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Pluck a guitar string, tighten the peg, pluck again, and the note rises. Press the string against a fret and it rises again. Every sonometer question in Class 11 asks you to predict changes like these, and the rules behind them were written down almost four hundred years ago by Marin Mersenne. This post explains who he was, what his laws say, how they come out of the formula in your NCERT book, and how to use them without the usual slips.
Who was Marin Mersenne?
The MacTutor biography from the University of St Andrews records that Mersenne was born on 8 September 1588 in Oizé, France, joined the Order of the Minims in 1611 and died in Paris on 1 September 1648. He was a friar, a mathematician and, for decades, the person through whom European scientists shared their work. MacTutor lists Descartes, Fermat, Gassendi and both Pascals among those who met him or wrote to him, and notes that letters from 78 different correspondents were found in his cell after his death.
He also had, in MacTutor's words, "a strong interest in music" and spent a lot of time on acoustics. His large book on music and sound, Harmonie universelle, was printed in Paris in 1636-37; a scan of the Bibliothèque nationale de France copy is freely available.
What are Mersenne's laws?
MacTutor states the laws as Mersenne published them: a string's frequency "is proportional to the square root of the tension, and inversely proportional to the length, to the diameter and to the square root of the specific weight of the string, provided all other conditions remain the same when one of these quantities is altered."
So there are four laws:
Frequency ∝ 1/L (length).
Frequency ∝ √T (tension).
Frequency ∝ 1/d (diameter).
Frequency ∝ 1/√ρ (density of the material).
How do the laws follow from the NCERT formula?
The NCERT Class 11 chapter Waves gives the speed of a transverse wave on a string as v = √(T/μ), where μ is mass per unit length. For a string fixed at both ends, the fundamental frequency is f = v/2L. Put them together:
f = (1/2L) √(T/μ)
That already contains the length and tension laws. Now write μ in terms of the wire. A wire of diameter d and density ρ has μ = ρπd²/4. Substituting:
f = (1/(Ld)) √(T/(πρ))
Now the diameter law (f ∝ 1/d) and the density law (f ∝ 1/√ρ) appear as well. Mersenne found by experiment what this one line of algebra shows. The JEE (Main) 2026 syllabus lists "standing waves in strings and organ pipes, fundamental mode and harmonics", so check the current year's syllabus, but expect this formula.
A worked sonometer example
Sample numbers: a sonometer wire 0.50 m long has mass per unit length 5.0 g/m and is under a tension of 80 N.
Convert units first: μ = 5.0 g/m = 5.0 × 10⁻³ kg/m.
Wave speed: v = √(80 / 0.0050) = √16,000 = 126.5 m/s.
Fundamental: f = v/2L = 126.5 / 1.0 = 126.5 Hz, about 126 Hz.
Now change one thing at a time, as Mersenne insisted.
Double the tension to 160 N. f rises by √2, to 178.9 Hz. It does not double.
Halve the length to 0.25 m. f doubles, to 253 Hz.
Use a wire of the same metal with twice the diameter. μ goes up four times, so √μ doubles and f halves, to 63.2 Hz.
To double the frequency by tension alone, you would need four times the tension, 320 N.
What mistakes cost marks here?
Using total mass instead of mass per length. μ is kg/m. If a question gives a 2.0 m wire of mass 10 g, μ = 0.010 / 2.0 = 0.0050 kg/m.
Forgetting the square root. Tension and density enter under a root. Length and diameter do not.
Mixing grams and kilograms. Convert to SI before you substitute.
Changing the diameter but forgetting μ. μ depends on d², so a thicker wire is heavier per metre.
If the link between v = √(T/μ) and the harmonics still feels abstract, read our guide to standing waves in strings and pipes. Standing waves are easier to see than to read about, and the Fermi Simulation Lab has interactive simulations for ideas like this one.
What else did Mersenne leave behind?
Mersenne is also remembered for numbers of the form 2ᵖ − 1, now called Mersenne primes. MacTutor records that in 1644 he listed the primes p up to 257 for which he believed 2ᵖ − 1 is prime, and that he turned out to be wrong about five of them: he included 67 and 257, and missed 61, 89 and 107. Even his errors set others working for centuries.
What can a student take from Mersenne?
Change one quantity and hold the rest. That is the whole method behind his four laws.
State results as proportions. "f ∝ √T" tells you more, faster, than a table of readings.
Share your work. Much of what Mersenne did was getting results from one person to the next. A study partner who checks your method plays the same role.
This week, take any three sonometer questions and, before solving, write which of L, T, d and ρ change and which stay fixed.
