Question

When both source and listener are approaching each other the observed frequency of sound is given by ($V_L$ and $V_S$ is the velocity of listener and source respectively, $n_0 = \text{radiated frequency}$)

• A. $n=n_{0}\left[\frac{V+V_{L{V-V_{s\right]$
• B. $n=n_{0}\left[\frac{V-V_{L{V+V_{s\right]$
• C. $n=n_{0}\left[\frac{V-V_{L{V-V_{s\right]$
• D. $n=n_{0}\left[\frac{V+V_{L{V+V_{s\right]$

Hint:

Logic Tip: Approaching = higher pitch. To get a higher frequency $n$, the fraction multiplier must be greater than 1. This means maximizing the numerator ($+$) and minimizing the denominator ($-$). Thus, $\frac{V+V_L}{V-V_S}$ is the only logical choice!

Answer 1

The Correct Option is A

Concept:
The Doppler Effect formula for the apparent frequency $n$ observed by a listener is given by: $$n = n_0 \left[ \frac{V \pm V_L}{V \mp V_S} \right]$$ where $V$ is the speed of sound in the medium. The signs in the numerator and denominator depend on the relative direction of motion.
Step 1: Determine the sign for the numerator (Listener's motion).
The numerator dictates the effect of the listener's motion. If the listener moves \textit{towards} the source, the relative speed of the sound waves hitting the listener increases to $(V + V_L)$. This results in an increase in the apparent frequency. Thus, the numerator must be $(V + V_L)$.
Step 2: Determine the sign for the denominator (Source's motion).
The denominator dictates the effect of the source's motion. If the source moves \textit{towards} the listener, it compresses the sound waves, decreasing the effective wavelength to $(V - V_S)/n_0$. This also results in an increase in the apparent frequency. For the overall frequency to increase, the denominator must decrease. Thus, the denominator must be $(V - V_S)$.
Step 3: Combine the terms.
Combining both effects (since both are approaching each other and both effects act to increase the apparent frequency): $$n = n_0 \left[ \frac{V + V_L}{V - V_S} \right]$$