What Beats Are
When two harmonic sound waves of close — but not equal — frequencies are heard at the same time, the ear registers a sound at roughly their average frequency, together with a slow, audibly distinct rise and fall in intensity. Each cycle of maximum loudness followed by near-silence is one beat. The frequency at which these loud–soft alternations occur is the beat frequency, and NCERT establishes that it equals the difference between the two source frequencies.
The phenomenon is a special case of interference in time rather than in space. Where two waves of identical frequency redistribute energy across positions to give a fixed pattern, two waves of slightly different frequency drift in and out of phase at a single point, so the loudness at that point cycles between reinforcement and cancellation.
Two harmonic waves of nearly equal frequency (here 11 Hz and 9 Hz, the NCERT illustration) gradually shift from in-phase to out-of-phase and back.
Superposition of Two Close Frequencies
Consider two waves of equal amplitude $a$ arriving at a point, written as displacements in time with angular frequencies $\omega_1=2\pi\nu_1$ and $\omega_2=2\pi\nu_2$:
$$s_1 = a\cos\omega_1 t, \qquad s_2 = a\cos\omega_2 t$$
By the superposition principle the resultant displacement is the sum $s = s_1 + s_2$. Applying the identity $\cos A + \cos B = 2\cos\!\frac{A-B}{2}\cos\!\frac{A+B}{2}$ gives
$$s = \left[\,2a\cos\!\left(\frac{\omega_1-\omega_2}{2}\,t\right)\right]\cos\!\left(\frac{\omega_1+\omega_2}{2}\,t\right)$$
Because $\nu_1$ and $\nu_2$ are close, $\tfrac{\omega_1+\omega_2}{2}$ is large while $\tfrac{\omega_1-\omega_2}{2}$ is small. The result is therefore a fast oscillation at the average (carrier) frequency $\frac{\nu_1+\nu_2}{2}$ — this sets the pitch we hear — multiplied by a slowly varying amplitude, the term in square brackets. That slow factor is the envelope; it swells and collapses, and its variation is what the ear perceives as the throbbing of loudness.
Superposing waves (a) and (b) from Figure 1 gives a fast carrier oscillation whose amplitude is bounded by a slow envelope. The envelope reaches its loud maxima twice per second — a beat frequency of 2 Hz.
Beat Frequency and the Audibility Limit
Loudness peaks whenever the envelope factor $\cos\!\left(\tfrac{\omega_1-\omega_2}{2}t\right)$ reaches $\pm 1$, because the intensity depends on amplitude regardless of its sign. A maximum at $+1$ and the next at $-1$ are both loud points, so the envelope produces two intensity maxima per cycle of itself. Working this through, the number of beats per second comes out to the plain difference of the frequencies. Since $\omega=2\pi\nu$, NCERT writes the result (Eq. 14.48) as
$$\nu_{beat} = \nu_1 - \nu_2$$
In practice we count beats, never negative numbers, so the working form is the absolute value $\nu_{beat}=|\nu_1-\nu_2|$. The NIOS treatment frames the same idea physically: if fork B completes $n$ more vibrations each second than fork A, then B gains one full vibration over A every $1/n$ second, the two forks pass through one complete in-phase / out-of-phase cycle in that time, and so exactly $n$ beats are heard per second.
The beat frequency is an absolute difference
A fork of 510 Hz sounded with one of 514 Hz, and the same 510 Hz fork sounded with one of 506 Hz, both give 4 beats per second. The beat count alone never tells you which fork is higher — it only fixes the gap. Two unknown frequencies are always consistent with a single observed beat count.
Rule: $\nu_{beat}=|\nu_1-\nu_2|$. From a beat count alone an unknown fork has two possible frequencies, $\nu_{known}\pm\nu_{beat}$.
Beats are only heard distinctly when the frequency difference is small. The human ear cannot resolve two intensity changes separated by less than about one-tenth of a second, so once the difference exceeds roughly 10 Hz, more than ten beats occur each second and the separate waxing and waning blur into a single steady tone. This is why the technique is restricted, in NCERT and in problems, to forks that differ by only a few hertz.
