Hw. 4 notes

Homework #4: Graphs and Wave Concepts

1. Key Concepts in Wave Mechanics

• Amplitude (A): The maximum displacement of a wave from its rest position, measured in centimeters (cm).

• Wavelength (λ): The distance between two successive identical points on the wave, measured in centimeters (cm).

• Period (T): The time taken to produce one complete wave, with SI unit in seconds (s).

• Frequency (f): The number of complete waves produced in one second, with SI unit in hertz (Hz).

• Wave Speed (v): The distance travelled by a wave in one second, measured in meters per second (m/s).

2. Wave Equation

• Equation: The relationship between wave speed, frequency, and wavelength is given by the equation: v = fλ where:

  • v is the wave speed (m/s)

  • f is the frequency (Hz)

  • λ is the wavelength (m)

3. Example Calculations

• Reference data from above graphs:

  • Amplitude, A = 2.0 cm

  • Wavelength, λ = 4.0 cm

  • Period, T = 4.0 s

  • Frequency Calculation:

  • f = rac{1}{T} = rac{1}{4.0s} = 0.25 Hz

  • Velocity Calculation:

  • v = fλ = 0.25 Hz imes 4.0 cm = 1.0 cm/s

4. Wavefront Definition

• Wavefront: An imaginary line on a wave that connects all identical adjacent points that are in phase.

• The distance between two adjacent wavefronts is the wavelength (λ).

• The time taken for the wavefront to move to the next adjacent wavefront is known as the period (T).

5. Sound Waves and Standing Waves

5.1 Propagation of Sound

• Sound waves propagate through fluids, such as air. In this context:

  • Compression: Regions of high air density.

  • Rarefaction: Regions of low air density.

• The wavelength of a sound wave can be illustrated graphically by pressure variations.

5.2 Bulk Modulus

• Definition: The bulk modulus (B) characterizes how a material resists uniform changes in volume. It is defined as:
  B = rac{ ext{Change in pressure due to compression} ( riangle P)}{ ext{Volumetric Strain} ( rac{ riangle V}{V})}

• Measured in Pascals (Pa), which is the SI unit for pressure.

• Relation to properties:

  • B represents the elastic property of the material.

  • 
    ho (density) indicates the inertial property of the material.

5.3 Bulk Modulus Values for Common Materials

6. Speed of Sound in Various Materials

7. Doppler Effect for Sound Waves

7.1 Definitions and Formulae

• Observed Frequency (f'): The frequency perceived by the observer when there is relative motion between the source of the sound and the observer.

• Actual Frequency (f): The true frequency of the sound produced by the source.

• Formula: f' = rac{(V + V1)}{(V + V2)}f where:

  • V = velocity of sound waves

  • V_1 = velocity of the observer

  • V_2 = velocity of the source

7.2 Understanding the Doppler Effect

• Describes the change in perceived frequency due to relative motion.

• Key scenarios:

  • Static Source, Moving Observer: The frequency increases as the source moves closer and decreases as it moves away.

• The source and observer之间的相对运动改变了音波的波长及频率。

• Types of frequency shifts:

  • Lower frequency when the source moves away.

  • Higher frequency when the source approaches.

7.3 Detector Stationary - Source Moving

• If the detector is stationary while the source moves towards it, an increase in frequency is observed due to wave compression:
  f' = rac{V{sound}}{V{sound} - V_{source}} f

• Conversely, if the source moves away from the detector:
  f' = rac{V{sound}}{V{sound} + V_{source}} f

7.4 Detector and Source Moving

• If both the observer and the source are in motion, the source moving towards the observer:
  f' = rac{V{sound} + V{observer}}{V{sound} - V{source}} f

• If both the source and the observer are moving away:
  f' = rac{V{sound} - V{observer}}{V{sound} + V{source}} f

• The relationships vary according to the direction of motion and their relative speeds.