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Waves and Sound

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Educational

Main ideas and lessons from the lecture (Waves and Sound)

1) What waves are

  • A wave is a traveling disturbance that moves through space.
  • A wave transports energy from one location to another (the medium carries the energy as the disturbance propagates).
  • Example: slinky
    • Shaking one end creates repeated disturbances that move away from the hand.
    • The hand’s motion supplies energy, which is carried through the medium by the wave.

2) Two main categories of waves

A. Transverse waves

  • Definition (key feature): particles of the medium oscillate perpendicular to the direction the wave travels.
  • Examples
    • Slinky (shaken up and down): slinky particles move up/down while the disturbance travels along the slinky.
    • Rope: points on the rope move vertically while the wave propagates horizontally.
  • Interpretation
    • Crest/trough positions correspond to particle motion at those points.

B. Longitudinal waves

  • Definition (key feature): particles of the medium oscillate parallel to the direction the wave travels.
  • Example: slinky pushed/pulled
    • Produces alternating:
      • compression (closer particles)
      • stretched/rarefied regions (more spread out)
  • Real-life example
    • Sound waves are longitudinal waves.

3) Other wave relationships and mixed behavior

  • Water waves can combine both:
    • a longitudinal component
    • a transverse component (circular particle motion)

4) Periodic waves and wave quantities used to describe them

Conditions for periodic waves (identical repeated cycles)

A wave is periodic when:

  • Multiple disturbances are exactly identical
  • They are created with equal time intervals between disturbances

In that case, the wave is produced by simple harmonic motion of the source with:

  • same amplitude
  • same oscillation speed
  • same time per full oscillation

Quantities defined (waveform on 2D axes)

  1. Amplitude (A)

    • Maximum displacement from equilibrium.
    • Measured vertically (distance from the midline to crest or trough).

  2. Wavelength (λ)

    • Horizontal distance over which the wave pattern repeats.
    • Labeled with λ.

  3. Period (T)

    • Time for one complete cycle of the waveform.
    • Labeled with T (seconds).

  4. Frequency (f)

    • Number of cycles per second.

    • Relationship: [ f = \frac{1}{T} ]

    • Units: hertz (Hz) (inverse seconds)

5) Wave speed relationships

Core relationship between speed, wavelength, and frequency

  • Think of speed as distance/time:
    • distance ↔ wavelength
    • time ↔ period

  • [ v = \frac{\lambda}{T} ]

  • Using ( f = 1/T ), the commonly used form is: [ v = f\lambda ]

Example application: radio waves (electromagnetic, transverse)

  • Radio waves travel at the speed of light: [ c = 3 \times 10^8 \ \text{m/s} ]

  • Formula used: [ \lambda = \frac{v}{f} ]

  • Given frequencies and computed wavelengths (as transcribed):

    • AM: computed (\lambda = 244 \ \text{m})
    • FM: computed (\lambda = 3.26 \ \text{m})
  • Conclusion drawn:
    • Longer wavelengths (AM) make interference/noise more likely → noisier sound
    • Shorter wavelengths (FM) interfere less → cleaner sound

6) How wave speed depends on the medium (ropes/strings/solids)

Rope/string dependence

For waves on a rope/string, speed depends on:

  • tension
  • linear mass density (mass per unit length), (\mu)

Stated formula: [ v = \sqrt{\frac{T}{\mu}} ]

Guitar string example (transverse waves)

  • Given:
    • String lengths: 0.628 m
    • Masses:
      • High E string: 0.208 g
      • Low E string: 3.32 g
    • Tension on each: 226 N
  • Results (as given):
    • High E string speed: 826 m/s
    • Low E string speed: 207 m/s
  • Interpretation:
    • Lighter string → higher wave speed → higher pitch (higher frequency)
    • Heavier string → lower wave speed → lower pitch (lower frequency)

Sound waves section (nature of sound)

1) How sound propagates in gases

  • Sound is modeled as longitudinal waves in a gas.
  • Example: loudspeaker in a tube of air
    • Loudspeaker membrane moves forward → pushes air → condensation (higher density / higher pressure region)
    • Membrane moves backward → pulls air → rarefaction (lower density / lower pressure region)
  • Repeating motion produces alternating condensation and rarefaction regions moving through the gas.

2) Motion of the molecules vs the wave

  • Gas molecules oscillate back and forth (they don’t overall move away).
  • The wave pattern of compressions/rarefactions travels through the gas.
  • Collisions transfer energy through the medium.

3) Wavelength, period, frequency for sound

  • Wavelength (λ): distance between similar points in successive compressions (e.g., middle of one condensation to middle of the next).
  • Period (T): time for one wavelength to pass a point.
  • Use wavelength/period to determine frequency (as with waves generally).

