What equation is used in real life

Life on the coast

Kalle lives in the village of Deichblick on the North Sea coast. It measures the water level every hour on a day and enters it in a coordinate system.
x-axis: time in hours
y-axis: water level in m



Kalle has connected his entered points: If that doesn't look like a sine function!

The sine function has the general equation $$ f (x) = a * sin (b * (x-c)) + d $$.

Kalle wants to determine the parameters. Then he could calculate the water level for any point in time (insert x, calculate y).

Yeah, in reality the curve doesn't look exactly like that. :-) The period length of the tides is actually 12.44 hours. Therefore, the tides shift backwards by about an hour from day to day. In addition to the position of the moon, there are other influences. But the sine curve is always recognizable.

Image: U. Muuß

People who live with ebb and flow need the times of high and low tide every day. It can look like this:

Image: Günter Schmidt

Parameter $$ a $$

The parameter $$ a $$ indicates how much the curve is stretched in the y-direction.



The height difference in the red water level curve is twice as great as in the simple sine curve. With the simple sine curve, $$ a = 1 $$. This means that $$ a = 2 $$ for the red curve.

calculate a
Find the distance between the maximum and minimum values ​​of the curve. Then divide by 2.

$$ a = (Max - Mi n) / 2 = (6-2) / 2 = 2 $$

The Parameter $$ a $$ you determine by subtracting the smallest from the largest function value and then dividing the result by 2.

$$ a = (Max - Mi n) / 2 $$

General functional equation: $$ f (x) = a * sin (b * (x-c)) + d $$

Parameter $$ d $$

The parameter $$ d $$ indicates how much the curve is shifted in the y direction.



Look at how the zeros of the simple sinusoid are shifted. The red curve is shifted 4 units upwards.

d calculate
Calculate the average water level. To do this, add the minimum and maximum water levels (you have just used the two values) and divide the result by 2.

$$ d = (Max + Mi n) / 2 = (6 + 2) / 2 = 4 $$

The Parameter d you determine by adding the largest function value and the smallest and then dividing the result by 2.

$$ d = (Max + Mi n) / 2 $$

General functional equation: $$ f (x) = a * sin (b * (x-c)) + d $$

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Parameter $$ b $$

The parameter $$ b $$ indicates how much the curve is compressed in the x-direction. To do this, determine the period length.



b calculate
The period of the simple sine curve is $$ 2 pi $$. The period length of the red curve is 12.
b you calculate like this:

$$ b = (2pi) / text {period length} = (2 * pi) / 12 = pi / 6 $$

The Parameter $$ b $$ you determine by measuring the period length and then dividing $$ 2pi $$ by this measured value.

$$ b = (2pi) / text {period length} $$

General functional equation: $$ f (x) = a * sin (b * (x-c)) + d $$

Why does $$ b = (2pi) / text {period length} $$ apply? The period length of the simple sine curve is $$ 2pi $$. If the parameter b had the value $$ 2pi $$, the period length of the compressed curve would be 1. As with the rule of three, you now start from this new curve with period length 1 and stretch it by a factor of 12 in the example.

Parameter $$ c $$

The parameter $$ c $$ indicates how much the curve is shifted in the x-direction.
Usually the sine curve is shifted by a multiple of a quarter period length. Here you can see the examples:




There are several ways to determine the displacement:

First option:
You are looking for the point on the curve that corresponds to $$ sin (0) $$ on the "original sine". In our curve this is e.g. -3 or 9 (sine is periodic!). That is exactly your $$ c $$, and you get with $$ c = -3 $$

$$ f (x) = 2 * sin (pi / 6 (x + 3)) + 4 $$.

Second option:
In the red curve, there is a maximum at x = 0. Therefore you shift the whole curve by $$ (3pi) / 2 $$. All you have to do is move the argument $$ bx $$ and you will get it as a new argument

$$ f (x) = 2 * sin (pi / 6x-3/2 pi) + 4 $$.

General functional equation: $$ f (x) = a * sin (b * (x-c)) + d $$

Excursion by boat

Now you have the complete functional equation of the red water level curve!
$$ f (x) = 2 * sin (pi / 6 (x + 3)) + 4 $$.

What can you do with it now?

Determine the water level for a point in time
Kalles sailboat has a draft of 3 meters and he would like to know if he can sail in 65 hours.
When you have the function equation, you can e.g. B. use the calculator to calculate how high the water level is at the corresponding time. This would be the function value for
x = 65.

$$ f (65) approx 2.27 $$

This means that the water level is 2.3 m high after 65 hours and Kalle cannot leak.



Determine the point in time when a specified water level is reached
Kalle wants to show his niece, who has not come from the coast, what it looks like at low tide in two days. To do this, he has to know when the water is at its lowest. This would be the search for an x-value for which the water level is f (x) = 2 m. This leads to the following equation.

$$ f (x) = 2 $$

$$ 2 * sin (pi / 6 (x + 3)) + 4 = 2 $$


The solutions are then, since there are two low tides, Kalle has to go to the dike with his niece either at around hour 54 or at hour 66. You are looking for the solutions that lie between 48 and 72 hours, since then the day after next is (if you assume that x = 0 at 0 o'clock).

Image: fotolia.com (philipus)

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