MHT-CET Physics · Superposition of Waves

Beats, and Tuning a Sonometer

Two sounds of slightly different frequency drift in and out of step, so their loudness swells and fades |f₁ − f₂| times a second; with a sonometer, whose frequency goes as 1/length, beats pin down a tuning fork's frequency.

All Superposition of Waves notes MHT-CET Physics strategy

Why this matters

22 PYQs, nine HARD — the densest-HARD page in the chapter, all of them sonometer or pipe puzzles. Two shapes: beat frequency and its timing (maxima, minima, loudness ratio), and a fork compared with a sonometer at two lengths — solving two equations with f × l constant.

Concept 1 of 2: Beat Frequency, Timing and Loudness

Two tones a few hertz apart come back into step that many times each second. Loud moments (waxing) are 1/Δf apart, and the quiet moment falls halfway between. The loudest sound has amplitude a₁ + a₂, the quietest a₁ − a₂.

Definition

Beats per second $= |f_1 - f_2|$ ; from $\sin\omega t$ forms, $\dfrac{\Delta\omega}{2\pi}$ .
Time between maxima $= \dfrac{1}{\Delta f}$ ; maximum to next minimum $= \dfrac{1}{2\Delta f}$ .
Waxing : waning intensity $= \left(\dfrac{a_1 + a_2}{a_1 - a_2}\right)^2$ (4 and 3 ⇒ 49 : 1).
Which way? If raising the tension on string Y cuts the beats with X, Y was BELOW X.
Two open pipes of lengths $l$ and $l + l_1$ : beats $\dfrac{v}{2}\left(\dfrac{1}{l} - \dfrac{1}{l + l_1}\right) \approx \dfrac{vl_1}{2l^2}$ .

Beats

f_{\text{beat}} = |f_1 - f_2|, \qquad T_{\text{beat}} = \frac{1}{|f_1 - f_2|}

Worked example

Forks of 512 Hz and 516 Hz sound together. Beats per second, time between maxima, and from a maximum to the next minimum?

Practice this conceptself-check · 2 quick reps

The same idea in a real exam question:

MHT-CET · 2023 · 11th May Shift 2 · Q39Easy

Example 1 · Superposition of Waves · Beats and Sonometer

If the two waves of same amplitude, having frequencies 340 Hz and 335 Hz, are moving in same direction, then the time interval between two successive maxima formed (in second) is

Reading ω as the frequency

\sin 316t

, 316 is ω, not f. The beat frequency is

\frac{316 - 310}{2\pi} = \frac{3}{\pi}

, not 6.

Drill 10 more on beat frequency, timing and loudness

Concept 2 of 2: A Tuning Fork Against a Sonometer

A sonometer's frequency goes as 1/length, so f × l is fixed. If the same number of beats is heard at two lengths, the fork's frequency lies between the wire's two frequencies — above one by the beat count and below the other by the same count.

Definition

Same wire, same tension: $f_1l_1 = f_2l_2$ ; the shorter length gives the higher frequency.
Two forks in unison with lengths $l_1, l_2$ and giving $b$ beats: $f_1l_1 = f_2l_2$ , $|f_1 - f_2| = b$ .
Same beats at $l$ and $l - \Delta l$ : $(f - b)l = (f + b)(l - \Delta l)$ .
Hanging weight of specific gravity $d$ immersed: $\dfrac{n}{n - x} = \sqrt{\dfrac{d}{d - 1}}$ .
Two wires at equal frequency: $\dfrac{1}{d}\sqrt{\dfrac{T}{\rho}}$ must match, so $T \times 2$ , $d \times 2$ needs $\rho \div 2$ .

Sonometer

f \propto \frac{1}{l}\;\Rightarrow\; f_1l_1 = f_2l_2

Worked example

A fork gives 5 beats with 60 cm of a sonometer wire, and still 5 beats when the wire is shortened to 59 cm. The fork's frequency?

Practice this conceptself-check · 2 quick reps

The same idea in a real exam question:

MHT-CET · 2024 · 14th May Shift 2 · Q8Moderate

Example 2 · Superposition of Waves · Beats and Sonometer

A sonometer wire 49 cm long is in unison with a tuning fork of frequency

n

. If the length of the wire is decreased by 1 cm and it is vibrated with the same tuning fork, 6 beats are heard per second. The value of

n

Pairing the higher frequency with the longer length

A SHORTER wire vibrates faster. Solve

f_1l_1 = f_2l_2

with the larger f on the smaller l, or the two frequencies come out swapped.

Drill 10 more on a tuning fork against a sonometer

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Beat Frequency, Timing and Loudness
Beats
$f_{\text{beat}} = |f_1 - f_2|, \qquad T_{\text{beat}} = \frac{1}{|f_1 - f_2|}$
A Tuning Fork Against a Sonometer
Sonometer
$f \propto \frac{1}{l}\;\Rightarrow\; f_1l_1 = f_2l_2$

Watch out for (2)

Reading ω as the frequency
→ Beat Frequency, Timing and Loudness
Pairing the higher frequency with the longer length
→ A Tuning Fork Against a Sonometer

Test yourself on Superposition of Waves

20 past MHT-CET questions from this chapter, timed at 18 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Take the Superposition of Waves chapter test Drill all 22 questions

Test yourself on Superposition of Waves

Related notes