JEE MainPhysics

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Determining Sound Velocity from Beat Frequency and Wavelengths

Q: Why is beat frequency f_1 - f_2 and not f_2 - f_1?

Because frequency is inversely proportional to wavelength (f = v/λ), the smaller wavelength λ_1 = 4.08m gives the higher frequency f_1.

The velocity of sound in a gas. in which two wavelengths 4.08m and 4.16m produce 40 beats in 12s, will be:

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A2.82.8ms^(-1)B175.5ms^(-1)C353.6ms^(-1)D707.2ms^(-1)

Step-by-step solutionView

Correct answer

Using the beat frequency $f_{beat} = \frac{40}{12} s^{- 1} = \frac{10}{3} Hz$ and the relationship $v (\frac{1}{λ_{1}} - \frac{1}{λ_{2}}) = f_{beat}$ , the velocity of sound is calculated to be $707.2 {m s}^{- 1}$ .

Option analysis

Why each option works or fails

A · 2.82.8ms^(-1)

You made an arithmetic slip in placing the decimal point while dividing the product of wavelengths by their difference. Ensure proper scale estimation before calculating: dividing approximately $17$ by $0.08$ yields a number greater than $200$ , which after multiplying by beat frequency results in several hundred ${m s}^{- 1}$ , not $2.828 {m s}^{- 1}$ .

B · 175.5ms^(-1)

You divided by an extra factor of $4$ or miscalculated the beat frequency by dividing $40$ by $48$ instead of multiplying by $\frac{40}{12}$ . Beat frequency is strictly the number of beats divided by total time: $f_{beat} = \frac{40}{12} Hz$ .

C · 353.6ms^(-1)

You introduced an unintended factor of $1 / 2$ , confusing beat frequency with the modulation frequency of the wave envelope. The beat frequency heard by an observer is $∣ f_{1} - f_{2} ∣$ , not $\frac{1}{2} ∣ f_{1} - f_{2} ∣$ .

D · 707.2ms^(-1)

This is the correct calculation: $v = f_{beat} \cdot \frac{λ_{1} λ_{2}}{λ_{2} - λ_{1}} = \frac{10}{3} \cdot \frac{4.08 \times 4.16}{0.08} = 707.2 {m s}^{- 1}$ . This calculation is completely correct.

Reviewed route

Solution

StepWorking

01Given

Two wavelengths $λ_{1} = 4.08 m$ and $λ_{2} = 4.16 m$ . Beat count = $40$ beats in $Δ t = 12 s$ .

02Find

Velocity of sound in the gas, $v$ .

03Visualise

Two sound waves of slightly different wavelengths travel in the same medium at speed $v$ . Their frequencies $f_{1} = v / λ_{1}$ and $f_{2} = v / λ_{2}$ superpose to produce beats with beat frequency $f_{b} = f_{1} - f_{2}$ .

04Strategise

Beat frequency is beats per second: $f_{b} = 40 / 12 s^{- 1} = 10 / 3 Hz$ . Using $f = v / λ$ , set up $f_{b} = v (1 / λ_{1} - 1 / λ_{2})$ and solve for $v = \frac{f_{b} \cdot λ_{1} λ_{2}}{λ_{2} - λ_{1}}$ .

05Execute

Substitute values: $v = \frac{(40 / 12) \times 4.08 \times 4.16}{4.16 - 4.08} = \frac{(10 / 3) \times 16.9728}{0.08} = \frac{10 \times 5.6576}{0.08} = 707.2 m/s$ .

✓Verify

Units are $s^{- 1} \times m^{2} / m = m/s$ . A speed around $700 m/s$ is well within physical range for lighter gases (like helium or hot air).

✓ Source and academic review

Question type: Single correct
Exam relevance: JEE Main · Physics
Academic status: Answer verified against the official NTA key
Source: Previous-year question
Editorial review: 22 September 2026

Quick checks

Students also ask

Why is beat frequency

f_{1} - f_{2}

and not

f_{2} - f_{1}

Because frequency is inversely proportional to wavelength ( $f = v / λ$ ), the smaller wavelength $λ_{1} = 4.08 m$ gives the higher frequency $f_{1}$ .

Answer

Using the beat frequency $f_{beat} = \frac{40}{12} s^{- 1} = \frac{10}{3} Hz$ and the relationship $v (\frac{1}{λ_{1}} - \frac{1}{λ_{2}}) = f_{beat}$ , the velocity of sound is calculated to be $707.2 {m s}^{- 1}$ .

