JEE Mains Physics · Waves

Beats and the Doppler Effect

Two close frequencies give beats at their difference |f₁ − f₂|; relative motion of source and observer shifts the heard frequency to f(v ± vₒ)/(v ∓ vₛ).

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Why this matters

Twenty-one PYQs, ten of them asking for a number, and one from 2026. Eight are beats: tuning forks loaded with wax or filed, two pipes or two wavelengths sounding together, a retuned sonometer. Nine are Doppler shifts from moving sources and observers, two of them for light from a receding galaxy. Four are echoes from a wall or a hill. The formulas are short; the marks go on the signs and on which frequency reaches whom.

Concept 1 of 3: Beat frequency of two close frequencies

Two notes of nearly equal frequency drift in and out of step. Each time they come back into step the sound swells, so loudness rises and falls at the difference of the two frequencies. The beat count tells you how far apart they are, but not which is higher; a small change to one fork settles that.

Definition

Beat frequency $= |f_1 - f_2|$ . "n beats in t seconds" is $n/t$ beats per second.
Loading a fork with wax lowers its frequency; filing its prongs raises it.
Decide the sign by testing both values: if changing fork A moves it towards the other fork, the beats fall; away, they rise.
$x = a\cos\Delta t\,\cos\bar{\omega}t$ is the sum of two waves at $\bar{\omega} \pm \Delta$ . The beat frequency is $\dfrac{2\Delta}{2\pi} = \dfrac{\Delta}{\pi}$ Hz.
Beats from two pipes, two strings or two wavelengths: write each frequency from its own formula, then subtract.
A row of forks each n beats above the one before: the kth fork is $f_1 + (k - 1)n$ .

Beats

f_{\text{beat}} = |f_1 - f_2| \qquad f_{\text{beat}} = v\left|\frac{1}{\lambda_1} - \frac{1}{\lambda_2}\right|

Worked example

Fork P gives 6 beats per second with a 256 Hz fork. When P is filed, the beats rise to 8 per second. Find the frequency of P before filing.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2026 · 28 Jan 2026 Shift 2 · Q21Moderate

Example 1 · Waves · Beats and the Doppler Effect

Two tuning forks A and B are sounded together giving rise to 8 beats in 2 s . When fork A is loaded with wax, the beat frequency is reduced to 4 beats in 2 s . If the original frequency of tuning fork B is 380 Hz , then the original frequency of tuning fork A is

\_\_\_\_

Hz.

Beats in a time are not beats per second

Ten beats in 2 s is a beat frequency of 5 Hz. Using 10 as the frequency difference doubles every answer that follows.

Test both candidate frequencies

A beat count gives two possible frequencies, one above and one below. Only the change after loading or filing tells which one is right; never take the higher one by default.

The envelope frequency is half the beat frequency

In a cos(Δt) cos(ω̄t) the envelope cos(Δt) has frequency Δ/2π, but the loudness peaks twice in each of its cycles. The beat frequency is Δ/π.

Drill 7 more on beat frequency of two close frequencies

Concept 2 of 3: Doppler effect for a moving source and observer

A source moving towards you squeezes the waves together, and you moving towards the source meet the waves more often. Either way the pitch rises; moving apart, it falls. Only motion along the line joining them counts, and if both move together with no relative motion, nothing changes.

Definition

$f' = f\,\dfrac{v \pm v_o}{v \mp v_s}$ . Choose each sign so that motion towards the other raises f′: + on top for an observer moving towards the source, − below for a source moving towards the observer.
Source and observer moving together at the same velocity: $f' = f$ . A passenger on the train hears the train's own whistle at its true frequency.
One car chasing another: the observer moves towards the source (+ on top) and the source moves away from the observer (+ below).
Approach and recession at speed u: $\dfrac{f_{\text{app}}}{f_{\text{rec}}} = \dfrac{v + u}{v - u}$ .
Light from a receding galaxy ( $v \ll c$ ): $\dfrac{\Delta\lambda}{\lambda} = \dfrac{v}{c}$ , a red shift.

