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Physics Chapter 9 : Topic 3
Doppler Effect Formula
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In the Doppler Effect formula, $\mathbf{f}_{\mathbf{o}}$ represents:
Reasoning: $f_{o}$ is the apparent frequency heard by the stationary observer.
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In the equation $\mathbf{f}_{\mathbf{o}}\mathbf{=}$ $\frac{\mathbf{f}_{\mathbf{s}}\mathbf{\ v}}{\mathbf{v\ + \ }\mathbf{v}_{\mathbf{s}}\mathbf{\ }}$, $\mathbf{f}_{\mathbf{s}}$ represents:
Reasoning: $\mathbf{f}_{\mathbf{s}}$ is the actual frequency produced by the vibrating source.
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In the Doppler formula, $\mathbf{(v)}$ represents:
Reasoning: $\mathbf{(v)}$ is the speed of sound in the medium.
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The quantity $\mathbf{v}_{\mathbf{s}}$ in the Doppler formula represents:
Reasoning: $\mathbf{v}_{\mathbf{s}}$ is the velocity of the sound source relative to the medium.
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When the sound source approaches a stationary observer, the correct formula is:
Reasoning: For approaching source, wavelength decreases and observed frequency increases.
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When the source approaches the observer, the denominator in the Doppler equation:
Reasoning: For approach: $\mathbf{v -}\mathbf{v}_{\mathbf{s}}$ is used, making the denominator smaller.
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For a source moving away from a stationary observer, the formula becomes:
Reasoning: A receding source increases wavelength, causing lower observed frequency.
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A smaller denominator in the Doppler formula causes:
Reasoning: Frequency is inversely proportional to the denominator.
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A larger denominator in the Doppler formula causes:
Reasoning: A larger denominator reduces the value of $f_{o}$.
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If $\mathbf{v}_{\mathbf{s}}\mathbf{= 0}$, the Doppler formula gives:
Reasoning: $f_{o} =$ $\frac{\mathbf{f}_{\mathbf{s}}\mathbf{\ v}}{\mathbf{v}}\mathbf{=}\mathbf{f}_{\mathbf{s}}$ $\mathbf{\Longrightarrow}$ No relative motion means no Doppler effect.
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A source of frequency $\mathbf{500\ Hz}$ approaches an observer. If speed of sound is $\mathbf{340\ m/s}$ and source speed is $\mathbf{20\ m/s}$, observed frequency is:
Reasoning: $f_{o} =$ $\frac{\mathbf{500\ (340)}}{\mathbf{340\ - \ 20}}\mathbf{=}\frac{\mathbf{170000}}{\mathbf{320}}\mathbf{= 531.25\ Hz}$
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A source of frequency $\mathbf{600\ Hz}$ moves away at $\mathbf{30\ m/s}$. Speed of sound is $\mathbf{330\ m/s}$. Observed frequency is:
Reasoning: $\mathbf{f}_{\mathbf{o}} =$ $\frac{\mathbf{600\ (330)}}{\mathbf{330\ + \ 30}}\mathbf{= 550\ Hz}$
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A stationary observer hears $\mathbf{680\ Hz}$ from a source producing $\mathbf{640\ Hz}$. Thesource is:
Reasoning: Observed frequency greater than source frequency indicates approach.
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A source emits $\mathbf{400\ Hz}$ sound. The observer hears $\mathbf{360\ Hz}$. The source is:
Reasoning: A lower observed frequency indicates recession.
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A source moves toward an observer at $\mathbf{50\ m/s}$. If sound speed is $\mathbf{350\ m/s}$, the denominator in the formula is:
Reasoning: For approaching source $\mathbf{\Longrightarrow}$ $\mathbf{v -}\mathbf{v}_{\mathbf{s}}\mathbf{= 350 - 50 = 300\ }\mathbf{m/s}$
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The observed frequency is greater than source frequency when:
Reasoning: Subtracting $\mathbf{v}_{\mathbf{s}}$ reduces denominator and increases frequency.
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The Doppler formula shows that observed frequency depends on:
Reasoning: The formula includes source velocity relative to observer.
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If the speed of the source increases toward the observer, $\mathbf{f}_{\mathbf{o}}$:
Reasoning: Increasing $\mathbf{v}_{\mathbf{s}}$ reduces denominator $\mathbf{v -}\mathbf{v}_{\mathbf{s}}$.
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If a source moving away increases its speed, the observed frequency:
Reasoning: For recession, denominator $\mathbf{v +}\mathbf{v}_{\mathbf{s}}$ increases.
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The Doppler Effect disappears when:
Reasoning: Without relative motion $\mathbf{\Longrightarrow}$ $f_{o}\mathbf{=}\mathbf{f}_{\mathbf{s}}$
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A source moving toward an observer produces a higher frequency because:
Reasoning: Compression reduces wavelength and increases observed frequency.
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In the Doppler Effect, the actual frequency of the source:
Reasoning: Only observed frequency changes due to relative motion.
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If source speed is small compared with sound speed, the Doppler shift is:
Reasoning: Small velocity difference produces a small frequency change.
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A source moving away from an observer gives a denominator greater than $\mathbf{v}$ because:
Reasoning: For a receding source $\mathbf{\Longrightarrow}$ $\mathbf{v +}\mathbf{v}_{\mathbf{s}}$ is used.
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Which condition gives maximum observed frequency?
Reasoning: Maximum compression of wavefronts gives maximum frequency.
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A source produces $\mathbf{1000\ Hz}$ sound and approaches at $\mathbf{40\ m/s}$. If sound speed is $\mathbf{360\ m/s}$, observed frequency is:
Reasoning: $f_{o}\mathbf{=}$ $\frac{\mathbf{1000\ (360)}}{\mathbf{330\ - \ 40}}\mathbf{= 1125\ Hz}$
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A source produces $\mathbf{800\ Hz}$ sound and moves away at $\mathbf{40\ m/s}$. Sound speed is$\mathbf{\ }\mathbf{320\ m/s}$. Observed frequency is:
Reasoning: $f_{o}\mathbf{=}$ $\frac{\mathbf{800\ (320)}}{\mathbf{320\ + \ 40}}\mathbf{= 711\ }\mathbf{Hz}\mathbf{\approx 720\ }\mathbf{Hz}$
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If $\mathbf{f}_{\mathbf{o}}\mathbf{=}\mathbf{f}_{\mathbf{s\ }}$, then:
Reasoning: No Doppler shift occurs without relative motion.
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The Doppler formula for a moving source assumes:
Reasoning: This expression applies specifically to a moving source and stationary observer.
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The main reason for using $\mathbf{v -}\mathbf{v}_{\mathbf{s}}$ for an approaching source is:
Reasoning: Approaching motion compresses wavefronts, increasing observed frequency.
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