Why each option works or fails

A: 2.82.8ms^(-1) - You made an arithmetic slip in placing the decimal point while dividing the product of wavelengths by their difference. Ensure proper scale estimation before calculating: dividing approximately $17$ by $0.08$ yields a number greater than $200$ , which after multiplying by beat frequency results in several hundred ${m s}^{- 1}$ , not $2.828 {m s}^{- 1}$ .
B: 175.5ms^(-1) - You divided by an extra factor of $4$ or miscalculated the beat frequency by dividing $40$ by $48$ instead of multiplying by $\frac{40}{12}$ . Beat frequency is strictly the number of beats divided by total time: $f_{beat} = \frac{40}{12} Hz$ .
C: 353.6ms^(-1) - You introduced an unintended factor of $1 / 2$ , confusing beat frequency with the modulation frequency of the wave envelope. The beat frequency heard by an observer is $∣ f_{1} - f_{2} ∣$ , not $40$ .
D · correct: 707.2ms^(-1) - This is the correct calculation: $v = f_{beat} \cdot \frac{λ_{1} λ_{2}}{λ_{2} - λ_{1}} = \frac{10}{3} \cdot \frac{4.08 \times 4.16}{0.08} = 707.2 {m s}^{- 1}$ . This calculation is completely correct.

Step-by-step solution

Given: Two wavelengths $λ_{1} = 4.08 m$ and $λ_{2} = 4.16 m$ . Beat count = $40$ beats in $Δ t = 12 s$ .
Find: Velocity of sound in the gas, $v$ .
Visualise: Two sound waves of slightly different wavelengths travel in the same medium at speed $v$ . Their frequencies $f_{1} = v / λ_{1}$ and $f_{2} = v / λ_{2}$ superpose to produce beats with beat frequency $f_{b} = f_{1} - f_{2}$ .
Strategise: Beat frequency is beats per second: $f_{b} = 40 / 12 s^{- 1} = 10 / 3 Hz$ . Using $f = v / λ$ , set up $f_{b} = v (1 / λ_{1} - 1 / λ_{2})$ and solve for $v = \frac{f_{b} \cdot λ_{1} λ_{2}}{λ_{2} - λ_{1}}$ .
Execute: Substitute values: $v = \frac{(40 / 12) \times 4.08 \times 4.16}{4.16 - 4.08} = \frac{(10 / 3) \times 16.9728}{0.08} = \frac{10 \times 5.6576}{0.08} = 707.2 m/s$ .
Verify: Units are $s^{- 1} \times m^{2} / m = m/s$ . A speed around $700 m/s$ is well within physical range for lighter gases (like helium or hot air).

Shortcut:

When to use it: When $Δ λ ≪ λ$ , using differential calculus gives an extremely fast estimate.

Given: Mean wavelength $λ \approx 4.12 m$ , $Δ λ = 4.16 - 4.08 = 0.08 m$ , $Δ f = f_{b} = 10 / 3 Hz$ .

Find: Speed of sound $v$ .

Strategise: Differentiating $f = v / λ$ gives $∣ Δ f ∣ \approx \frac{v}{λ^{2}} Δ λ$ , hence $v \approx \frac{Δ f \cdot λ^{2}}{Δ λ}$ .

Execute: $v \approx \frac{(10 / 3) \times (4.12)^{2}}{0.08} = \frac{10 \times 16.9744}{3 \times 0.08} = \frac{169.744}{0.24} \approx 707.27 m/s$ . Matches $707.2 m/s$ .

Verify: Difference between $λ_{1} λ_{2}$ and $λ_{a v g}^{2}$ is negligible ( $16.9728$ vs $16.9744$ ), giving four-digit precision.

Determining Sound Velocity from Beat Frequency and Wavelengths

What feels right?

Using the beat frequency $f_{beat} = \frac{40}{12} s^{- 1} = \frac{10}{3} Hz$ and the relationship $v (\frac{1}{λ_{1}} - \frac{1}{λ_{2}}) = f_{beat}$ , the velocity of sound is calculated to be $707.2 {m s}^{- 1}$ .

Why each option works or fails

Solution

01Given

02Find

03Visualise

04Strategise

05Execute

✓Verify

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