Problem 44 - Precalculus Mathematics for Calculus

Question

Doppler Effect When a car with its horn blowing drives by an observer, the pitch of the horn seems higher as it approaches and lower as it recedes (see the figure on the next page). This phenomenon is called the Doppler effect. If the sound source is moving at speed \(v\) relative to the observer and if the speed of sound is \(v_{0}\), then the perceived frequency \(f\) is related to the actual frequency \(f_{0}\) as follows: $$ f=f_{0}\left(\frac{v_{0}}{v_{0} \pm v}\right) $$ We choose the minus sign if the source is moving toward the observer and the plus sign if it is moving away. Suppose that a car drives at \(110 \mathrm{ft} / \mathrm{s}\) past a woman standing on the shoulder of a highway, blowing its hom, which has a frequency of \(500 \mathrm{Hz}\). Assume that the speed of sound is \(1130 \mathrm{ft} / \mathrm{s}\). (This is the speed in dry air at \(70^{\circ} \mathrm{F}\).) (a) What are the frequencies of the sounds that the woman hears as the car approaches her and as it moves away from her? (b) Let \(A\) be the amplitude of the sound. Find functions of the form $$ y=A \sin \omega t $$ that model the perceived sound as the car approaches the woman and as it recedes.

Step-by-Step Solution

Answer

As the car approaches: 553.9 Hz; as it moves away: 455.65 Hz. Sound models: \(y = A \sin(553.9 \times 2\pi t)\) and \(y = A \sin(455.65 \times 2\pi t)\).

Step 1: Understand the Given Information

We are given: - The car's speed: \( v = 110 \text{ ft/s} \) - The actual frequency of the horn: \( f_0 = 500 \text{ Hz} \) - The speed of sound \( v_0 = 1130 \text{ ft/s} \) Our task is to find the frequencies \( f \) the woman hears when the car approaches and moves away, and to model the sound wave.

Step 2: Calculate the Frequency as the Car Approaches

When the car approaches, we use the minus sign in the formula: \[ f = f_0 \left( \frac{v_0}{v_0 - v} \right) \] Substitute the values: \[ f = 500 \left( \frac{1130}{1130 - 110} \right) = 500 \left( \frac{1130}{1020} \right) \] Calculate \( f \): \[ f \approx 500 \times 1.1078 \approx 553.9 \text{ Hz} \]

Step 3: Calculate the Frequency as the Car Moves Away

When the car recedes, we use the plus sign in the formula: \[ f = f_0 \left( \frac{v_0}{v_0 + v} \right) \] Substitute the values: \[ f = 500 \left( \frac{1130}{1130 + 110} \right) = 500 \left( \frac{1130}{1240} \right) \] Calculate \( f \): \[ f \approx 500 \times 0.9113 \approx 455.65 \text{ Hz} \]

Step 4: Model the Sound Wave as the Car Approaches

For the approaching car, the perceived frequency \( f \) is 553.9 Hz. Convert frequency to angular frequency \( \omega = 2\pi f \). \[ \omega = 2 \pi \times 553.9 \text{ rad/s} \] The sound wave function is: \[ y = A \sin(553.9 \times 2\pi t) \]

Step 5: Model the Sound Wave as the Car Moves Away

For the receding car, the perceived frequency is 455.65 Hz. Convert frequency to angular frequency \( \omega = 2\pi f \). \[ \omega = 2 \pi \times 455.65 \text{ rad/s} \] The sound wave function is: \[ y = A \sin(455.65 \times 2\pi t) \]

Key Concepts

Frequency Modulation

Frequency modulation is a concept that plays a significant role in the Doppler Effect. It is the change in frequency of a wave in relation to an observer moving relative to the source of the wave.
In simpler terms:

• If the source moves towards the observer, the waves compress, and the frequency increases.
• If the source moves away, the waves stretch, and the frequency decreases.

This concept is beautifully exemplified in the scenario where a moving car's horn sounds different to a stationary listener.
As the car with the blowing horn approaches, the frequency appears higher due to the compression of sound waves, known as a positive frequency shift. When the car recedes, the frequency lowers because the waves are stretched, illustrating a negative frequency shift.
This change in frequency is modeled using the formula:
\[ f = f_0 \left( \frac{v_0}{v_0 \pm v} \right) \]
Understanding this formula helps us grasp how waves interact with motion, which is the essence of frequency modulation in the real world.

Wave Function

Wave functions are a powerful tool to describe waves mathematically. In terms of sound, the wave function defines how sound waves propagate over time.
The general form of a wave function is:

• \( y = A \sin(\omega t) \)

Let's break it down:

• \(y\) represents the displacement of the wave at a given point and time.
• \(A\) is the amplitude, showing the wave's maximum displacement, correlating to the loudness of the sound.
• \(\omega\) is the angular frequency, which relates to the wave's speed of oscillation.
• \(t\) is time, indicating the wave's progression at that moment.

In the Doppler Effect scenario, as the car approaches:
\( y = A \sin(1107.8\pi t) \) expresses the higher frequency sound waves.
Conversely, for the receding car:
\( y = A \sin(911.3\pi t) \) shows the lower frequency, as the car moves away.
Wave functions effectively capture how the Doppler Effect translates these real-world events into mathematical representation.

Angular Frequency

Angular frequency is a vital concept when discussing wave behavior, particularly in wave functions. It is denoted by \(\omega\) and is essential in determining the wave's rate of oscillation.
The formula for angular frequency is:

• \(\omega = 2\pi f\)

where \(f\) is the frequency.
Angular frequency provides insight into how fast or slow a wave oscillates.
Consider a basic analogy: imagine a hand tracking around a clock face.

• The faster it moves, the higher the angular frequency.
• The slower it moves, the lower the angular frequency.

In the context of the car example, calculating \(\omega\) helps us model the sound waves accurately using the wave function.
For the approaching car with a frequency of approximately 553.9 Hz, the angular frequency becomes:
\( \omega = 2\pi \times 553.9 \approx 1107.8\pi \) rad/s.
While for the receding car at 455.65 Hz, it's:
\( \omega = 2\pi \times 455.65 \approx 911.3\pi \) rad/s.
This demonstrates how angular frequency reflects changes in pitch due to the Doppler Effect.