Doppler Effect — Revision Notes

⚡ 30-Second Revision

General Formula: — $f' = f_0 \left( \frac{v \pm v_o}{v \mp v_s} \right)$ \n- Sign Convention: \n * Observer towards source: $v_o$ is '+' (numerator). \n * Observer away from source: $v_o$ is '-' (numerator). \n * Source towards observer: $v_s$ is '-' (denominator). \n * Source away from observer: $v_s$ is '+' (denominator). \n- Key Principle: Apparent frequency changes due to relative motion, actual frequency ($f_0$) and speed of sound ($v$) in medium remain constant. \n- Reflection: Double Doppler effect. For source approaching stationary wall and hearing reflection: $f' = f_0 \left( \frac{v+v_s}{v-v_s} \right)$. \n- Wind: Adjust $v$ to $v_{eff} = v \pm v_{wind}$ (add if wind aids sound, subtract if opposes).

2-Minute Revision

The Doppler Effect is the apparent change in frequency of a wave due to relative motion between the source and the observer. The source's actual frequency ($f_0$) and the wave's speed ($v$) in the medium remain constant.

The general formula for sound waves is $f' = f_0 \left( \frac{v \pm v_o}{v \mp v_s} \right)$. The crucial part is the sign convention: if motion causes an *increase* in perceived frequency, use '+' for $v_o$ (numerator) and '-' for $v_s$ (denominator).

If motion causes a *decrease*, use '-' for $v_o$ and '+' for $v_s$. Remember that $v_o$ and $v_s$ are components of velocity along the line connecting source and observer. \nFor reflection problems, the Doppler Effect occurs twice.

First, the reflecting surface acts as an observer, receiving a shifted frequency. Second, it acts as a source, re-emitting this shifted frequency, which is then detected by the original observer. A common case is a source approaching a stationary wall and hearing its own reflection, where $f' = f_0 \left( \frac{v+v_s}{v-v_s} \right)$.

If wind is present, the effective speed of sound $v_{eff}$ must be used, which is $v \pm v_{wind}$ depending on the wind's direction relative to sound propagation.

5-Minute Revision

The Doppler Effect is a fundamental wave phenomenon where the perceived frequency ($f'$) of a wave differs from its emitted frequency ($f_0$) due to relative motion between the source and the observer.

It's vital to remember that the actual frequency of the source and the speed of the wave ($v$) in the medium are constant; only the *apparent* frequency changes. \n\nThe general formula for the apparent frequency of sound waves is: \n

$$f' = f_0 \left( \frac{v \pm v_o}{v \mp v_s} \right)$$

\nHere, $v$ is the speed of sound, $v_o$ is the speed of the observer, and $v_s$ is the speed of the source.

The critical aspect is the sign convention:\n* **Numerator ($v \pm v_o$):** Use '+' if the observer moves *towards* the source (increases frequency). Use '-' if the observer moves *away* from the source (decreases frequency).

\n* **Denominator ($v \mp v_s$):** Use '-' if the source moves *towards* the observer (compresses waves, increases frequency). Use '+' if the source moves *away* from the observer (stretches waves, decreases frequency).

\n\nExample 1: Source approaching stationary observer\n$f_0 = 1000\,\text{Hz}$, $v = 340\,\text{m/s}$, $v_s = 20\,\text{m/s}$, $v_o = 0$. \n$f' = 1000 \left( \frac{340}{340 - 20} \right) = 1000 \left( \frac{340}{320} \right) = 1000 \times \frac{17}{16} = 1062.

5\,\text{Hz}$. (Frequency increases)\n\n**Example 2: Observer receding from stationary source**\n$f_0 = 1000\,\text{Hz}$,$v = 340\,\text{m/s}$,$v_s = 0$,$v_o = 10\,\text{m/s}$. \n$f' = 1000 \left( \frac{340 - 10}{340} \right) = 1000 \left( \frac{330}{340} \right) = 1000 \times \frac{33}{34} \approx 970.

59\,\text{Hz}$. (Frequency decreases)\n\n**Reflection Problems (Double Doppler):** When sound reflects off a moving object, the Doppler Effect occurs twice. \n* **Stage 1:** The reflecting object acts as an observer, receiving a shifted frequency ($f_{rec}$).

\n* Stage 2: The reflecting object acts as a source, emitting the reflected sound at $f_{rec}$, and the original observer detects this. \nFor a source moving towards a stationary wall and hearing its own reflection, the formula simplifies to: \n

$$f' = f_0 \left( \frac{v+v_s}{v-v_s} \right)$$

\n\nEffect of Wind: If wind is blowing, the effective speed of sound relative to the ground changes.

If wind blows in the direction of sound, $v_{eff} = v + v_{wind}$. If against, $v_{eff} = v - v_{wind}$. This $v_{eff}$ replaces $v$ in the Doppler formula. \n\nMastering these concepts and practicing various scenarios, especially those involving reflection and relative motion of both source and observer, is crucial for NEET.

Prelims Revision Notes

The Doppler Effect is the apparent change in frequency of a wave due to relative motion between the source and the observer. \n\nKey Formula (Sound Waves): \n$f' = f_0 \left( \frac{v \pm v_o}{v \mp v_s} \right)$ \nWhere: \n* $f'$ = Apparent frequency \n* $f_0$ = Actual (source) frequency \n* $v$ = Speed of sound in the medium (constant) \n* $v_o$ = Speed of the observer relative to the medium \n* $v_s$ = Speed of the source relative to the medium \n\nSign Conventions (Crucial for NEET): \n* **Observer ($v_o$ in numerator):** \n * '+' if observer moves *towards* the source (increases frequency).

\n * '-' if observer moves *away* from the source (decreases frequency). \n* **Source ($v_s$ in denominator):** \n * '-' if source moves *towards* the observer (compresses waves, increases frequency).

\n * '+' if source moves *away* from the observer (stretches waves, decreases frequency). \n\nSpecial Cases: \n1. **Source moving, Observer stationary ($v_o = 0$):** \n * Towards: $f' = f_0 \left( \frac{v}{v - v_s} \right)$ \n * Away: $f' = f_0 \left( \frac{v}{v + v_s} \right)$ \n2.

**Observer moving, Source stationary ($v_s = 0$):** \n * Towards: $f' = f_0 \left( \frac{v + v_o}{v} \right)$ \n * Away: $f' = f_0 \left( \frac{v - v_o}{v} \right)$ \n\nReflection Problems (Double Doppler): \n* When sound reflects off a moving object, the Doppler Effect applies twice.

\n* Step 1: Calculate frequency received by the reflector ($f_{rec}$), treating the reflector as an observer. \n* Step 2: Calculate frequency heard by the original observer ($f'$), treating the reflector as a source emitting $f_{rec}$.

\n* Shortcut for Source approaching stationary wall and hearing its own reflection: $f' = f_0 \left( \frac{v+v_s}{v-v_s} \right)$ \n\nEffect of Wind: \n* If wind is present, the effective speed of sound relative to the ground changes.

\n* $v_{eff} = v + v_{wind}$ (if wind blows in direction of sound propagation). \n* $v_{eff} = v - v_{wind}$ (if wind blows opposite to sound propagation). \n* Replace $v$ with $v_{eff}$ in the Doppler formula.

\n\nImportant Points: \n* The speed of sound ($v$) is constant relative to the medium. \n* The actual frequency ($f_0$) of the source does not change. \n* Only the component of velocity along the line connecting source and observer matters.

\n* For light, the effect is called Redshift (moving away) and Blueshift (moving towards).

Vyyuha Quick Recall

Do Observers Shift Frequency? Yes!