Doppler Effect JEE Main: Formula & Solved Problems

The Doppler effect is one of the most application-rich topics in wave physics. JEE Main tests it directly (one to two questions per session) and indirectly through beats and sound intensity. Understanding the sign convention thoroughly is all that stands between a student and full marks on every Doppler question — the formula is the same every time, only the sign changes.

The Master Formula

Apparent frequency heard by an observer: f' = f × (v + v_o)/(v − v_s). Here v is the speed of sound in the medium (≈ 340 m/s at room temperature), v_o is the speed of the observer, v_s is the speed of the source, and f is the actual frequency. Sign convention: use the + sign in the numerator when the observer moves toward the source, and the + sign in the denominator when the source moves away from the observer. Memorise: "numerator changes with observer motion; denominator changes with source motion; toward → frequency increases."

Shorthand: observer approaching source → v_o positive numerator (f' > f). Observer receding → v_o negative numerator (f' < f). Source approaching observer → v_s positive denominator denominator decreases → f' > f. Source receding → v_s negative denominator → f' < f. Both approaching: f' = f × (v + v_o)/(v − v_s) — maximum observed frequency. For the broader waves chapter, see our Waves & Oscillations Guide.

Common JEE Main Scenarios

Scenario 1 — stationary source, moving observer: f' = f × (v ± v_o)/v. Frequency increases when observer approaches. Scenario 2 — moving source, stationary observer: f' = f × v/(v ∓ v_s). Scenario 3 — both moving: apply the full formula with correct signs. Scenario 4 — moving source approaches, passes and recedes: f' drops from f × v/(v − v_s) to f × v/(v + v_s) as source passes. JEE Main has asked for the ratio of these two frequencies: f'_approach/f'_recede = (v + v_s)/(v − v_s). Take a free mock test on wave optics and sound to practise Doppler problems under timed conditions.

Doppler Effect with Beats

Beat frequency = |f₁ − f₂|. If a source of frequency f moves toward a wall with speed v_s, the source "hears" its reflection with a Doppler-shifted frequency f' = f(v + v_s)/(v − v_s) (the wall acts as a stationary observer and then a stationary source for the reflection). Beat frequency = f' − f = f × 2v_s/(v − v_s). JEE Main uses this in the "echo beats" problem type — a car moving toward a wall honks and hears beats from the echo.

Doppler Effect for Light

For light (electromagnetic waves), the Doppler formula differs because no medium is required. Relativistic Doppler shift: f_observed = f_source × √[(1+β)/(1−β)] for source approaching, where β = v/c. For v << c (non-relativistic): Δf/f ≈ v/c. Red shift (source receding): wavelength increases. Blue shift (source approaching): wavelength decreases. JEE Main tests light Doppler qualitatively — identifying whether a star is approaching or receding based on its spectrum shift.

Reflection and Apparent Speed of Image

When a source moves toward a plane mirror: the image moves in the opposite direction at the same speed. The observer (at source) sees the image approaching at 2v_s (relative velocity). Apparent frequency from image: f' = f(v + v_s)/(v − v_s) — same as wall-echo Doppler. For sound reflecting off a moving wall (wall moving toward source at v_w): treat the wall as a moving observer that absorbs then re-emits. This two-step Doppler is JEE Advanced territory but occasionally appears in JEE Main integer-answer type.

Exam Tips

Write the master formula at the top of your rough work for every Doppler problem. Then identify: (1) Who is the source? (2) Who is the observer? (3) Which direction are they moving relative to each other? Apply the sign rule. Double-check: if they approach each other, f' > f — if your answer has f' < f, you have a sign error. For complete wave physics preparation, connect this with our Simple Harmonic Motion Guide. Upgrade for ₹149/month for 100+ wave problems sorted by JEE difficulty level.