The average depth of the water in a port on a tidal river is 5 m. At low tide, the depth is 2 m. One cycle is completed approximately every 12 hours. How can I write an equation to represent this?

Answer 1

Let "t" represent time in hours after the low tide (t = 0 is the low tide). Let d(t) represent the depth of the water. Can you see that the graph would look like this:

It is either a sine wave or a cosine wave. They are both alike except for their horizontal shift. But since this starts at the minimum at t = 0, it is most like a negative cosine wave. So:

d(t) = -Acos[B(t - C)] + D

where:

A is the amplitude, in this case A = 3 because the curve goes up and down from the midline by 3

B is derived from the period. The period is 12 and (for sine or cosine) B = 2π/Period so B = π/6

C is the horizontal shift which in this case is zero

D is the vertical shift, which in this case is 5

So: d(t) = -3cos[(π/6)t] + 5

Answer 2

D = 3sin(pi(t)/6) + 5

where D= depth in feet

t = time in hours

general formula for a sinusoidal curve is

y = Asin(Bx+C) + D

where A = amplitude,

period = 2pi/B,

C = phase shift,

D= midline= average depth

A=3, B = pi/6, C = 0, D=5. There is no phase shift. A = difference between average depth and minimum depth = 5-2 = 3

for a 12 hour cycle the period = 12 = 2pi/B and B= 2pi/12 = pi/6