The time interval between similar lunar phases-the synodic month-averages about

Question

The time interval between similar lunar phases-the synodic month-averages about 29.5 days Therefore, in those lunar calendars in which each month begins on the new moon, the full moon falls between the 14th or 15th of the lunar month. A cycle of waxing and waning moon takes 29.5 days approximately. Construct a periodic function to describe the changing phases, starting with a "new moon" (totally dark) and ending one cycle later. Write an introduction describing the problems highlighting the periodicity idea.

Explanation / Answer

Here we need to define a periodic function.

THis function will be a function of time 't' in days and the function that w need to define could be named L(t)

L(t) represents the fraction of moon, visible on any day 't'

We are given that the cycle of the whole process takes t = 29.5 days , after this the cycle repeate

HEnce the time period T = 29.5 = 2*pi/w , w is the angular frequency

Let C(t) represent the cosine function.

=> C(t) E [0,1] , here C(t) =1 would represents the middle of the given graph.

we know that the range of the cosine function is [-1,1] but out function C(t) shall be positive

=> we'll have to add a constant and scale the cosine as following

C(t) = 1/2*[1+cos(w*t)]

lets take a look at the above function , at t=0

C(0) = 1/2*[1+cos(0)] = 1, but as per the graph given at t=0 , the value of the function shall be =0

so we need to perform some changes to the function

Lets introduce a phase shift of pi radians

=> C(t) = 1/2*[1+cos(w*t + pi)]

now at t=0

C(0) = 1/2[1+cos(pi)] = 1/2*[1+(-1)] =0

this comse out to be right

hence the periodic function is :

C(t) = 1/2*[1 + cos(w*t + pi)]

where w = 2*pi/29.5

Related Questions

Navigate

• Browse (All)
• Browse T
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.