Chapter 6: Problem 111

Solve each problem. Wave Action The vertical position of a floating ball in an experimental wave tank is given by the equation \(x=2 \sin (\pi t / 3)\) where \(x\) is the number of feet above sea level and \(t\) is the time in seconds. For what values of \(t\) is the ball \(\sqrt{3} \mathrm{ft}\) above sea level?

### Short Answer

## Step by step solution

## - Understand the Given Equation

## - Set Up the Given Condition

## - Isolate \( \sin \left( \frac{\pi t}{3} \right) \)

## - Find the Solutions for the Sine Equation

## - Solve for \( t \)

## - Generalize the Solution

## Key Concepts

These are the key concepts you need to understand to accurately answer the question.

###### trigonometric equations

To begin with, we set \( x = \sqrt{3} \) and then manipulate the equation step by step to isolate the sine function.

Here's a breakdown:

- Set \( x = \sqrt{3} \): \( \sqrt{3} = 2 \, \text{sin} \left( \frac{\pi t}{3} \right) \)
- Divide by 2: \( \text{sin} \left( \frac{\pi t}{3} \right) = \frac{\sqrt{3}}{2} \)

###### sinusoidal wave

The sine function here oscillates between -1 and 1, meaning our wave oscillates vertically between -2 and 2 feet over time. The argument of the sine function, \( \frac{\pi t}{3} \), affects the wave's frequency. Dividing by 3 slows down the oscillation because the wave completes its periodic cycle more slowly.

Sinusoidal waves are everywhere around us:

- Sound waves
- Light waves
- Ocean waves

###### wave motion

Wave motion can be categorized as:

- Transverse waves: where the oscillations are perpendicular to the wave direction (e.g., water waves).
- Longitudinal waves: where the oscillations are parallel to the wave direction (e.g., sound waves).

This concept is critical in understanding how waves function in different environments and applications such as:

- Marine studies
- Communication technologies
- Energy transfer within physical systems