Study Summary: Pitch, Loudness, Quality/Timbre, Intensity of Sound, Decibel

1. Pitch

• Definition: The pitch of a sound wave is determined by its frequency.
• Examples:
  • Female voices generally have a higher pitch than male voices.
  • A soprano sings at the highest pitch in a choir.

2. Loudness, Intensity, and the Decibel Scale

2.1 Loudness and Intensity

• Loudness: A factor of the amplitude of the sound wave.
• Intensity (I): Directly proportional to the square of its amplitude.
  • Doubling the amplitude makes the sound 4 times louder (intensity increases by a factor of 4).
  • Sound intensity is the power carried by sound waves per unit area.

2.2 The Bel and Decibel (dB)

• Bel (B): A logarithmic unit expressing the ratio between two levels of signal power, voltage, or current. Named after Alexander Graham Bell.
• Decibel (dB): One-tenth of a Bel (1 dB = 0.1 B).
  • More commonly used because the smallest difference in loudness detectable by the human ear is approximately 1 dB.
  • A nonlinear, logarithmic scale is appropriate because the ear is more sensitive to sounds of very small amplitude than very large amplitude.

2.3 Decibel Formula and Calculations

The loudness (L) in decibels is given by:

L = 10 log10 (I / I0)

• I: Sound intensity in Watts per square metre (W/m²).
• I₀: Reference intensity (threshold of human hearing).
• I₀ value: Approximately 10-12 W/m².

Examples:

1. Loudness at reference intensity (I = I₀):

   L = 10 log10 (I0 / I0) = 10 log10 (1) = 0 dB

   The loudness at the reference intensity is 0 dB.

2. Loudness for I = 10^-11 W/m²:

   L = 10 log10 (10-11 / 10-12) = 10 log10 (10) = 10 dB

3. Increase from 10^-1 W/m² to 1 W/m²:

   Loudness at I1 = 10-1 W/m²:

   L1 = 10 log10 (10-1 / 10-12) = 10 log10 (1011) = 110 dB

   Loudness at I2 = 1 W/m²:

   L2 = 10 log10 (1 / 10-12) = 10 log10 (1012) = 120 dB

   The increase in loudness is L2 - L1 = 120 dB - 110 dB = 10 dB.

   Alternatively, the increase can be calculated as:

   Lincrease = 10 log10 (I2 / I1) = 10 log10 (1 / 10-1) = 10 log10 (10) = 10 dB

2.4 Other Units of Sound Intensity

Other dimensionless units include the neper (np) and the bel (B):

• Neper (np): L = (1/2) ln (I / I0) np
• Bel (B): L = log10 (I / I0) B

Conversion Factors:

• 1 bel = (ln 10) / 2 np ≈ 1.151 np (Note: The document states `1 bel = 1/2 ln 10 np` which seems to be a typo given the example, it should be `1 bel = 2 / ln 10 np` or more commonly, `1 np = 10 / ln 10 dB ≈ 4.343 dB`)
• 1 decibel (dB) = (1/20) ln 10 np ≈ 0.115 np (Correct from calculation: (1/20) * 2.3026 = 0.115 np)

Example: Convert 10 decibels to neper

Given 1 dB = 0.115 np

10 dB = 10 * 0.115 np = 1.15 np

Therefore, 10 decibel = 1.15 np = 1 bel.

3. Variation of Intensity of Sound with Distance

• Inverse-Square Law: The intensity of a sound wave falls with distance in an inverse-square manner.
  • I ∝ 1/r², or I = k/r², where r is the distance from the source and k is a constant.
  • This is analogous to the behavior of electric fields or gravitational fields, where energy spreads out spherically from a point source.

Problem Example:

A sound source is at P. Calculate the ratio of the sound intensity at A, B, and C.

Given distances from P:

• Distance to A: √(6² + 4²) = √52 m
• Distance to B: √(3² + 4²) = √25 = 5 m
• Distance to C: 6 m (assuming C is 6m from P along the horizontal axis, and A, B are offset vertically by 4m from that line based on the diagram context, or if P is at (0,4) and C is at (-6,0) so distance is 6. Given the solution, it seems the distances from P are directly 5, 3, 9 in the OCR text explanation for the example solution, which contradicts the diagram given the diagram: A at (-6,4), B at (-3,4), C at (0,0) and P at (0,4). Let's follow the solution's implicit distances for the problem.)

Let's use the distances directly implied by the problem's solution: 5, 3, 9. Assuming these are direct distances `r_A=5`, `r_B=3`, `r_C=9` from source P.

IA = k / (5²)

IB = k / (3²)

IC = k / (9²)

Ratio IA : IB : IC = k/25 : k/9 : k/81

= 1/25 : 1/9 : 1/81

To get whole numbers, multiply by the least common multiple of denominators (25 * 9 * 81 = 18225):

= (18225/25) : (18225/9) : (18225/81)

= 729 : 2025 : 225

4. Quality/Timbre

• Definition: Depends on the number of overtones present in the sound and their relative intensities.
• Function: Enables differentiation between musical instruments or voices producing the same fundamental frequency.
• Example:
  • A tube open at one end produces only odd harmonics.
  • A tube open at both ends produces both odd and even harmonics.
  • The latter has a more complex waveform, resulting in a different timbre.