Q204-Mechanical-Waves

Waves and Their Classifications

  • Wave is defined as a disturbance or oscillation that travels through matter or space, accompanied by the transfer of energy.

Mechanical Waves

  • Mechanical waves require a medium to propagate.

    • Types of Mechanical Waves:

      • Longitudinal Waves: Particle's oscillation is parallel to the wave's propagation (e.g., sound waves).

      • Transverse Waves: Particle's oscillation is perpendicular to the wave's propagation (e.g., waves on a string).

      • Surface Waves: Have both transverse and longitudinal properties (e.g., water waves).

Wave Characteristics

  • Waves transport energy but not matter.

  • Each wave has distinctive features:

    • Crest: The highest point on a wave.

    • Trough: The lowest point between two waves.

    • Amplitude: Maximum displacement from the equilibrium position.

    • Wavelength: Distance between two successive crests or troughs.

    • Frequency (f): Number of crests passing a fixed point per second (SI unit: Hertz).

    • Period (T): Time taken for one complete wave cycle.

    • Wave Speed (v): Speed of wave propagation determined by medium properties.

  • General wave equation:

    • ( v = f \times \lambda )

Wave Functions

  • Wave function describes the position of any particle in the medium at any time.

  • Common forms of wave functions include sinusoidal waves.

    • General linear mechanical wave function:

    • ( y(x,t) = A \cos(kx - \omega t) )

    • Where A is amplitude, k is wave number, and ( \omega ) is angular frequency.

Superposition Principle

  • The resulting wave is the sum of individual waves:

    • Constructive Interference: Occurs when waves add together to create a larger amplitude.

    • Destructive Interference: Occurs when waves cancel each other out.

Standing Waves

  • Formed by the interference of two waves traveling in opposite directions.

  • Characterized by nodes (points of no displacement) and antinodes (points of maximum displacement).

  • Mathematical representation:

    • ( y(x,t) = A_{SW} \sin(kx) \sin(\omega t) ) where ( A_{SW} = 2A ) captures both waves' effects.

Mersenne’s Laws of String Instruments

  • Frequency of vibration of a string fixed at both ends follows:

    • ( f_n = \frac{n}{2L} \sqrt{\frac{T}{\mu}} )

    • Where L is length, T is tension, and ( \mu ) is linear mass density.

Sound Waves

  • Longitudinal waves that require a medium (air) for propagation.

  • Speed of sound depends on temperature; at 20°C, speed is approximately 344 m/s.

Doppler Effect

  • Describes the change in frequency observed when a source moves relative to an observer.

    • Formula: ( \frac{f_L}{f_S} = \frac{v \pm v_L}{v \pm v_S} )

    • Where ( f_L ) is the frequency heard, ( f_S ) is the source frequency, and v is the speed of sound.

Sonic Boom

  • Occurs when an object travels through air at a speed greater than that of sound (Mach 1).

  • Categories of speeds:

    • Subsonic (M < 1.0), Transonic (M = 1.0), Supersonic (M > 1.0), Hypersonic (M > 5.0).

Seismic Waves

  • Produced by earthquakes, causing significant damage and can include both transverse and longitudinal waves.

Summary of Activities and Examples

  • Exercises include calculations of wave properties and applications of concepts like sound waves, wave superposition, and analysis of wave mediums.

References

  1. Young, H., Freedman, R., and Ford, A. (2016) University Physics with Modern Physics, 14e, Pearson

  2. Hewitt, P. (2013) Conceptual Physics, 12e, Addison-Wesley

  3. Giancoli, D. (2013) Physics: Principles with Application, Addison-Wesley