / (2L), where n = 1, 2, 3, ...
• •All harmonics present
• •Wavelengths: lambda_n = 2L/n

Closed pipe (closed at one end):

• •Node at closed end, antinode at open end
• •Harmonics: f_n = n * v / (4L), where n = 1, 3, 5, ... (odd only)
• •Only odd harmonics present
• •Fundamental wavelength: lambda_1 = 4L

Resonance

A system oscillates with maximum amplitude when driven at its natural frequency.

Examples:

• •Musical instruments (strings, air columns)
• •Tuning forks
• •Breaking glass with voice

Equations

[1] v = 343 m/s (speed of sound in air at 20C)
[2] v = 331 + 0.6T (speed of sound in air, T in Celsius)
[3] beta = 10 * log10(I/I_0), where I_0 = 10^(-12) W/m^2 (decibel level)
[4] I = P/(4*pi*r^2) (intensity from point source)
[5] f' = f * (v +/- v_o) / (v -/+ v_s) (Doppler effect)
[6] f_beat = |f_1 - f_2| (beat frequency)
[7] f_n = n * v / (2L), n = 1, 2, 3, ... (open pipe harmonics)
[8] f_n = n * v / (4L), n = 1, 3, 5, ... (closed pipe harmonics)

Worked Examples

Example 1: Speed of Sound

Problem: What is the speed of sound in air at 35C?

Solution:

1. •Use the temperature formula: v = 331 + 0.6T
2. •v = 331 + 0.6(35) = 331 + 21 = 352 m/s

Example 2: Decibel Calculation

Problem: A sound has intensity 5.0 x 10^(-6) W/m^2. What is its decibel level?

Solution:

1. •Use decibel formula: beta = 10 * log10(I/I_0)
2. •beta = 10 * log10(5.0 x 10^(-6) / 10^(-12))
3. •beta = 10 * log10(5.0 x 10^6)
4. •beta = 10 * 6.7 = 67 dB

Example 3: Doppler Effect

Problem: A car horn emits 400 Hz. If the car approaches at 30 m/s, what frequency does a stationary observer hear? (v_sound = 343 m/s)

Solution:

1. •Source approaching, observer stationary
2. •f' = f * v / (v - v_s)
3. •f' = 400 * 343 / (343 - 30)
4. •f' = 400 * 343 / 313 = 438 Hz

Example 4: Standing Waves in Closed Pipe

Problem: A closed pipe is 0.5 m long. What are the first two resonant frequencies? (v = 343 m/s)

Solution:

1. •Closed pipe: only odd harmonics, f_n = n * v / (4L)
2. •First harmonic (n=1): f_1 = 1 * 343 / (4 * 0.5) = 343/2 = 171.5 Hz
3. •Third harmonic (n=3): f_3 = 3 * 343 / (4 * 0.5) = 3 * 171.5 = 514.5 Hz

Explanation Patterns

1. •Identify the type of problem: speed, intensity, Doppler, beats, or resonance
2. •Draw a diagram showing wave direction, motion of source/observer, or pipe configuration
3. •Establish the reference frame and sign conventions for Doppler problems
4. •Check if pipe is open or closed - this determines which harmonics exist
5. •Use appropriate formula and substitute values carefully
6. •Verify units and check if answer is physically reasonable

Common Problem Types

1. •Speed of sound: Calculate v at different temperatures or in different media
2. •Intensity/decibels: Convert between intensity (W/m^2) and decibel level, or use inverse square law
3. •Doppler effect: Find observed frequency with moving source and/or observer
4. •Beat frequency: Find the beat frequency or one of the original frequencies
5. •Open pipe resonance: Find harmonics with antinodes at both ends
6. •Closed pipe resonance: Find odd harmonics with node at closed end
7. •Wavelength in pipes: Relate wavelength to pipe length for standing waves