Sine Wave Tidal written question

Question

would appreciate some help working out the sine equation for the following question please:

Depth of water is 6m at low tide and 16m at high tide, which is 6 hours later. Assuming the motion of the water is a simple harmonic, draw a graph to show how the water varies with time over a period of 12 hours, commencing at low tide.

Answer 1

We have the general form

$$y = a\sin b(t-d)+c$$

where:

• $\vert a\vert$ is the amplitude.
• $b$ is the frequency.
• $(d, c)$ is the horizontal and vertical shift.

Ignoring units for now, at low tide, the depth is $6$ and at high tide, the depth is $16$. Therefore, the middle point/equilibrium point is at $\frac{16+6}{2} = \frac{22}{2} = 11$. From here, you can deduce that:

• $a$ will be the height of the crest or the depth of the trough relative to the equilibrium point. Hence, $a = 16-11 = 5$ or $a = \vert 6-11\vert = \vert -5\vert = 5$.
• The graph is shifted vertically. Initially, the low-tide would be located at $-5$ (negative amplitude) but it’s now at $+6$ (low-tide), so $c = 11$.
• It takes $6$ hours to go from low-tide to high-tide. This means it would take $6$ hours to go back to low-tide and complete the cycle. Hence, there is one cycle in $12$ hours. We will treat $12$ as our $2\pi$ in normal trigonometry, so we can obtain $b$ through a ratio:

$$\frac{1}{12} = \frac{b}{2\pi} \iff b = \frac{\pi}{6}$$

• The graph is shifted horizontally. In a regular sine graph would start from the point of equilibrium. Here, however, you start from the low-tide. The shift must be by $\frac{T}{4} = 3$, so $c = 3$.

Adding all this together, you get

$$y = 5\sin\frac{\pi}{6}(x-3)+11$$

which becomes

$$y = 5\sin\left(\frac{\pi}{6}x-\frac{\pi}{2}\right)+11$$

If you’ve studied cosine graphs as well, you would notice $c = 3$ is really just $\frac{T}{4}$ (a quarter of the period), and the shift $\left(x-\frac{T}{4}\right)$ would produce a negative cosine graph, so you could also use

$$y = -5\cos\left(\frac{\pi}{6}x\right)+11$$

Here are the plots of the graphs.