Lecture Notes - Waves on a String and Sound Waves

Wave Speed (vWAVE): The speed at which the disturbance propagates through a medium is given by the formula:
   $v_{ ext{WAVE}} = ext{λ} imes f$
    - Where:
      - λ (Wavelength): The distance between successive crests of the wave.
      - f (Frequency): The number of cycles per second.
      - Amplitude: Does not affect speed; it is solely dependent on the medium.
Properties of Sound Waves:
- The speed of sound at room temperature is approximately constant regardless of amplitude and frequency.
- Temperature Factor: The speed of sound varies depending on the medium's temperature (higher temperatures increase speed).

Molecules vs. Wave Speed: Molecules do not travel across the medium at the wave speed; they oscillate in place while the wave propagates.
Key Fact 1: The wave speed is determined solely by the properties of the medium, such as density and tension, rather than the wave's characteristics.

Universal Wave Equation Rearranged:
   $v_{ ext{WAVE}} = rac{T}{ ext{μ}}$
    - T: Tension in the string.
    - μ (Mass Density): Mass per unit length of the string.
  - How to control frequency:
    - Wiggling the end of a string faster.
    - Splashing water more rapidly.
    - Controlling vocal cord vibrations.
    - More generally, adjusting the oscillating source of the wave.

Key Parameters:
- Tension (T): Determines how strongly the molecules are pulling on each other.
- Mass Density (μ): Affects how slowly the mass accelerates under force. Higher mass density results in slower wave speed.
General Formula for Wave Speed on a String:
$v_{ ext{WAVE}} = rac{T}{μ}$
- Rearranged to: $v_{ ext{WAVE}} = rac{ ext{T}}{ ext{μ}}$

When a wave reaches a boundary, some or all of the wave is reflected.
Fixed End Reflection: When the wave is reflected from a fixed end, it inverts.
Free End Reflection: When reflected from a free end, the wave remains non-inverted.

Fixed End: Reflects inverted wave.
Free End: Reflects non-inverted wave.
Transition Between Different Densities: An incident wave from a lower density string to a higher density string will reflect and transmit differently:
- Reflected Wave: Inverted and smaller in amplitude.
- Transmitted Wave: Non-inverted and smaller in amplitude.

General Definition: Impedance quantifies the ability of a medium to respond to a driving force, similar to resistance in electrical circuits.
Key Concept:
  - A wave encountering a boundary with a mismatch in impedance will experience reflection.
    - Small impedance mismatch → Small reflection.
    - Large impedance mismatch → Large reflection.
Wave Reflection: Amplitude of reflection depends on impedance ( $Z$ ), calculated by:
$Z = rac{T}{μ}$

Mechanical waves transmit energy and momentum, not matter.
Disturbances result in oscillations of medium molecules without net displacement.
The speed of waves depends on the medium's properties (tension & density) independently of frequency & amplitude.
Waves follow the principle of superposition:
- Arbitrary waves can be decomposed into sinusoidal functions.
Impedance is affected by both the density and elasticity of the material.
- A significant mismatch leads to substantial reflection.

Type: Longitudinal waves composed of pressure variations.
Propagation Medium: Can travel through liquid, gas, or solid.
As with all mechanical waves, the speed depends on the properties of the material (e.g., air, water, and steel).
Speed Formula for Sound Waves:
   $v = rac{B}{ρ}$
    - Where:
      - B (Bulk Modulus): Measure of the material's resistance to compression.
      - ρ (Density): Mass density of the medium.

Sound speed is affected by the temperature via the formula:
   $v = 331 ext{ m/s} imes ext{ } ext{√} rac{T}{273K}$
    - Example Values:
      - Speed of sound in air at room temperature: 343 m/s
      - Speed of sound in water: 1480 m/s

The pitch perceived relates directly to the frequency of sound:
- Lower frequencies imply lower pitches; higher frequencies indicate higher pitches.
Example Frequencies:
  - Middle C (C4): 256 Hz
  - A0: 27.5 Hz
  - C8: 4186 Hz
  - Human Hearing Range: Typically from 20 Hz to 20 kHz, with variations across species and applications.

Each physical wave type exhibits unique properties and behaviors, governed by fundamental laws. The study of waves includes understanding their speed, reflections, impedance, and frequencies across various mediums, essential for grasping broader physical concepts in both sound and mechanical waves.