SH Wave Equation and Simple Harmonic Motion

SH Wave Equation Derivation

• The wave equation describes waves like sound, water, and seismic waves.
• All waves share properties: reflection, refraction, diffraction, interference, wavelength, frequency, speed, amplitude.
• A wave is an energy-transferring disturbance.
• SH wave equation example: y = a sin(vt – x) where 'y' is displacement, 'a' is amplitude, 'v' is velocity, 't' is time, and 'x' is distance.
• Differentiating y = a sin(vt – x) with respect to time (t) twice yields the acceleration: y'' = -v²y.
• Differentiating y = a sin(vt – x) with respect to distance (x) twice yields the rate of change of compression: y'' = -v²y.
• Comparing the second-order time and distance derivatives gives the SH wave differential equation: y'' = -v²y.

Simple Harmonic Motion (SHM) Energy

• Potential Energy (PE) in SHM: PE = ½ mω²y² (m = mass, ω = angular velocity, y = displacement).
• Kinetic Energy (KE) in SHM: KE = ½ mω²(a² - y²) (a = amplitude).
• Total Energy (E) in SHM: E = PE + KE = ½ mω²a² = 2π²υ²ma² (υ = frequency).

SHM Problems

• Question 1: Doubling the amplitude of an SH oscillator:
  • Frequency remains unchanged.
  • Maximum velocity doubles.
  • Maximum acceleration doubles.
• Question 2: At an extreme position, the vibrating particle's velocity is zero.
• Question 3: Deriving the SHM differential equation, y = a sin(ωt + φ), involves differentiating twice with respect to time to show y'' + ω²y = 0.
• Question 4: A 2 x 10⁻⁴ second period translates to a resonant frequency of 5000 Hz (Hertz = cycles per second).
• Question 5: For a 12 m long, 2.1 kg steel wire to have transverse wave speed matching the speed of sound (343 m/s), the tension (T) needs to be approximately 2.06 x 10⁴ N.