Sound Waves Study Notes

Introduction to sound waves with a practical example: three musicians playing the alpenhorn in Valais, Switzerland.
The chapter deeply explores the behavior of sound waves, focusing on the characteristics and properties of the waves generated by these large musical instruments.

Illustration:
- Motion of a longitudinal pulse in a compressible gas illustrated in Figure 17.1.
- The darker region in the figure indicates the compression produced by a moving piston.
- A longitudinal wave propagating through a gas-filled tube is created by an oscillating piston.
Mathematical Representation of Sinusoidal Sound Waves:
- Position Variation:
  $s(x, t) = S_{max} ext{Cos}(kx - heta t)$
- Where:
  - $S_{max}$ = maximum displacement of the medium.
  - $k$ = wave number, related to the wavelength.
  - $ heta$ = angular frequency.
- Pressure Variation from Equilibrium Value:
  $riangle P = P_{max} ext{sin}(kx - heta t)$
- Where:
  - $P_{max}$ = pressure amplitude.
  - Indicates that the pressure variation wave is 90° out of phase with the displacement wave.
Relationship between Displacement Amplitude and Pressure Amplitude:
$P<em>{max} = ho v imes S</em>{max}$
- Where:
  - $
    ho$ = density of the medium.
  - $v$ = speed of sound in the medium.

Key Factors Influencing the Speed of Sound:
- Bulk Modulus (B):
- Describes the medium's resistance to compression, indicating its elastic properties.
- SI Unit of Bulk Modulus is Pascal (Pa).
- Density ($ ho$):
- The inertial property of the medium, indicating how mass is distributed.
- SI Unit of Density is kg/m³.
Dependence on Temperature:
- The speed of sound in gases generally increases with temperature. The relationship forms part of the equation defining the speed of sound:
  $v = ext{const} imes ext{T}^{1/2}$
- Where:
  - $v$ = speed of sound.
  - T = absolute temperature.

Graphical Representation:
- Figure 17.6 illustrates spherical waves emitted by a point source; circular arcs represent concentric spherical wave fronts.
- Area of Spherical Wave Front (A):
  $A = 4 imes ext{pie} imes r^2$
- Where:
  - $r$ = radius of the spherical wave front.
Intensity ($I$):
- Defined as the power per unit area carried by the wave. The SI unit of intensity is Watts per square meter (W/m²).
Sound Level in Decibels (dB):
- Reference intensity threshold:
- $I_0 = 1 imes 10^{-12} ext{W/m}^2$
- Corresponds to $eta = 0 ext{ dB}$ (threshold of hearing).
- Intensity at threshold of pain:
- $I = 1 ext{W/m}^2$
- Corresponds to $eta = 120 ext{ dB}$.

Phenomenon Description:
- The frequency of sound changes for an observer moving relative to a stationary sound source.
- Example: A cyclist moving towards a stationary point source (e.g., a truck's horn) perceives a higher frequency as they approach and a lower one as they recede.
Mathematical Expression for Frequency Change:
- Let:
- $f$ = original frequency of the source.
- $v$ = speed of sound waves.
- $v_o$ = speed of the observer.
- $v_s$ = speed of the source.
- General Formula:
  $f' = rac{f(v + v<em>o)}{(v - v</em>s)}$
- Conditions List:
  - $v_o o + W$ when the observer moves toward the source.
  - $v_o o - W$ when the observer moves away from the source.
  - $v_s o + W$ when the source moves toward the observer.
  - $v_s o - W$ when the source moves away from the observer.
**Classification of Waves:
- Infrasonic:** Frequencies below 20 Hz.
- Sonic: Frequencies between 20 Hz to 20 kHz.
- Ultrasonic: Frequencies above 20 kHz.