Beats are interference in time. The general adding-of-waves rule is set out in Superposition of Waves.
The Wax-Loading Tuning-Fork Logic
Because the beat count gives two candidate frequencies, an extra observation is needed to choose. The standard move is to alter one fork by a known amount and watch whether the beats speed up or slow down. The physics of that alteration is fixed: loading a prong with wax adds mass and lowers the fork's natural frequency; filing the prongs removes mass and raises it. For strings the analogue is tension — increasing tension raises frequency.
An unknown fork gives $b$ beats per second with a known fork of frequency $\nu_0$, so the unknown is either $\nu_0+b$ or $\nu_0-b$. Load the unknown fork with wax (lowering it) and read off this table.
The reasoning behind the table is worth saying in words, because the trap is easy to fall into. The unknown fork is either above or below the known one by the beat amount. Loading pushes the unknown down. If it was already the lower fork, pushing it further down widens the gap, so the beats grow more rapid. If it was the higher fork, pushing it down moves it toward the known frequency, narrowing the gap, so the beats slow. The direction of change in the beat count therefore resolves the ambiguity completely.
Loading lowers; filing raises — and read the change carefully
Two errors recur. First, candidates forget that wax lowers the frequency and filing raises it; reverse this and every conclusion flips. Second, after correctly lowering the loaded fork, they misread the consequence: if the beats increase after loading, the loaded fork was the lower one — not the higher. The string version (NEET 2020) substitutes tension for wax: decreasing tension lowers the string's frequency, exactly like loading.
Rule: loading / lower tension ⇒ frequency falls. Beats rise after this ⇒ the changed source was the lower of the pair.
Two sitar strings A and B are slightly out of tune and produce 5 beats per second. The tension of B is slightly increased and the beat frequency falls to 3 Hz. If A is 427 Hz, find the original frequency of B.
Increasing tension raises B's frequency. Had B been higher than A, raising it further would have increased the beat frequency — but the beats decreased. Therefore B was the lower fork, $\nu_B<\nu_A$. With $\nu_A-\nu_B=5$ Hz and $\nu_A=427$ Hz, the original frequency of B is $427-5=\mathbf{422}$ Hz.
Applications
The chief use of beats is in tuning. A musician sounds an instrument's string or pipe against a reference of the desired pitch and adjusts it until the beats slow to zero; the disappearance of beats signals that the two frequencies have become exactly equal. The same principle, run in reverse, measures an unknown frequency: the beat count gives the gap, and the loading test gives the sign.
| Use | What is known | What beats reveal |
|---|---|---|
| Tuning an instrument | A reference frequency | Adjust until beats vanish ⇒ frequencies equal |
| Measuring an unknown frequency | A standard fork of frequency $\nu_0$ | Beat count $b$ ⇒ unknown is $\nu_0\pm b$ |
| Resolving the $\pm$ sign | Direction of frequency change (wax / file / tension) | How the beat count shifts fixes higher vs lower |
A tuning fork of unknown frequency gives 5 beats per second with a fork of 500 Hz. What is the unknown frequency?
The beat count fixes only the gap: $\nu' = 500 \pm 5$, so the unknown is either $\mathbf{495}$ Hz or $\mathbf{505}$ Hz. A second observation — loading or filing one fork and watching the beats — is needed to choose between them.
Beats in one screen
- Beats = periodic waxing and waning of loudness from superposing two close, unequal frequencies.
- Resultant $s = 2a\cos\!\left(\tfrac{\omega_1-\omega_2}{2}t\right)\cos\!\left(\tfrac{\omega_1+\omega_2}{2}t\right)$ — a carrier at the average frequency under a slow envelope.
- Beat frequency $\nu_{beat}=|\nu_1-\nu_2|$; pitch heard is the average $(\nu_1+\nu_2)/2$.
- Beats are perceived only if the difference is small (under about 10 Hz), since the ear cannot resolve faster alternations.
- Wax loading lowers a fork's frequency; filing raises it; tension raises a string's frequency.
- A beat count gives two candidates $\nu_0\pm b$; the way beats change after a known adjustment fixes which.