Chapter 5.6, Problem 33E

Tides The Bay of Fundy in Nova Scotia has the highest tides in the world. In one 12-h period the water starts at mean sea level, rises to 21 ft above, drops to 21 ft below, then returns to mean sea level. Assuming that the motion of the tides is simple harmonic, find an equation that describes the height of the tide in the Bay of Fundy above mean sea level. Sketch a graph that shows the level of the tides over a 12-h period.

To determine

The equation that describes the height of the tide in the Bay of Fundy above mean sea level and sketch the graph.

Answer to Problem 33E

The equation that describes the height of the tide in the Bay of Fundy above mean sea level is $y=21\sin \frac{\pi }{6}t$ and graph is shown in Figure (1).

Explanation of Solution

Given:

In $12$ hour duration, water starts at mean sea level, rises to $21\text{ft}$ above and drops to $21\text{ft}$ below, and returns again to mean sea level.

Formula used:

The equations for displacement of objects in simple harmonic motion are,

$y=a\sin \omega t$ (1)

And,

$y=a\cos \omega t$ (2)

Where, $\left|a\right|$ is the amplitude, $\frac{2\pi }{\omega }$ is the time period and $\frac{\omega }{2\pi }$ is the frequency.

Calculation:

At $t=0$, the water starts from mean position.

It means at $t=0$, value of y is 0 which is possible in sine wave form.

Therefore use sine form of equation and not cosine form.

Obtain the values of a and $\omega$.

The water reaches to a maximum height of $21\text{ft}$ above sea level.

Time period is $12$ hours as one cycle is completed in $12$ hours of time.

Formula to calculate value of $\omega$ is,

$\text{Time}\text{period}=\frac{2\pi }{\omega }$

Substitute $12$ for time period in the above formual.

$\begin{array}{c}12=\frac{2\pi }{\omega }\\ \omega =\frac{2\pi }{12}\\ \omega =\frac{\pi }{6}\end{array}$

Substitute $21$ for a, $\frac{\pi }{6}$ for $\omega$, in equation (1).

$y=21\sin \frac{\pi }{6}t$

The equation that describes the height of tide in the Bay of Fundy above mean sea level is $y=21\sin \frac{\pi }{6}t$.

For function $y=21\sin \frac{\pi }{6}t$, the amplitude is $21$, therefore draw the graph with maximum displacement of $21$ above and below the x-axis. The time period is $12$, therefore, first cycle will begin at origin and complete at $12$ unit on time axis.

Figure (1)

Therefore, the graph is shown in Figure (1) and the equation that describes the height of tide in the Bay of Fundy above mean sea level is $y=21\sin \frac{\pi }{6}t$.