Problem 50 - Calculus Early Transcendentals: Pearson New International Edition

Question

. The circular frequency \(v\) of oscillation of a point is given by \(v=\frac{2 \pi}{\text { period }}\). What happens when you add two motions that have the same frequency or period? To investigate, we can graph the functions \(y(t)=2 \sin (\pi t / 5)\) and \(y(t)=\sin (\pi t / 5)+\) \(\cos (\pi t / 5)\) and look for similarities. Armed with this information. we can investigate by graphing the following functions over the interval \([-5,5]:\) (a) \(y(t)=3 \sin (\pi t / 5)-5 \cos (\pi t / 5)+2 \sin ((\pi t / 5)-3)\) (b) \(y(t)=3 \cos (\pi t / 5-2)+\cos (\pi t / 5)+\cos ((\pi t / 5)-3)\)

Step-by-Step Solution

Answer

When functions with the same frequency are added, the resulting function retains the original frequency, showing combined amplitude and phase effects.

Step 1: Understand the Problem

The problem is asking us to add two trigonometric functions with the same frequency or period, analyze a specific set of functions, and determine patterns or similarities. To approach this, we must recognize the common components in the functions and utilize their respective sinusoidal forms.

Step 2: Analyze the Given Functions

The exercise provides two initial functions: \( y_1(t) = 2 \sin(\pi t / 5) \) \( y_2(t) = \sin(\pi t / 5) + \cos(\pi t / 5) \). Both functions have components with a period of \( 10 \) since the circular frequency \( v = \frac{2\pi}{\pi/5} = 10 \). This hints at a potential investigation into the constructive or destructive interference of similar frequencies.

Step 3: Graph the Functions

Plot the given functions over the interval \([-5, 5]\) to visually inspect the patterns. This will aid in understanding how functions with similar frequencies interfere with each other. Notice that both the sine and cosine components oscillate with the same periodicity.

Step 4: Identify Patterns or Interferences

By graphing the two functions, identify whether they exhibit constructive interference (increasing amplitude) or destructive interference (decreasing amplitude). Consider checking plots generated using graphing tools like Desmos or a graphing calculator.

Step 5: Investigate the New Functions

Analyze the new functions: (a) \( y_3(t) = 3\sin(\pi t / 5) - 5\cos(\pi t / 5) + 2\sin((\pi t / 5)-3) \) (b) \( y_4(t) = 3\cos(\pi t / 5 - 2) + \cos(\pi t / 5) + \cos((\pi t / 5)-3) \).Notice that despite their different coefficients and phase shifts, they all have the same core frequency of \( \frac{2\pi}{10} \).

Step 6: Compare Results

Evaluate the amplitude changes and potential phase shifts across functions. After graphing, check if multiple wave components might result in a single consistent periodic pattern, demonstrating synchronized oscillation due to the shared frequency.

Step 7: Draw Conclusions

Having observed the graphs, conclude that when multiple periodic functions with the same frequency are added, their superposition results in a new wave that has the same frequency. The resulting function demonstrates features dependent on the amplitudes and phases of the added waveforms.

Key Concepts

Circular Frequency

Circular frequency is a fundamental concept in understanding how oscillations work. It represents how often a wave completes a full cycle of oscillation in radians per unit time. The formula to calculate circular frequency \( v \) is given by \( v = \frac{2\pi}{T} \), where \( T \) is the period of the oscillation.

This measure is crucial when dealing with periodic phenomena because it remains constant regardless of how one transforms time (stretching or compressing the timeline). For example, in an exercise where functions like \( y(t) = 2\sin(\pi t / 5) \) and \( y(t) = \sin(\pi t / 5) + \cos(\pi t / 5) \) are analyzed, the circular frequency is constant, even if you change the amplitude or something else in the waveform.

Understanding circular frequency helps in assessing how quickly a wave repeats itself, and this uniform measure allows for easy comparison between waves.

Oscillation Period

The oscillation period is the time it takes for a wave to make one complete cycle. It is the reciprocal of frequency, helping to define how many oscillations occur within a specific time. Period \( T \) is particularly important when you want to determine the properties of a wave or how two waves interact together.

For example, in the exercise provided, both \( y(t) = 2\sin(\pi t / 5) \) and \( y(t) = \sin(\pi t / 5) + \cos(\pi t / 5) \) are calculated to have a period of \( 10 \), given their circular frequency formula \( v = \frac{2\pi}{\pi/5} = 10 \). This shared period implies these functions will align and repeat themselves identically every 10 time units.

Recognizing the period helps identify patterns, such as when two waves might peak or trough together, a feature crucial for exploring interference effects.

Constructive and Destructive Interference

Interference, a fundamental concept in wave theory, occurs when two or more waves overlap. The result of this overlap depends on the phase and amplitude of the individual waves. The two main types of interference are constructive and destructive interference.

• **Constructive Interference**: When waves meet in phase, their amplitudes add together, resulting in a wave with a larger amplitude. This effect often leads to increased overall intensity, as observed when two synchronized sound waves become louder.
• **Destructive Interference**: When waves meet out of phase, their amplitudes subtract from one another, which can lead to a decrease in overall intensity, sometimes even canceling each other out completely.

In our exercise, by graphing the functions over the interval \([-5, 5]\), you can visually see changes in amplitude, which can be attributed to constructive or destructive interference between the wave components with the same frequency.

Sine and Cosine Functions

Sine and cosine functions form the backbone of trigonometric analysis. Both are periodic functions with specific properties useful in modeling oscillatory behavior such as waves.

The sine function, \( \, sin(x) \), starts at zero, rises to 1 by \( \frac{\pi}{2} \), and then completes its cycle by \( 2\pi \), creating a smooth, wave-like curve. The cosine function, \( \, cos(x) \), follows a similar pattern but starts at 1 when \( x = 0 \) and is essentially a phase-shifted sine wave by \( \frac{\pi}{2} \).

In the exercise, functions that combine sine and cosine, such as \( y(t) = 2\sin(\pi t / 5) \) and \( y(t) = \sin(\pi t / 5) + \cos(\pi t / 5) \), demonstrate how these trigonometric elements can blend to form complex waveforms.

These functions allow the representation of movements through various dimensions and are critical for understanding interference because they clearly display oscillations as players in the wave's behavior.