Problem 44 - Algebra and Trigonometry

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 horn, 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

(a) Approaching: 553.92 Hz, Receding: 455.65 Hz. (b) Approaching: \(y = A \sin(2\pi \times 553.92 \times t)\); Receding: \(y = A \sin(2\pi \times 455.65 \times t)\).

Step 1: Calculate Approaching Frequency

The original frequency of the horn is given as \(f_0 = 500\, \text{Hz}\) and the speed of the car is \(v = 110\, \text{ft/s}\). The speed of sound is \(v_0 = 1130\, \text{ft/s}\). Using the formula for the perceived frequency when the source is approaching: \[f = f_0 \left( \frac{v_0}{v_0 - v} \right)\] Substitute the given values: \[f = 500 \left( \frac{1130}{1130 - 110} \right) = 500 \left( \frac{1130}{1020} \right) \approx 553.92 \text{Hz}\]Thus, as the car approaches, the woman hears \(553.92\, \text{Hz}\).

Step 2: Calculate Receding Frequency

Using the same formula, but with the source moving away, the expression for frequency is: \[f = f_0 \left( \frac{v_0}{v_0 + v} \right)\] Substitute the given values: \[f = 500 \left( \frac{1130}{1130 + 110} \right) = 500 \left( \frac{1130}{1240} \right) \approx 455.65 \text{Hz}\]Thus, as the car moves away, the woman hears \(455.65\, \text{Hz}\).

Step 3: Model Approaching Sound Wave

The perceived frequency when the car approaches is \(553.92\, \text{Hz}\). The angular frequency \(\omega\) is related to frequency by \(\omega = 2\pi f\). Therefore, \(\omega = 2\pi \times 553.92\). The function for the sound is: \[y = A \sin(2\pi \times 553.92 \times t)\] This models the sound the woman hears as the car approaches.

Step 4: Model Receding Sound Wave

The perceived frequency when the car is moving away is \(455.65\, \text{Hz}\). Again, \(\omega = 2\pi f\). Thus, \(\omega = 2\pi \times 455.65\). The function that models this sound is: \[y = A \sin(2\pi \times 455.65 \times t)\] This is the model for the sound the woman hears as the car recedes from her.

Key Concepts

Perceived Frequency

The concept of perceived frequency is central to understanding the Doppler Effect. When a sound source moves relative to an observer, the frequency of the sound appears different from the actual frequency emitted. This happens because of the compression or elongation of sound waves.

As a car approaches you with honking horns, you hear a higher pitch because the sound waves are compacted. This results in a perceived frequency higher than the original. Conversely, as the car moves away, the sound waves are stretched, and the perceived frequency is lower than the actual frequency.

• Approaching source: The perceived frequency formula is \( f = f_0 \left( \frac{v_0}{v_0 - v} \right) \), which leads to a higher frequency.
• Receding source: The formula becomes \( f = f_0 \left( \frac{v_0}{v_0 + v} \right) \), resulting in a lower frequency.

This perceptive change is not because the sound waves themselves change, but because the distance they travel changes. This alteration in perceived frequency is why the Doppler Effect is so impactful in everyday acoustics.

Sound Wave Modeling

Sound wave modeling translates the complex interactions of sound frequencies into understandable mathematical functions. These models help us analyze how the sound from a moving source is heard by a stationary observer.

In our exercise, sound is modeled using sinusoidal wave equations: \( y = A \sin(\omega t) \). This function represents sound waves with amplitude \( A \) and angular frequency \( \omega \). The angular frequency is directly linked to how fast the wave oscillates, which is affected by the perceived frequency.

• Approaching Wave: The perceived frequency is higher, so the angular frequency \( \omega \) that determines wave oscillation is also higher, computed as \( \omega = 2\pi \times 553.92 \).
• Receding Wave: The oscillation is lower due to a lower perceived frequency, leading \( \omega \) to be \( 2\pi \times 455.65 \).

By plugging in values derived from the Doppler Effect equations into these models, you effectively visualize the sound change as the car moves.

Angular Frequency

Angular frequency is a crucial concept for relating a wave’s oscillation to its perceived frequency. Distinct from linear frequency, angular frequency \( \omega \) denotes how many radians a wave completes per second.

In mathematical terms, angular frequency is linked to regular frequency (\( f \)) by the equation \( \omega = 2\pi f \), where \( 2\pi \) converts the cycle frequency into radians. In the context of sound waves, angular frequency offers a precise way to model and visualize the changes in the wave pattern as a sound source moves.

• When the source is approaching: The angular frequency increases because the perceived frequency is higher, reflecting faster oscillations.
• When the source is receding: Conversely, it decreases due to a lower perceived frequency.

This change in angular frequency directly impacts the mathematical model of the sound wave, influencing its representation as it approaches, passes by, and moves away from a listener.