Sine Curves

February 18, 2014

Frequency.

Frequency.


I’ve slapped together this simple VPython program to introduce sinusoidal functions to my pre-Calculus students.

  • Left and right arrow keys increase and decrease the frequency;
  • Up and down arrow keys increase and decrease the amplitude;
  • “a” and “s” keys increase and decrease the phase.
Amplitude.

Amplitude.

The specific functions shown on the graph are based on the general function:

y =  A \sin{F x + P}

where:

  • A — amplitude
  • F — frequency
  • P — phase
Phase. Note how the curve seems to move backward when the phase increases.

Phase. Note how the curve seems to move backward when the phase increases.

When I first introduce sinusoidal functions to my pre-Calculus students I have them make tables of the functions (from -2π to 2π with an interval of π/8) and then plot the functions. Then I’ll have them draw sets of sine functions so they can observe different frequencies, amplitudes, and phases.

from visual import *

class sin_func:
    def __init__(self, x, amp=1., freq=1., phase=0.0):
        self.x = x
        self.amp = amp
        self.freq = freq
        self.phase = phase

        self.curve = curve(color=color.red, x=self.x, y=self.f(x), radius=0.05)
        self.label = label(pos=(xmin/2.0,ymin), text="Hi",box=False, height=30)
        
    def f(self, x):
        y = self.amp * sin(self.freq*x+self.phase)
        return y

    def update(self, amp, freq, phase):
        self.amp = amp
        self.freq = freq
        self.phase = phase
        self.curve.y = self.f(x)
        self.label.text = self.get_eqn()

    def get_eqn(self):
        if self.phase == 0.0:
            tphase = ""
        elif (self.phase > 0):
            tphase = u" + %i\u03C0/8" % int(self.phase*8.0/pi)
        else:
            tphase = u" - %i\u03C0/8" % int(abs(self.phase*8.0/pi))
        print self.phase*8.0/pi

        txt = "y = %ssin(%sx %s)" % (simplify_num(self.amp), simplify_num(self.freq), tphase)
        return txt
    
def simplify_num(num):
    if (num == 1):
        snum = ""
    elif (num == -1):
        snum = "-"
    else:
        snum = str(num).split(".")[0]+" "
    return snum
        
amp = 1.0
freq = 1.0

damp = 1.0
dfreq = 1.0

phase = 0.0
dphase = pi/8.0

xmin = -2*pi
xmax = 2*pi
dx = 0.1

ymin = -3
ymax = 3

scene.width=640
scene.height=480

xaxis = curve(pos=[(xmin,0),(xmax,0)])
yaxis = curve(pos=[(0,ymin),(0,ymax)])

x = arange(xmin, xmax, dx)
#y = f(x)

func = sin_func(x=x)
func.update(amp, freq, phase)

while 1: #theta <= 2*pi:
    rate(60)

    if scene.kb.keys: # is there an event waiting to be processed?
        s = scene.kb.getkey() # obtain keyboard information
        #print s
        if s == "up":
            amp += damp
        if s == "down":
            amp -= damp
        if s == "right":
            freq += dfreq
        if s == "left":
            freq -= dfreq

        if s == "s":
            phase += dphase
        if s == "a":
            phase -= dphase

        func.update(amp, freq, phase)
        #update_curve(func, y)
        

Citing this post: Urbano, L., 2014. Sine Curves, Retrieved July 27th, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

Working with Climate Data

March 14, 2013

Monthly climatic data from the Eads Bridge, from 1893 to the 1960’s. It’s a comma separated file (.csv) that can be imported into pretty much any spreadsheet program.

135045.csv

The last three columns are mean (MMNT), minimum (MNMT), and maximum (MXMT) monthly temperature data, which are good candidates for analysis by pre-calculus students who are studying sinusoidal functions. For an extra challenge, students can also try analyzing the total monthly precipitation patterns (TPCP). The precipitation pattern is not nearly as nice a sinusoidal function as the temperature.

Students should try to deconstruct the curve into component functions to see the annual cycles and any longer term patterns. This type of work would also be a precursor the the mathematics of Fourier analysis.

This data comes from the National Climatic Data Center (NCDC) website.

Citing this post: Urbano, L., 2013. Working with Climate Data, Retrieved July 27th, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

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