Problem 48 - University Physics with Modern Physics

Question

Moving Source vs. Moving Listener. (a) A sound source producing \(1.00-\mathrm{kHz}\) waves moves toward a stationary listener at one-half the speed of sound. What frequency will the listener hear? (b) Suppose instead that the source is stationary and the listener moves toward the source at one- half the speed of sound. What frequency does the listener hear'? How does your answer compare to that in part (a)? Explain on physical grounds why the two answers differ.

Step-by-Step Solution

Answer

(a) 2000 Hz, (b) 1500 Hz. Different outcomes due to how motion affects wave crests.

Step 1: Understand the Doppler Effect Formula for Moving Source

For a moving source, the Doppler Effect formula is given by:\[ f' = \frac{f}{1 - \frac{v_s}{v}} \]where \( f' \) is the observed frequency, \( f \) is the source frequency, \( v_s \) is the speed of the source, and \( v \) is the speed of sound. Given \( f = 1.00 \text{kHz} \) and \( v_s = \frac{v}{2} \), plug in these values to solve for \( f' \).

Step 2: Calculate Observed Frequency for Moving Source

Substituting the given values into the formula:\[ f' = \frac{1000}{1 - \frac{1}{2}} = \frac{1000}{0.5} = 2000 \text{ Hz} \]Therefore, the listener hears a frequency of \( 2000 \text{ Hz} \) when the source moves toward the stationary listener at half the speed of sound.

Step 3: Utilize Doppler Effect Formula for Moving Listener

For a moving listener, the Doppler Effect formula is:\[ f' = f \left( 1 + \frac{v_l}{v} \right) \]where \( v_l \) is the speed of the listener. With \( v_l = \frac{v}{2} \) and \( f = 1000 \text{ Hz} \), substitute these into the formula to find \( f' \).

Step 4: Calculate Observed Frequency for Moving Listener

Plug in the values into the formula:\[ f' = 1000 \left( 1 + \frac{1}{2} \right) = 1000 \times 1.5 = 1500 \text{ Hz} \]Thus, the listener hears a frequency of \( 1500 \text{ Hz} \) when moving toward the stationary source at half the speed of sound.

Step 5: Compare and Explain Differences

The frequency heard by the listener when the source is moving is \( 2000 \text{ Hz} \), while it's \( 1500 \text{ Hz} \) when the listener is moving. This difference occurs because the relative motion affects wave compression differently in each scenario: a moving source increases wave crests physically reaching the listener, whereas a moving listener simply increases the frequency of encountering existing wave crests.

Key Concepts

Moving Source

When a sound source moves towards a listener, the waves it emits compress in the direction of the motion. This causes an increase in the frequency of the sound waves reaching the listener. The phenomenon is known as the Doppler Effect. In the formula for a moving source, \( f' = \frac{f}{1 - \frac{v_s}{v}} \), the observed frequency \( f' \) depends on the speed of the source \( v_s \). As the source approaches, the denominator decreases, making \( f' \) larger. For instance, when the source's speed is half the speed of sound, the frequency heard doubles. Thus, in a moving source scenario, more crests of the sound waves hit the listener per second.

Imagine a train with a whistle moving towards you; the sound pitch goes higher than when it was static. The motion squeezes the sound waves moving toward you, making sound waves catch up to each other faster.

This is why the frequency heard by the stationary listener from the moving source is higher, exemplified by a frequency shift to 2000 Hz."},{