Astrolabe Roadstead tides - Sinusoidal Functions

In summary, the conversation discusses the correct equation for the height of tides using the given information of high and low tides, period, amplitude, vertical shift, and phase shift. The final equation is -2.1 sin (29(x-0.4)) +2.4, with a phase shift of 0.4 hours. The conversation also mentions the rule of twelfths and the importance of using an accurate equation when determining the best time for a fisherman to come in to avoid getting stuck on sandbars.
  • #1
Evangeline101
112
5

Homework Statement


upload_2016-8-13_0-13-9.png


Homework Equations


upload_2016-8-13_0-13-43.png


3. The Attempt at a Solution
a) The height of the high tide is 4.5 m

b) The height of the low tide is 0.25 m

c)

Period = 12.5 hours k= 360/12.5 = 28.8
amplitude = 2.125 m
vertical shift = 2.375 m
phase shift = it doesn't look like there is any phase shift (correct if i am wrong please)

So the equation is:

-2.125 sin(28.8x) + 2.375

Is my equation correct?

d)

Determine height of tide at 5:00 pm:

y= -2.125 sin (28.8x) + 2.375

y= -2.125 sin (28.8(17)) + 2.375

y = -2.125 sin(489.6) +2.375

y = 0.737 m

The height of the tide at 5:00 pm is 0.74 m.

When would be a better time for the fisherman to come in?

If the fisherman comes into shore at 5:00 p.m. there will be a low tide with a height of 0.74 m. Since sandbars occur at a low tide (I am assuming) it would be best to come in when the tides are high to avoid getting stuck.

For example:
The fisherman could come in a few hours after 5:00 pm, between 8:00 pm to 11:00 pm, to catch the high tide. Please look over my answers and point out any mistakes :)

Thanks.



 

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  • #2
The low tide looks about 0.3m to me but otherwise your answers look OK as far as I can see.
I presume you are studying maths, not seamanship, but as a matter of interest, there is a "rule of twelfths", which give quite good answers without using sines at the same time as steering and keeping lookout! The overall tidal rise is 4.25 m, so in the first hour it rises about 1/12 x 4.25 = 0.35m, giving a total height of 0.25+0.35 = 0.6m. Not bad.
The full rule is as follows:
Hour 1, rise is 1/12
Hour 2, rise is 2/12
Hour 3, rise is 3/12
Hour 4, rise is 3/12
Hour 5, rise is 2/12
Hour 6, rise is 1/12..
 
  • #3
Evangeline101 said:
Period = 12.5 hours
12.4 is closer, and I agree with tech99 that the minimum looks more like 0.3 than 0.25.
Evangeline101 said:
it doesn't look like there is any phase shift (correct if i am wrong please)
What height does your equation give for midnight? What does the graph show?
 
  • #4
tech99 said:
The low tide looks about 0.3m to me but otherwise your answers look OK as far as I can see.

haruspex said:
12.4 is closer, and I agree with tech99 that the minimum looks more like 0.3 than 0.25.

Ok so by changing the height of the low tide to 0.3 m and the period to 12.4, the new equation so far would be:

y= -2.1 sin (29x) + 2.4

haruspex said:
What height does your equation give for midnight? What does the graph show?

At midnight my equation gives a height of 3.25 m, which is obviously not what the graph shows.

So here is the new equation:

-2.1 sin (29(x-0.4)) +2.4

Using this equation the height at midnight is 3.6 m, which is what the graph shows.

Does this look right?
 
  • #5
Evangeline101 said:
Ok so by changing the height of the low tide to 0.3 m and the period to 12.4, the new equation so far would be:

y= -2.1 sin (29x) + 2.4
At midnight my equation gives a height of 3.25 m, which is obviously not what the graph shows.

So here is the new equation:

-2.1 sin (29(x-0.4)) +2.4

Using this equation the height at midnight is 3.6 m, which is what the graph shows.

Does this look right?
Not sure how you get the -.4. Looks too much subtraction to me. I plugged in y(0)=2.6 and got x-0.2. The height at midnight is not 3.6m, more like 3.1m. I think you did not follow the right line of dots up.
 
