Problem 74 - Precalculus: Functions and Graphs

Question

Identify the damping factor \(f(x)\) for the damped wave. Sketch graphs of \(y=\pm f(x)\) and the equation on the same coordinate plane for \(-2 \pi \leq x \leq 2 \pi\) $$y=3^{-x / 3} \cos 2 x$$

Step-by-Step Solution

Answer

The damping factor is \( f(x) = 3^{-x/3} \).

Step 1: Understand the Damped Wave Equation

The given equation is a damped wave equation of the form \( y = f(x) \cos(2x) \) where \( f(x) \) is the damping factor. In this context, we need to identify \( f(x) \).

Step 2: Isolate the Damping Factor

In the equation \( y = 3^{-x/3} \cos(2x) \), the damping factor is represented by the function multiplied by the \( \cos(2x) \). Hence, \( f(x) = 3^{-x/3} \).

Step 3: Sketch \( y = \pm f(x) \)

Plot the graphs of \( y = f(x) = 3^{-x/3} \) and \( y = -f(x) = -3^{-x/3} \). These curves decay exponentially as \( x \) increases because they are exponential functions with the base less than 1. The graph should show a curve starting from the top (for \( f(x) \)) and bottom (for \( -f(x) \)), decaying towards zero.

Step 4: Sketch the Damped Wave Equation

Sketch the graph of \( y = 3^{-x/3} \cos(2x) \) on the same coordinate plane. This graph oscillates between \( 3^{-x/3} \) and \( -3^{-x/3} \) since it is the product of the sinusoidal function \( \cos(2x) \) modulated by the damping factor \( 3^{-x/3} \). The wave will show clear oscillations damped by the envelope curves \( y = \pm f(x) \).

Step 5: Combine and Label the Graphs

Ensure the graph shows \( y = 3^{-x/3} \cos(2x) \) between its damping boundaries \( y = 3^{-x/3} \) and \( y = -3^{-x/3} \), clearly labeling each curve in the range \(-2\pi \leq x \leq 2\pi\).

Key Concepts

Damping Factor

In the study of wave equations, a damping factor is crucial. It determines how a wave's amplitude decreases over time or space. When we talk about the damped wave equation, we're looking specifically at how this factor affects the wave itself. The formula given in the exercise, \[ y = 3^{-x/3} \cos(2x) \] identifies the damping factor as \[ f(x) = 3^{-x/3} \]. This means that as the value of \( x \) grows larger, the term \( 3^{-x/3} \) becomes smaller, leading to a reduction in wave amplitude. This occurs because any number less than one raised to an increasing power reduces in value. Understanding the role of a damping factor helps in fields like acoustics or engineering, where controlling vibrations is critical.

Exponential Decay

Exponential decay describes a process where quantities reduce at a rate proportional to their size. This shows up clearly in the damping factor we've identified as \( f(x) = 3^{-x/3} \). Here, the base of the exponent, 3, is crucial. Since \( 3^{-x/3} \) decreases as \( x \) increases, it demonstrates exponential decay. This behavior is crucial in the wave equation: \[ y = 3^{-x/3} \cos(2x) \]. The decay affects how the wave's amplitude diminishes over time, resulting in smaller peaks and troughs as \( x \) expands.Key points about exponential decay in this context:

• Rapid reduction for initial values of \( x \).
• Approaches zero without ever reaching it.
• Defines the outer bounds of the oscillating wave.

Oscillations

Oscillations refer to repeated variations in a wave's values over a regular interval of time or space. In the damped wave equation \[ y = 3^{-x/3} \cos(2x) \], oscillations are defined by the cosine component \( \cos(2x) \). This trigonometric function inherently oscillates between -1 and 1. However, the presence of the damping factor \( f(x) = 3^{-x/3} \) modulates these oscillations.Here's how the interactions work:

• The wave starts at a maximum amplitude at \( x = 0 \).
• As \( x \) increases, oscillations remain frequent but become less pronounced.
• The damping factor reduces the effect of oscillations over time, showing diminishing peaks and valleys.

These modulated oscillations are key in understanding wave behavior where energy loss, such as friction or resistance, is a factor.

Wave Modulation

Wave modulation involves changing the properties of a wave, like amplitude or frequency, through another influence. In our damped wave equation \( y = 3^{-x/3} \cos(2x) \), the modulation is driven by the damping factor \( 3^{-x/3} \). This modulation makes the wave's amplitudes smaller as \( x \) grows.Key aspects of wave modulation include:

• The dampening effect alters only amplitude—not the frequency, which remains consistent due to the \( \cos(2x) \) portion.
• Envelope curves \( y = \pm 3^{-x/3} \) show limits of the wave oscillations due to this modulation.
• This concept helps in technology contexts, like in signal processing, where understanding how waves change over distance or time is needed.

In essence, wave modulation in damped waves highlights how external factors can control a wave's behavior, essential for numerous practical applications.