Standing Wave on a Thin Rope: Analyzing Harmonics

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SUMMARY

The discussion focuses on the analysis of a standing wave on a thin rope, specifically using the wave function y(x,t)=2.30mmcos[(6.98rad/m)x+(742rad/s)t]. The standing wave function is derived as y(x,t)=4.6cos(6.98x)-cos(72t), indicating the superposition of the traveling wave and its reflection. The harmonic oscillation is identified as N=π/2K=2π, confirming the wave's harmonic nature and its relationship to the wave number.

PREREQUISITES
  • Understanding of wave functions and their components
  • Knowledge of harmonic motion and standing waves
  • Familiarity with wave parameters such as amplitude, wave number, and angular frequency
  • Basic skills in trigonometric identities and their applications in wave analysis
NEXT STEPS
  • Study the derivation of standing wave equations in fixed boundary conditions
  • Explore the relationship between wave number and frequency in wave mechanics
  • Learn about the implications of harmonic oscillation in physical systems
  • Investigate the effects of mass and tension on wave speed in ropes
USEFUL FOR

Students of physics, particularly those studying wave mechanics, as well as educators and anyone interested in the mathematical modeling of physical phenomena involving waves.

Faraz
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Homework Statement



A fellow student of mathematical bent tells you that the wave function of a traveling wave on a thin rope is y(x,t)=2.30mmcos[(6.98rad/m)x+(742rad/s)t]. Being more practical-minded, you measure the rope to have a length of 1.35 m and a mass of 3.38 grams. Assume that the ends of the rope are held fixed and that there is both this traveling wave and the reflected wave traveling in the opposite direction.
a)What is the wavefunction y(x,t) for the standing wave that is produced?
b)In which harmonic is the standing wave oscillating?

Homework Equations


y(x,t)=Acos(kx-ωt)-Acos(kx+ωt)

The Attempt at a Solution


a)4.6cos(6.98x)-cos(72t)
b)N=PI/2K=2PI
 
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Great. How did you arrive at those answers snd what do they mean?
 

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