Problem 77 - Precalculus Mathematics for Calculus

Question

Height of a Wave As a wave passes by an offshore piling, the height of the water is modeled by the function $$ h(t)=3 \cos \left(\frac{\pi}{10} t\right) $$ where \(h(t)\) is the height in feet above mean sea level at time \(t\) seconds. (a) Find the period of the wave. (b) Find the wave height, that is, the vertical distance between the trough and the crest of the wave. (Figure can't copy)

Step-by-Step Solution

Answer

(a) The period is 20 seconds. (b) The wave height is 6 feet.

Step 1: Understand the Function

The function given is \( h(t) = 3 \cos \left( \frac{\pi}{10} t \right) \). This is a cosine function with an amplitude and frequency that affect the wave's height and period respectively.

Step 2: Find the Period of the Wave

The cosine function has a general form \( a \cos(b t) \), whose period is given by \( \frac{2\pi}{b} \). Here \( b = \frac{\pi}{10} \). So, the period \( T \) is calculated as follows:\[T = \frac{2\pi}{\frac{\pi}{10}} = \frac{2\pi \times 10}{\pi} = 20\]Thus, the period of the wave is 20 seconds.

Step 3: Determine the Amplitude

The amplitude of the wave, which is the distance from the mean position to either the crest or trough, is given by the coefficient of the cosine function. Thus, the amplitude \( a = 3 \).

Step 4: Calculate the Wave Height

The wave height is the vertical distance between the trough and the crest. This is twice the amplitude. Therefore, the wave height is:\[2 \times 3 = 6\]So, the wave height is 6 feet.

Key Concepts

Periodic Functions

Periodic functions are functions that repeat their values in regular intervals or periods. These functions are common in many real-world situations, such as waves, oscillations, and circular movements. A classic example of a periodic function is the sine or cosine function, used frequently in trigonometry.

The main characteristics of a periodic function include:

• Repetition: The function values recur at regular intervals.
• Period: The length of an interval after which the function values repeat.

Understanding periodic functions is crucial in analyzing cycles in nature, like sound waves or ocean tides. In our wave function example, the periodicity helps determine when a wave crest will appear at the same height again.

Wave Amplitude

The wave amplitude is the maximum extent of a vibration or wave, measured from the position of equilibrium. In the context of waves, it is half of the wave height, the distance from the crest to the trough.

To find the amplitude in our wave function, look at the coefficient of the cosine function. For the equation \( h(t) = 3 \cos\left(\frac{\pi}{10} t\right) \), the amplitude is 3. This means that the displacement from the mean sea level to the crest (or trough) is 3 feet.

Amplitude is significant because:

• It indicates the energy of the wave—the greater the amplitude, the more energy it carries.
• It affects the wave's impact, especially in phenomena like noise levels or ocean waves hitting a shore.

Cosine Function

The cosine function is a trigonometric function that describes the position of points on a unit circle. It is known for its smooth periodic oscillation. One full cycle of the cosine wave represents 360 degrees or \( 2\pi \) radians.

In equations, the cosine function is often represented as \( a \cos(b t + c) \), where:

• \( a \) is the amplitude, dictating the peak values.
• \( b \) influences the frequency of repetition.
• \( c \) adjusts the wave's phase shift or horizontal displacement.

For our wave example \( h(t) = 3 \cos\left(\frac{\pi}{10} t \right) \), the cosine function plays a pivotal role in determining the behavior of the wave, defining both its height fluctuation and the periodic intervals.

Wave Height

Wave height refers to the vertical distance from the wave's trough to its crest. It can be calculated by doubling the amplitude. In our example, with an amplitude of 3, the wave height is \( 2 \times 3 = 6 \) feet.

Wave height is a crucial measurement in understanding wave dynamics and energy. It influences:

• Navigation and safety for vessels: Larger waves can be more hazardous.
• Coastal erosion and potential flooding.

The greater the wave height, the more energy the wave carries, often leading to significant impact on land and marine environments.

Wave Period

The wave period is the time taken for two successive crests (or troughs) to pass a fixed point. It reflects how long a cycle of a wave completes before repeating again.

To calculate the wave period in a cosine function like \( h(t) = 3 \cos\left(\frac{\pi}{10} t \right) \), use the formula for period: \( \frac{2\pi}{b} \). Here, \( b = \frac{\pi}{10} \), so the period is \( \frac{2\pi}{\frac{\pi}{10}} = 20 \) seconds.

Understanding the wave period is important as it helps predict the timing of wave arrivals, crucial for:

• Coastal engineering and creating structures that withstand wave forces.
• Fisheries and shipping, optimizing scheduling around wave patterns.