Problem 52 - University Physics with Modern Physics

Question

The siren of a fire engine that is driving northward at 30.0 \(\mathrm{m} / \mathrm{s}\) emits a sound of frequency 2000 \(\mathrm{Hz}\) . A truck in front of this fire engine is moving northward at 20.0 \(\mathrm{m} / \mathrm{s}\) . (a) What is the frequency of the siren's sound the fire engine's driver hears reflected from the back of the truck? (b) What wavelength would this driver measure for these reflected sound waves?

Step-by-Step Solution

Answer

(a) 2154.5 Hz; (b) 0.159 meters.

Step 1: Understanding the Doppler Effect

The Doppler Effect describes the change in frequency or wavelength of a wave in relation to an observer moving relative to the wave source. For this exercise, we're analyzing the frequency and wavelength changes as perceived by the fire engine driver from the sound that bounces back from the truck.

Step 2: Identify Given Values

The frequency of the siren emitted by the fire engine is 2000 Hz. The speed of both the fire engine and the truck are 30.0 m/s and 20.0 m/s, respectively.

Step 3: Calculate Frequency Heard by the Truck

First, we calculate the frequency heard by the truck driver, using the formula for the Doppler Effect: \[ f' = f \frac{v + v_t}{v + v_s} \]where:- \( f = 2000 \) Hz (frequency of the sound emitted),- \( v = 343 \) m/s (speed of sound in air),- \( v_t = 20 \) m/s (speed of truck),- \( v_s = 30 \) m/s (speed of the fire engine).After substituting the values:\[ f' = 2000 \frac{343 + 20}{343 + 30} \approx 2076.3 \text{ Hz} \]

Step 4: Calculate Frequency Reflected Back to the Fire Engine

Now, considering the truck as a source, reflecting the frequency \( f' \), we use the formula:\[ f'' = f' \frac{v + v_s}{v + v_t} \]Substitute \( f' = 2076.3 \) Hz, \( v_s = 30 \) m/s, \( v_t = 20 \) m/s:\[ f'' = 2076.3 \frac{343 + 30}{343 + 20} \approx 2154.5 \text{ Hz} \]

Step 5: Calculate the Wavelength

The wavelength of the reflected sound waves is given by:\[ \lambda = \frac{v}{f''} \]Using \( v = 343 \) m/s and \( f'' = 2154.5 \) Hz:\[ \lambda = \frac{343}{2154.5} \approx 0.159 \text{ meters} \]

Step 6: Verify and Conclude

We've calculated the frequency of the sound the fire engine driver hears after reflection at approximately 2154.5 Hz and the wavelength of this sound to be roughly 0.159 meters, which is consistent with the characteristics of the Doppler Effect when both source and observer are in motion.

Key Concepts

Sound Frequency

Sound frequency is a fundamental concept in understanding how sound waves work and how we perceive them. When a sound is emitted, its frequency is measured in hertz (Hz), which counts the number of wave cycles per second. In the case of the fire engine, the siren emits a frequency of 2000 Hz.
This is an important aspect of sound because frequency determines the pitch that we hear. A higher frequency means a higher pitch, while a lower frequency equates to a lower pitch. For moving vehicles, like a fire engine, the relative motion between the source of the sound (the siren) and the observer (the truck or the truck driver) influences the observed frequency due to the Doppler Effect.

Wave Reflection

Wave reflection occurs when a wave bounces back after hitting an object. In this scenario, the sound wave from the fire engine's siren hits the back of the truck, and reflects back towards the fire engine.
This reflection causes an additional Doppler shift in the frequency of the sound perceived by the fire engine driver. When waves reflect, their properties can change based on the relative motion of the reflecting object. Essentially, the truck acts like a new source of the sound after the siren's frequency has shifted once upon reaching it. This whole process is crucial to calculate the final frequency heard by the fire engine driver.

Speed of Sound

The speed of sound is vital for calculations involving sound waves. In air, under normal conditions, the speed of sound is approximately 343 meters per second (m/s).
This speed is the basis for determining how quickly the sound travels between the fire engine and the truck, as well as affecting how frequency alterations are perceived due to the Doppler Effect.

• It connects the changes in frequency to the physical distance covered by the sound wave within a specific time frame.
• A faster speed than the normal 343 m/s in a medium might lead to less pronounced Doppler shifts.
• Any calculation of wavelength or frequency shift relies on knowing this speed precisely.

Wavelength Calculation

Wavelength is the distance between consecutive points of a wave, usually crests, in a cycle of waves. It relates inversely to frequency: as frequency increases, wavelength decreases, and vice versa.
In this exercise, after obtaining the reflected frequency that the fire engine driver hears, the wavelength is calculated using:\[ \lambda = \frac{v}{f''} \]where \( v \) is the speed of sound in air (343 m/s), and \( f'' \) is the reflected frequency (2154.5 Hz).
The resulting wavelength of approximately 0.159 meters corresponds to the precision of increased frequency due to the Doppler shift observed. Understanding how wavelength varies with frequency helps in grasping how sound waves behave in different motion scenarios.