Problem 28 - Algebra 2

Question

After a wave is created by a boat, the height of the wave can be modeled using \(y=\frac{1}{2} h+\frac{1}{2} h \sin \frac{2 \pi t}{P},\) where \(h\) is the maximum height of the wave in feet, \(P\) is the period in seconds, and \(t\) is the propagation of the wave in seconds. If \(h=3\) and \(P=2,\) write the equation for the wave and draw its graph over a 10-second interval.

Step-by-Step Solution

Answer

Wave equation: \( y = \frac{3}{2} + \frac{3}{2} \sin(\pi t) \). It oscillates from 0 to 3 every 2 seconds.

Step 1: Substitute Known Values

To start, substitute the given values of the maximum height and period into the wave equation. The known values are \( h = 3 \) and \( P = 2 \). Substitute these into the equation: \[ y = \frac{1}{2} \cdot 3 + \frac{1}{2} \cdot 3 \sin \frac{2\pi t}{2} \]. Simplify the expression to obtain: \[ y = \frac{3}{2} + \frac{3}{2} \sin(\pi t) \].

Step 2: Simplify the Equation

Extract the common factor from the sine term for further simplification. The equation is already mostly simplified:\[ y = \frac{3}{2} + \frac{3}{2} \sin(\pi t) \]. This is the form we can use to plot.

Step 3: Graph the Wave

Draw the graph of the equation \( y = \frac{3}{2} + \frac{3}{2} \sin(\pi t) \) over a 10-second interval. The wave is a sine wave beginning at \( y = \frac{3}{2} \) with an amplitude of \( \frac{3}{2} \) and a period of 2 seconds. Plot the points from \( t = 0 \) to \( t = 10 \), marking key points such as the maxima, minima, and zeros of the sine function.At \( t = 0, \ y = \frac{3}{2} \). At \( t = 1, \ y = 3 \). At \( t = 2, \ y = \frac{3}{2} \). At \( t = 3, \ y = 0 \), and so on. The graph is a wave oscillating between \( 0 \) and \( 3 \) with a period of 2 seconds.

Key Concepts

Wave Equation

A wave equation is a mathematical representation of a wave's motion and characteristics. It describes how the wave's height changes as time progresses. In the given exercise, the wave equation is expressed as:\[ y=\frac{1}{2} h+\frac{1}{2} h \sin \frac{2 \pi t}{P} \] In this equation:

• \( y \) denotes the wave height at any given time \( t \)
• \( h \) is the maximum height of the wave
• \( P \) represents the period of the wave

By substituting the boat's wave parameters \( h = 3 \) and \( P = 2 \) seconds into the wave equation, it transforms to:\[ y = \frac{3}{2} + \frac{3}{2} \sin(\pi t) \] This equation now makes it easy to plot a graph showing how the wave rises and falls over time. The sine function here contributes to the wave's oscillating nature, creating the familiar up-and-down motion of waves across a period.

Amplitude and Period

Amplitude and period are crucial parameters in describing waves. Understanding these terms helps to comprehend the wave's behavior represented in the equation.Amplitude refers to the maximum deviation from the wave's central point and determines the wave's height or strength.

• In the equation \( y = \frac{3}{2} + \frac{3}{2} \sin(\pi t) \), the amplitude is \( \frac{3}{2} \), which indicates how high the wave rises above its main position at \( y = \frac{3}{2} \).

Period is the time it takes for the wave to complete one full cycle of rise and fall.

• With \( P = 2 \) seconds in our equation, it signifies that the wave repeats its complete cycle every 2 seconds.

Together, the amplitude and period shape the graph of a wave, influencing the appearance of its peaks and troughs as well as how frequently they occur.

Sine Function

The sine function is a key component used to model wave patterns mathematically. It introduces the vital oscillating behavior found in waves. In a trigonometric context, sine functions exhibit regular and repeating cycles that mirror wave motion. In the wave equation \( y = \frac{3}{2} + \frac{3}{2} \sin(\pi t) \), the sine part \( \sin(\pi t) \) is what causes the wave to oscillate. Let's break it down:

• The function creates a repeating wave-like pattern, which begins at zero, rises to a peak, falls back to zero, descends to a trough, and finally returns to zero again.
• As \( t \) changes, the sine value fluctuates between -1 and 1, resulting in the wave's continuous rise and fall.

In this exercise, the sine function adds dynamism to the wave equation by ensuring that the wave has peaks and valleys, contributing to the sinusoidal nature of many real-world phenomena, like sound and light waves.