Doppler effect

f' = f\,\frac{v \pm v_o}{v \mp v_s} \qquad \frac{\Delta\lambda}{\lambda} = \frac{v}{c}

Worked example

An ambulance with a 500 Hz siren moves at

30\ \text{m/s}

towards a cyclist who rides towards it at

10\ \text{m/s}

. Sound travels at

340\ \text{m/s}

. What does the cyclist hear before and after they pass?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2023 · 11 April 2023 · Q92Moderate

Example 2 · Waves · Beats and the Doppler Effect

A car

P

travelling at

20{\text{ }ms}^{- 1}

sounds its horn at a frequency of

400\text{ }Hz

. Another car Q is travelling behind the first car in the same direction with a velocity

40{\text{ }ms}^{- 1}

. The frequency heard by the passenger of the car

Q

is approximately [Take, velocity of sound

= 360{\text{ }ms}^{- 1}

]

Set each sign by 'towards raises'

Do not memorise one sign pattern. For the observer term, motion towards the source adds to v on top; for the source term, motion towards the observer subtracts from v below. Check that the answer rises when they close in.

A chase is not an approach

When the observer follows the source in the same direction, the observer moves towards the source but the source moves away from the observer. Both signs are +, and the two effects partly cancel.

Drill 8 more on doppler effect for a moving source and observer

Concept 3 of 3: Doppler effect for an echo from a wall

An echo gets two shifts. First the wall is a stationary observer hearing the moving source. Then the wall re-sends that frequency as a stationary source, heard by the moving driver. Approaching the wall, both shifts raise the pitch, so the echo is well above the horn.

Definition

Step 1, wall as observer: $f_1 = f\,\dfrac{v}{v - u}$ for a source approaching at u.
Step 2, wall as source, driver as observer approaching at u: $f_2 = f_1\,\dfrac{v + u}{v}$ .
Together: $f_{\text{echo}} = f\,\dfrac{v + u}{v - u}$ . The change $f_{\text{echo}} - f = f\,\dfrac{2u}{v - u}$ .
The driver hears the horn itself at f (moving with it), so the beats between horn and echo are $f_{\text{echo}} - f$ .
A listener between a source behind and a wall ahead hears the direct sound lowered and the reflected sound raised; the beats are their difference.

Echo from a wall

f_{\text{echo}} = f\,\frac{v + u}{v - u}

Worked example

A bat flies at

10\ \text{m/s}

towards a wall, sending out 40 kHz. Sound travels at

340\ \text{m/s}

. What frequency does the bat hear in the echo?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2023 · 6 April 2023 · Q25Moderate

Example 3 · Waves · Beats and the Doppler Effect

A person driving car at a constant speed of

15\text{ }m/s

is approaching a vertical wall. The person notices a change of

40\text{ }Hz

in the frequency of his car's horn upon reflection from the wall. The frequency of horn is___ Hz. (Given: Speed of sound:

330\text{ }m/s

)

An echo is shifted twice

Applying the Doppler formula once, for the source only, misses the second shift as the driver moves into the reflected sound. The echo heard by an approaching driver is f(v + u)/(v − u).

The driver hears his own horn unshifted

The driver moves with the horn, so the direct sound reaches him at its true frequency. A beat or a 'change' between horn and echo is measured from f, not from a shifted value.

Drill 3 more on doppler effect for an echo from a wall

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 (3)

Beat frequency of two close frequencies
Beats
$f_{\text{beat}} = |f_1 - f_2| \qquad f_{\text{beat}} = v\left|\frac{1}{\lambda_1} - \frac{1}{\lambda_2}\right|$
Doppler effect for a moving source and observer
Doppler effect
$f' = f\,\frac{v \pm v_o}{v \mp v_s} \qquad \frac{\Delta\lambda}{\lambda} = \frac{v}{c}$
Doppler effect for an echo from a wall
Echo from a wall
$f_{\text{echo}} = f\,\frac{v + u}{v - u}$

Watch out for (7)

Beats in a time are not beats per second
→ Beat frequency of two close frequencies
Test both candidate frequencies
→ Beat frequency of two close frequencies
The envelope frequency is half the beat frequency
→ Beat frequency of two close frequencies
Set each sign by 'towards raises'
→ Doppler effect for a moving source and observer
A chase is not an approach
→ Doppler effect for a moving source and observer
An echo is shifted twice
→ Doppler effect for an echo from a wall
The driver hears his own horn unshifted
→ Doppler effect for an echo from a wall

Test yourself on Waves

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