Problem 45 - University Physics with Modern Physics

Question

Two train whistles, \(A\) and \(B\) , each have a frequency of 392 \(\mathrm{Hz} . A\) is stationary and \(B\) is moving toward the right (away from \(A\) ) at a speed of 35.0 \(\mathrm{m} / \mathrm{s}\) . A listener is between the two whistles and is moving toward the right with a speed of 15.0 \(\mathrm{m} / \mathrm{s}\) (Fig. E16.45). No wind is blowing. (a) What is the frequency from A as heard by the listener? (b) What is the frequency from \(B\) as heard by the listener? (c) What is the beat frequency detected by the listener?

Step-by-Step Solution

Answer

(a) 409.17 Hz; (b) 374.12 Hz; (c) 35.05 Hz.

Step 1: Understanding the Doppler Effect Formula

The Doppler effect formula for sound involving a stationary source and a moving observer is \( f' = f \frac{v + v_o}{v} \), where \( f' \) is the observed frequency, \( f \) is the source frequency, \( v \) is the speed of sound in air (approximately 343 m/s), and \( v_o \) is the observer's speed. The sign is positive because the observer is moving towards the source.

Step 2: Calculate Frequency from Train A

Train A is stationary, and the listener is moving toward it with a speed of 15.0 m/s. Using the Doppler effect formula, the frequency heard by the listener is \( f'_A = 392 \frac{343 + 15}{343} \approx 409.17 \text{ Hz} \).

Step 3: Calculate Frequency from Train B

For train B, which is moving away from the listener, use the Doppler effect formula in the form for a moving source and moving observer: \( f'_B = f \frac{v + v_o}{v + v_s} \), where \( v_s \) is the speed of the source (35 m/s). We compute: \( f'_B = 392 \frac{343 + 15}{343 + 35} \approx 374.12 \text{ Hz} \).

Step 4: Calculate the Beat Frequency

The beat frequency is the absolute difference between the two frequencies heard by the listener. It is \( f_{beat} = |f'_A - f'_B| = |409.17 - 374.12| = 35.05 \text{ Hz} \).

Key Concepts

Frequency

Frequency refers to how many times a wave repeats itself in one second. It's measured in Hertz (Hz), and in the context of sound waves, it determines the pitch of a sound we hear. Imagine frequency like the ticking of a clock; if the clock ticks faster (higher frequency), it sounds different than if it ticks slower (lower frequency). For example, on a piano, the middle C has a frequency of roughly 261.63 Hz, and a higher C note has a higher frequency.

Beat Frequency

When two sound waves of slightly different frequencies meet, they produce a phenomenon called beats. The beat frequency is simply the difference between the frequencies of these two sound waves. For instance, if you have one sound at 400 Hz and another at 405 Hz, the beat frequency will be 5 Hz. This beat frequency manifests as a pulsing or throbbing sound that you might hear when tuning musical instruments.

Sound Waves

Sound waves are a type of mechanical waves that travel through a medium like air, water, or solids. They're caused by vibrations from a source, like a speaker or a musical instrument, and they travel through the air as compressions and rarefactions. Think of sound waves like ripples in water; as you drop a stone in a pond, the ripples move outward from the source, which is similar to how sound waves spread from their origin.

Stationary Source and Moving Observer

In scenarios where a sound source is stationary and the observer moves towards it, the observer experiences a higher frequency than the original emitting frequency. This is due to the Doppler effect, where the sound waves compress as the observer approaches, creating a higher pitch. If instead, the observer moves away from the source, the waves are stretched, resulting in a lower frequency. Consider standing still while a car approaches you blaring its horn; as it gets closer, the pitch of the horn seems higher.

Moving Source and Moving Observer

This scenario is a bit trickier as both the source of the sound and the observer is in motion. The Doppler effect becomes more complex because the observed frequency depends on the speed and direction of both the source and the observer. For instance, if both the source and observer move towards each other, the frequency increases more than it would if only one were moving. Conversely, if they move apart, the frequency decreases. This dual movement creates a fascinating interplay of sound compression and expansion.