  • #6
haruspex said:
The height at midnight is not 3.6m, more like 3.1m. I think you did not follow the right line of dots up.

midnight = 12 am = 24 hours

At 24 hours, the height of the tide is about 3.6 m

(the green line is the line of dots i followed)
upload_2016-8-14_19-14-35.png


That's how i got 3.6 m.
 

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  • #7
Accidentally posted the same response twice
 
  • #8
Evangeline101 said:
midnight = 12 am = 24 hours
How true o:)
You are probably close enough. To get a better fit would require a proper regression analysis. This is not difficult with a sine curve once you have the period reasonably accurate.
 
  • #9
haruspex said:
Not sure how you get the -.4. Looks too much subtraction to me.

I got 0.4 as the phase shift by looking at the graph and estimating, i took the midpoint (2.4) and looked at how far the curve was from the y-axis (if that makes any sense). I would say that the phase shift is 0.4 hours (24 minutes) to the right??
 
  • #10
Evangeline101 said:
I got 0.4 as the phase shift by looking at the graph and estimating, i took the midpoint (2.4) and looked at how far the curve was from the y-axis (if that makes any sense). I would say that the phase shift is 0.4 hours (24 minutes) to the right??
I understand that makes it work well at the 24 mark, but it's a bit off at the 0 mark. As I wrote, you are probably close enough.
 
  • #11
haruspex said:
I understand that makes it work well at the 24 mark, but it's a bit off at the 0 mark. As I wrote, you are probably close enough.

Your right, 0.4 is a bit off at 0 (2.8) , but 0.2 works well at 0 and is a bit off at the 24 mark (3.4)

I know you said I'm probably close enough, but which do you think is better to use? .
 
  • #12
Evangeline101 said:
Your right, 0.4 is a bit off at 0 (2.8) , but 0.2 works well at 0 and is a bit off at the 24 mark (3.4)

I know you said I'm probably close enough, but which do you think is better to use? .
For the last part of the question, you need the accuracy to be in the last six hours, so use 0.4.
 
  • #13
haruspex said:
For the last part of the question, you need the accuracy to be in the last six hours, so use 0.4.

Ok makes sense.

One more question, when I am stating the amplitude, should I say that the amplitude is -2.1 m or the amplitude is 2.1 m?
 
  • #14
Evangeline101 said:
Ok makes sense.

One more question, when I am stating the amplitude, should I say that the amplitude is -2.1 m or the amplitude is 2.1 m?
An amplitude is a magnitude, like a speed, so is never negative.
 
  • #15
haruspex said:
An amplitude is a magnitude, like a speed, so is never negative.
Ok, thanks for all the help! :biggrin:
 

1. What is an astrolabe?

An astrolabe is an ancient astronomical instrument used to measure the altitude of celestial bodies, such as the sun and stars. It was commonly used by sailors and navigators to determine their location at sea.

2. How does an astrolabe measure tides in a roadstead?

An astrolabe measures tides in a roadstead by using the altitude of the moon or a specific star to calculate the height of the tide. This is done by measuring the angle between the horizon and the celestial body, which can then be converted into a tidal height using trigonometric functions.

3. What is the significance of sinusoidal functions in astrolabe roadstead tides?

Sinusoidal functions are used to model the periodic rise and fall of tides in a roadstead. The shape of the tide curve closely resembles a sine wave, hence the use of sinusoidal functions to accurately describe and predict this phenomenon.

4. How accurate are astrolabe roadstead tide predictions?

The accuracy of astrolabe roadstead tide predictions depends on various factors, such as the quality of the instrument, the skill of the user, and the specific conditions of the location. However, with proper use and calibration, astrolabes can provide relatively accurate tide predictions.

5. Are astrolabes still used to measure tides in roadsteads today?

Astrolabes are no longer commonly used for tide measurements in roadsteads, as more advanced technology and methods have been developed. However, they are still used in some traditional and historical contexts, and their use in navigation and astronomy continues in some cultures.

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