Fixed-String Harmonics and Sonometer Relations
For a uniform ideal string of length L fixed at both ends, allowed modes satisfy L=n lambda_n/2 and have frequencies f_n=n sqrt(T/mu)/(2L), where n is a positive integer.
Why this shows up in the exam
Sonometer scaling · Harmonic and overtone identification · Common-frequency and resonance problems
Learn the idea
A string fixed at both ends fits an integer number of half-wavelengths. Both ends must remain nodes, so only wavelengths that tile the length by half-wave loops can persist as normal modes.
🧠 Memory hook: Fixed-fixed strings count half-waves: one, two, three, and so on.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- lambda_n = 2L/n — allowed wavelength of the nth fixed-string mode
- f_n = n/(2L) sqrt(T/mu) — harmonic frequencies of a fixed uniform string
How to approach it
- 1Mark nodes at both active ends
- 2Count half-wave loops to determine n
- 3Use ratios before numerical substitution
Common slip-ups that cost marks
- •Calling the first overtone the first harmonic
- •Forgetting that length is between active bridges
- •Applying one uniform-string formula across a composite string
🌟 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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