Problem 17 - Technical Mathematics with Calculus

Question

Plot each sine wave. $$y=3 \sin 377 t$$

Step-by-Step Solution

Answer

The sine wave can be plotted on a graph with a maximum height of 3 and a period of \(\frac{2\pi}{377}\). Connect the dots smoothly to visualize the sine wave pattern.

Step 1 - Understanding the Sine Wave Equation

The general form of a sine wave is given by the equation: \(y = A \sin(Bt + C) + D\), where \(A\) is the amplitude, \(B\) affects the period of the wave, \(C\) is the phase shift, and \(D\) is the vertical shift. In the given equation \(y = 3 \sin(377t)\), the amplitude \(A\) is 3, and there is no phase shift or vertical shift, which means \(C = 0\) and \(D = 0\). The term \(377t\) affects the frequency of the wave.

Step 2 - Determining the Period of the Wave

The period \(T\) of the sine wave is determined by the formula \(T = \frac{2\pi}{B}\), where \(B\) is the coefficient of \(t\) inside the sine function. In this case, \(B = 377\), so the period is \(T = \frac{2\pi}{377}\).

Step 3 - Plotting the Sine Wave

Plot the sine wave on a graph with the horizontal axis representing time \(t\) and the vertical axis representing \(y\). Start by plotting points for one period from \(t = 0\) to \(t = \frac{2\pi}{377}\), using the fact that the sine wave reaches its maximum at \(\frac{1}{4}\) of the period, its minimum at \(\frac{3}{4}\) of the period, and crosses the t-axis at \(0\), \(\frac{1}{2}\) of the period and the period itself. Multiply the sine values by the amplitude \(3\) to scale accordingly. Connect the points smoothly to show the periodic nature of the sine wave.

Key Concepts

Sine Wave Equation

Understanding the sine wave equation is fundamental when learning about waves and oscillations. It represents harmonic oscillations, which can describe various phenomena, from the vibrations of a guitar string to light waves. The general form is given by the formula:

\[y = A \sin(Bt + C) + D\]

where:\[A\] is the amplitude, which determines the peak value of the wave;\[B\] affects the wave's period, or how long it takes to complete one full cycle;\[C\] indicates the phase shift, or how much the wave is horizontally shifted; and\[D\] represents the vertical shift, or the baseline from which the wave oscillates. In our exercise, the sine wave equation is \[y = 3 \sin(377t)\], highlighting an amplitude of 3, a factor that will influence the frequency of the wave, with no phase or vertical shifts present.

Amplitude

Amplitude is a measure of how far the wave reaches from its equilibrium or rest position. It’s the height of the wave and can be thought of as the 'strength' or 'intensity' of the wave. Higher amplitudes represent stronger or more intense waves. In our example, the amplitude is represented by the number 3 in the equation \(y = 3 \sin(377t)\), indicating that the wave oscillates 3 units above and below its equilibrium position. This tells us that at its most extreme points, the sine wave will reach 3 units from the center line of the graph when plotted.

Wave Period

The wave period of a sine wave is the time it takes for one full cycle to occur and is denoted by \(T\). Mathematics has given us a handy formula to find the period:

\[T = \frac{2\pi}{B}\]

where \(2\pi\) represents a full circle in radians and \(B\) is the coefficient of \(t\) within the sine function. Thus, for our example with \(B = 377\), the wave period is\[T = \frac{2\pi}{377}\]. This number tells us how quickly the sine wave cycles, and since it’s based on the properties of a circle, it reflects the natural oscillatory patterns found in circular motion and waves.

Frequency of Wave

Frequency of a wave speaks to how often the cycles of the wave occur over a period of time, typically one second. It's commonly measured in hertz (Hz), accounting for the number of times the wave repeats itself within a second. The relationship between period \(T\) and frequency \(f\) is given by:

\[f = \frac{1}{T}\]

Since we've already determined the period \(T\) of our sine wave is \(\frac{2\pi}{377}\), we calculate its frequency as:\[f = \frac{377}{2\pi}\]. This indicates how fast the wave oscillates, which can relate to the pitch of a sound wave or the color of a light wave in different contexts. For this high-frequency wave of our exercise, you'd expect the graph to show very tight, closely-spaced oscillations, reflecting its rapid cycle rate.