Problem 81 - University Physics with Modern Physics

Question

A woman stands at rest in front of a large, smooth wall. She holds a vibrating tuning fork of frequency \(f_{0}\) directly in front of her (between her and the wall). (a) The woman now runs toward the wall with speed \(v_{\mathrm{W}}\) . She detects beats due to the interference between the sound waves reaching her directly from the fork and those reaching her after being reflected from the wall. How many beats per second will she detect? (Note: If the beat frequency is too large, the woman may have to use some instrumentation other than her ears to detect and count the beats.) (b) If the woman instead runs away from the wall, holding the tuning fork at her back so it is between her and the wall, how many beats per second will she detect?

Step-by-Step Solution

Answer

(a) The beat frequency is \( \frac{2 v_{\text{w}} f_{0}}{v - v_{\text{w}}} \). (b) The beat frequency is \( \frac{2 v_{\text{w}} f_{0}}{v + v_{\text{w}}} \).

Step 1: Understanding Beat Frequency

When two waves of slightly different frequencies interfere with each other, they produce beats. The beat frequency is the absolute difference between the frequencies of the two waves. In this problem, the woman's motion affects the frequency of the sound waves by creating a Doppler effect.

Step 2: Doppler Effect Moving Towards the Wall (Part a)

When the woman runs towards the wall, the frequency of the sound waves she perceives directly from the tuning fork is not affected by her motion. However, the frequency of the sound waves reflected from the wall is affected by her motion. This is due to the Doppler effect. The frequency of the reflected sound waves can be calculated using the formula: \( f' = f_{0} \left(\frac{v + v_{\text{w}}}{v - v_{\text{w}}}\right) \), where \( f_{0} \) is the frequency of the tuning fork, \( v \) is the speed of sound, and \( v_{\text{w}} \) is the speed of the woman.

Step 3: Calculating Beat Frequency Moving Towards the Wall (Part a)

The beat frequency \( f_{\text{beat}} \) is given by the absolute difference between the frequency of the direct sound \( f_{0} \) and the reflected sound \( f' \). Using the equation from the Doppler effect, the beat frequency is \( f_{\text{beat}} = \left| f' - f_{0} \right| = \left| f_{0} \left( \frac{v + v_{\text{w}}}{v - v_{\text{w}}} \right) - f_{0} \right| \). Simplifying, \( f_{\text{beat}} = \frac{2 v_{\text{w}} f_{0}}{v - v_{\text{w}}} \).

Step 4: Doppler Effect Moving Away from the Wall (Part b)

When the woman runs away from the wall, holding the tuning fork at her back, the frequency of the sound waves she perceives directly is also affected. For the sound waves reaching her directly from the fork, the frequency measured will be \( f_{1} = f_{0} \left( \frac{v - v_{\text{w}}}{v} \right) \). For the reflected waves, the frequency is \( f'_{b} = f_{0} \left( \frac{v}{v + v_{\text{w}}} \right) \).

Step 5: Calculating Beat Frequency Moving Away from the Wall (Part b)

For part b, the beat frequency is the absolute difference between the two frequencies derived earlier: \( f_{\text{beat}} = \left| f_{1} - f'_{b} \right| = \left| f_{0} \left( \frac{v - v_{\text{w}}}{v} \right) - f_{0} \left( \frac{v}{v + v_{\text{w}}} \right) \right| \). This simplifies to \( f_{\text{beat}} = \frac{2 v_{\text{w}} f_{0}}{v + v_{\text{w}}} \).

Key Concepts

Beat Frequency

Beat frequency is an important concept in wave theory. It occurs when two waves of slightly different frequencies overlap and interfere with one another. This interference creates the phenomenon of beats, where the sound alternates between loud and soft. This happens because at some points, the waves add up constructively (making the sound louder), and at others, they cancel each other out partially (making the sound softer). Beat frequency, specifically, can be described as the number of beats per second, and it is calculated as the absolute difference between the two frequencies of the waves involved. For instance, if two tuning forks, one vibrating at 256 Hz and another at 260 Hz, are struck together, the beat frequency would be 4 Hz. This means 4 beats per second are heard.

Sound Waves

Sound waves are longitudinal waves that propagate through a medium such as air, water, or solids. Unlike light, which can travel through a vacuum, sound requires a physical medium to move. These waves are generated by vibrating objects and consist of compressions and rarefactions. The frequency of a sound wave determines its pitch, while the amplitude affects its loudness.
There are several key properties of sound waves that are critical in understanding phenomena like the Doppler Effect and beats. These properties include:

• Wavelength: The distance between two consecutive compressions or rarefactions.
• Frequency: The number of wavelengths that pass a given point per second, measured in Hertz (Hz).
• Speed: The rate at which the sound wave travels through the medium, influenced by factors such as temperature and medium density.
• Amplitude: The height of the wave, which is related to the energy and intensity of the sound.

Understanding these properties is fundamental when solving physics problems related to sound, especially those involving Doppler shifts or interference patterns.

Wave Interference

When two or more sound waves cross each other, they interfere. This interaction can be temporary and results in a new wave pattern. Understanding wave interference is crucial when studying phenomena like beat frequency. There are two main types of wave interference:

• Constructive Interference: When waves meet in phase (their peaks and troughs line up), they add together, increasing the amplitude and resulting in a louder sound. Think of this as synchronizing two identical metronomes.
• Destructive Interference: When waves meet out of phase (peaks align with troughs), they cancel each other out, decreasing the amplitude and resulting in a softer or even silent sound. This is like two opposite waves meeting in the ocean, canceling each other out.

In many physics problems, like the one described in the original exercise, wave interference combines with other concepts such as the Doppler Effect to produce detectable results, like beats, that can be used to draw conclusions about wave properties in motion.

Physics Problem Solving

Physics problems, like the one in this exercise, require a systematic approach to ensure successful and understandable solutions. Here’s a simplified guide for tackling physics exercises:

• Understand the Problem: Begin by carefully reading the problem statement to identify the knowns and unknowns. Pay attention to how different concepts interact, such as the Doppler Effect with moving objects.
• Visualize the Scenario: Draw diagrams or sketches to help visualize what’s happening. This could include labeling different frequencies, directions of motion, and wave paths.
• Identify the Concepts: Determine which physics principles apply, such as interference, sound wave properties, or the Doppler Shift.
• Develop a Strategy: Formulate a plan to solve the problem. This might involve using mathematical formulas to calculate frequencies or understanding how a concept like beats will manifest in a given situation.
• Execute and Review: Perform the necessary calculations and immediately check your work for consistency. Reviewing each step can prevent small mistakes from affecting the overall solution.

By following these steps, students can systematically approach complex physics problems, making them more manageable and understandable.