4) Loudness and pressure amplitude

  • Loudness depends primarily on pressure amplitude:
    • Higher pressure in condensations → larger amplitude → louder sound
    • Lower pressure in rarefactions → smaller amplitude → quieter sound
  • Example analogy:
    • speaking normally vs shouting → shouting produces larger pressure amplitude → louder sound

5) Speed of sound dependence

  • Sound speed depends on:
    • material
    • temperature
  • General ranking:
    • Gas: lowest
    • Liquid: medium
    • Solid: highest
  • Temperature example values (given):
    • Air at 0°C: 331 m/s
    • Air at 20°C: 343 m/s
  • Standard assumption for problems (when unspecified):
    • speed of sound in air at 20°C = 343 m/s

6) Alternative formulas for sound speed (medium properties)

  • Sound speed in gases (ideal gas model): [ v = \sqrt{\frac{\gamma k T}{m}} ] where:

    • (\gamma = 5/3) for monoatomic gas
    • (\gamma = 7/5) for diatomic gas
    • (k = 1.38 \times 10^{-23}\ \text{J/K}) (Boltzmann constant)
    • (T) = temperature in kelvin
    • (m) = molecular mass

  • Sound speed in liquids: [ v = \sqrt{\frac{\text{bulk modulus}}{\text{density}}} ]

  • Sound speed in solids: [ v = \sqrt{\frac{\text{Young’s modulus}}{\text{density}}} ]

7) Recap of two methods to compute sound speed

  • From wave properties:
    • (v = f\lambda) (or equivalent relations using period/distance)
  • From material properties:
    • gas/liquid/solid formulas using moduli, density, temperature, molecular mass, etc.

Sound intensity and decibels (measurement concepts)

1) Sound power and intensity

  • Power of a sound wave = energy transported per second (watts).

  • Sound intensity (I):

    • power passing perpendicularly through a surface divided by its area: [ I = \frac{P}{A} ]

    • Units: W/m²

Numerical example given (Intensity)

  • Given:
    • Power: 12 × 10⁻⁵ W
    • Surface 1 area: 4 m²
    • Surface 2 area: 12 m²

  • Results:

    • [ I_1 = \frac{12 \times 10^{-5}}{4} = 3 \times 10^{-5}\ \text{W/m}^2 ]

    • [ I_2 = \frac{12 \times 10^{-5}}{12} = 1 \times 10^{-5}\ \text{W/m}^2 ]

2) Hearing thresholds

  • Audible frequency range: 20 Hz to 20,000 Hz

  • Threshold of hearing (~1000 Hz tone):

    • [ I_0 = 1 \times 10^{-12}\ \text{W/m}^2 ]

  • Below this: not detectable.

3) Intensity in 3D (spherical spreading)

  • For an isotropic source: [ I = \frac{P}{4\pi r^2} ] (power spread over the surface area of a sphere)

4) Decibels and intensity level (log scale)

  • Decibel compares intensities using a logarithmic scale.
  • Intensity level (\beta): [ \beta = 10\ \text{dB}\cdot \log_{10}\left(\frac{I}{I_0}\right) ] where (I_0 = 1 \times 10^{-12}\ \text{W/m}^2).

Examples stated:

  • Threshold of hearing: (\beta = 0\ \text{dB})
  • Normal conversation: (\sim 3.2 \times 10^{-6}\ \text{W/m}^2) → (\beta \approx 65)
  • Threshold of pain: (I = 10\ \text{W/m}^2) → (\beta = 130\ \text{dB})

Intensity ratio example (two systems)

  • Given:
    • System 1: (\beta_1 = 90\ \text{dB})
    • System 2: (\beta_2 = 93\ \text{dB})

  • Computation shown:

    • (\beta_2 - \beta_1 = 3\ \text{dB})

    • [ 3 = 10\log_{10}\left(\frac{I_2}{I_1}\right)\Rightarrow 0.3 = \log_{10}\left(\frac{I_2}{I_1}\right) ]

    • [ \frac{I_2}{I_1} = 10^{0.3} \approx 2 ]

  • Conclusion:

    • System 2 intensity is twice System 1.

Doppler effect (change in perceived frequency)

1) Definition

  • Doppler effect: change in the frequency detected by an observer when the source and/or observer moves relative to the medium.

2) Qualitative behavior

  • Source moving toward observer:
    • wavelength decreases → detected frequency increases → higher pitch
  • Source moving away:
    • wavelength increases → detected frequency decreases → lower pitch
  • Example described:
    • Approaching ambulance/fire truck sounds have higher pitch
    • After passing, pitch drops

3) Doppler effect formulas

  • The lecture describes separate formulas for:
    • moving source, stationary observer
      • approaching: detected frequency increases (factor (>1))
      • receding: detected frequency decreases (factor (<1))

4) Example: train horn

  • Given:
    • Train speed: 44.7 m/s
    • Speed of sound: 343 m/s
    • Horn emitted frequency: 415 Hz
  • Results given:
    • Approaching: higher perceived frequency (\approx) “(415 \times \frac{1}{1-v_s/v_{\text{sound}}})” → computed as (4xx Hz) (partial transcript value)
    • Leaving: computed as 367 Hz

5) Doppler effect when observer moves

  • If the source is stationary and the observer moves:
    • moving toward the source → increased detected frequency
    • moving away → decreased detected frequency
  • The lecture emphasizes a general combined formula:
    • choose + or − based on whether source/observer move toward or away from each other
    • the goal is to match detected frequency to emitted frequency with correct sign conventions

Speakers / sources featured

  • Primary speaker: the video’s course instructor (unnamed in the transcript; uses phrases like “I will discuss,” “I defined,” “we discussed,” etc.).
  • No other speakers or external sources are explicitly identified in the subtitles.

Original video