Variable-Tension and Composite Strings
For a slowly varying string the local speed is v(x)=sqrt(T(x)/mu(x)) and travel time is the integral of dx/v(x); at ideal junctions frequency is common while wavelength and mode count adjust to each section's speed.
Why this shows up in the exam
Pulses on hanging ropes · Joined strings and node matching · Composite sonometer modes
Learn the idea
When tension or linear density varies with position, local wave speed must be integrated or matched section by section. A hanging rope is faster near the top because it supports more weight; a joined string changes wavelength at the junction while preserving oscillation frequency.
🧠 Memory hook: Same frequency crosses a junction; speed and wavelength adapt.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- dt = dx/v(x) — travel-time element in a nonuniform string
- v_i = sqrt(T_i/mu_i) — local speed in each uniform section
- f common, lambda_i = v_i/f — frequency continuity across a stationary junction
How to approach it
- 1Write T(x) and mu(x) for each region
- 2Integrate travel time or match wavelengths
- 3Enforce boundary and junction continuity
Common slip-ups that cost marks
- •Using one average speed without justification
- •Changing frequency at a stationary junction
- •Assuming the same tension when a hanging rope's own weight matters
🌟 That's the whole idea — you've got this. Try the practice set below; every question you attempt makes it stick a little harder.
Original chapter practice
Original questions for this chapter, not past-paper questions or an exact mapping to this individual concept.
A mass of 1 kg is attached to a spring of force constant 100 N/m. Find the angular frequency of small oscillations.
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