Problem 42 - Algebra and Trigonometry

Question

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). 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 ft/s past a woman standing on the shoulder of a highway, blowing its horn, which has a frequency of 500 Hz. Assume that the speed of sound is 1130 ft/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

Frequencies: 552 Hz (approaching), 456 Hz (receding). Models: \(y=A\sin(3467.02t)\) (approaching), \(y=A\sin(2868.51t)\) (receding).

Step 1: Understanding the Problem

We need to find what frequency a stationary observer perceives when a moving sound source travels towards and then away from them. We will use the Doppler effect formula to solve for both scenarios. Additionally, we'll create a sinusoidal function to model the sound waves perceived by the observer.

Step 1: Calculate Frequency as Car Approaches

The Doppler effect formula is \( f = f_0 \left( \frac{v_0}{v_0 ext{-} v} \right) \) when the car approaches the observer. Here, \( f_0 = 500 \text{ Hz} \), \( v_0 = 1130 \text{ ft/s} \), and \( v = 110 \text{ ft/s} \). Substitute these into the equation: \[ f = 500 \left( \frac{1130}{1130 - 110} \right) = 500 \left( \frac{1130}{1020} \right) \approx 551.96 \text{ Hz}. \] Thus, the frequency as the car approaches is approximately 552 Hz.

Step 2: Calculate Frequency as Car Moves Away

When the car recedes, the formula becomes \( f = f_0 \left( \frac{v_0}{v_0 + v} \right) \). Substitute the values: \[ f = 500 \left( \frac{1130}{1130 + 110} \right) = 500 \left( \frac{1130}{1240} \right) \approx 456.45 \text{ Hz}. \] So, the frequency as the car moves away is approximately 456 Hz.

Step 3: Model Sound with Sinusoidal Function as Car Approaches

The sinusoidal function for sound is \( y = A \sin \omega t \), with \( \omega = 2\pi f \). Use the frequency as the car approaches: \( \omega = 2\pi \times 551.96 \approx 3467.02 \). Thus, the function is \( y = A \sin(3467.02 t) \).

Step 4: Model Sound with Sinusoidal Function as Car Moves Away

When the car moves away, \( \omega = 2\pi f \) with \( f = 456.45 \). So, \( \omega = 2\pi \times 456.45 \approx 2868.51 \). The function becomes \( y = A \sin(2868.51 t) \).

Key Concepts

Frequency

Frequency is an essential element in understanding how sound is perceived, especially in the context of the Doppler Effect. Frequency refers to how often a sound wave cycles, typically measured in Hertz (Hz). Higher frequencies result in higher pitches, while lower frequencies produce lower pitches.

When a sound source, like a car horn, moves toward an observer, the sound waves are compressed, leading to a higher frequency and thus a higher pitch. Conversely, as the source moves away, the sound waves are stretched, lowering the frequency and the pitch heard by the observer.

In our case, the equation used to determine the perceived frequency when a car approaches or moves away is:

• Approaching: \( f = f_0 \left( \frac{v_0}{v_0 - v} \right) \)
• Receding: \( f = f_0 \left( \frac{v_0}{v_0 + v} \right) \)

Here, \( f_0 \) is the actual frequency emitted by the source, \( v_0 \) is the speed of sound, and \( v \) is the speed of the moving source.

Sinusoidal Function

Sinusoidal functions are mathematical functions that describe smooth, periodic oscillations. They are fundamental in modeling wave-like phenomena, such as sound waves. The general form of a sinusoidal function is \( y = A \sin(\omega t) \), where:

• \( A \) is the amplitude, which determines the peak value of the wave.
• \( \omega \) is the angular frequency, defined as \( 2\pi f \) where \( f \) is the frequency.
• \( t \) is time.

Sinusoidal functions are essential for representing sound waves, as they capture the periodic nature of these waves. Whether modeling the sound as the car approaches or recedes, the sinusoidal function changes based on the perceived frequency, reflecting how sound waves compress and stretch relative to the observer.

Sound Waves

Sound waves are longitudinal waves created by vibrating objects and propagated through a medium like air. They consist of compressions and rarefactions, transferring energy from the source to an observer. These waves are typically modeled using sinusoidal functions due to their periodic nature.

In the Doppler Effect scenario, the car horn is the sound source. As it moves towards or away from the observer, the observed frequency of the sound waves changes even though the actual frequency emitted by the horn remains constant. This change in perceived frequency is due to the Doppler Effect, which occurs because of the relative motion between the source and the observer, altering wave compression or extension.

Understanding sound waves' behavior helps in comprehending how frequency and waveform models such as sinusoidal functions capture these dynamics.

Speed of Sound

The speed of sound is the rate at which sound waves travel through a medium. In our case, it is given as 1130 ft/s, which is a typical speed in dry air at 70°F. The speed of sound depends on factors such as the medium's temperature, pressure, and density.

The speed of sound plays a crucial role in the Doppler Effect, as it is a key component of the formula used to calculate the perceived frequency when the sound source is moving. The formula takes into account the relative motion by comparing the speed of sound \( v_0 \) with the speed of the moving source \( v \).

Having a clear understanding of the speed of sound helps in comprehending how different speeds of the source impact sound perception, thus enhancing the grasp of concepts like the Doppler Effect.