General Sinusoid Properties.
Cuthbert Nyack
The applet here illustrates some of the basic properties of
Sinusoids. f(t) = a cos(wt + f) +
b sin(Wt + q).
Fn = 1 shows how a sinusoid with phase f1
can be represented as a sum of a sine(red) and a cosine(cyan).
f1 must be changed to see the different components.
Fn = 2 to 5 show special cases of Fn = 1.
Fn = 6 shows how adding a cosine with amplitude 'a' and a sine
with amplitude 'b' combine to result in a sinusoid. The magnitude
and phase of the resulting sinusoid is shown at the bottom in blue.
Fn = 7 to 10 show special cases of Fn = 6.
Fn = 11 shows how 1 period of a cosine can be represented as the
sum of 2 counterrotating vectors in the complex plane
ie cos(wt) = ½
(ejwt +
e-jwt).
'a' must be changed to see the vectors at different times.
e jwt is a counterclockwise rotating vector (orange) while
e -jwt is a clockwise rotating (cyan) vector. The resultant is shown by the horizontal red
line (½ size) and the
vertical red shows the corresponding point on the cosine.
Fn = 12 to 15 shows special cases of Fn = 11.
Fn = 16 shows how 1 period of a sine can be represented as the
sum of 2 counterrotating vectors in the complex plane
ie sin(wt) = 1/(2j)
(ejwt -
e-jwt).
'a' must be changed to see the vectors at different times.
The imaginary resultant is shown horizontally here.
Fn = 17 to 20 shows special cases of Fn = 16.
Fn = 21 shows how a sines and cosines are affected by
changing their phase.
The top plots show standard sine and cosines. The bottom plots
show their phase shifted counterparts. the phase shift is set by
f1 and may be +ve or -ve
corresponding to advanced or delayed sinusoids.
Fn = 21 to 25 shows special cases of Fn = 21.
Fn = 26 shows how a sines and cosines are affected by
Time Shifting.
The top plots show standard sine and cosines. The bottom plots
show their Time shifted counterparts. the Time shift is set by
T and may be +ve or -ve
corresponding to advanced or delayed sinusoids.
Fn = 27 to 30 shows special cases of Fn = 26.
Fn = 31 illustrates the orthogonality of 2 sines.
The applet shows sin(nwt)sin(mwt)
(w refers to
angular frequency omega) for different
n and m. The +ve part of product function is shown shaded red, and the -ve part shaded blue. The red and blue areas cancel out. When n = m
the product is always positive and averages 1/2. the
product averages zero whenever n is not equal to m.
The width
of the applet corresponds to the fundamental period T. Product must
be averaged over T(= 2
p/w)
(or nT) to get zero. The Average value is shown
on the applet. For m not equal to n, the error (< 2E-16) shown is numerical error.
Fn = 32 to 35 shows special cases of Fn = 31.
Fn = 36 illustrates the orthogonality of a sine and a cosine.
Fn = 37 to 40 shows special cases of Fn = 36.
Fn = 41 illustrates the orthogonality of 2 cosines.
Fn = 42 to 45 shows special cases of Fn = 41.
Fn = 46 illustrates the time domain addition of 2 sinusoids.
Fn = 47 to 51 shows what is considered to be 'consonant' frequencies.
Fn = 52 , 53 show imperfect 'consonant' frequencies.
Fn = 54 to 57 shows what is considered to be 'dissonant' frequencies.
Fn = 58 illustrates the time domain addition of 3 sinusoids.
Fn = 59 , 60 show 3 frequency 'chords'.
Fn = 61 illustrates the Lissajous figures. sin(n1wt) is applied to the H axis and
sin(n2wt) +
sin(n3wt) applied to the V axis.
Fn = 62 to 77 shows special cases of Fn = 61.
Fn = 78 shows a parametric plot of the functions
x = a sin(t), y = b cos(t).
Fn = 79 shows the case where a = b. The result is a circle
and this is the reason why sines and cosines are referred to
as circular functions.
Fn = 80 and 81 shows other cases of Fn = 78.
Fn = 82 shows a parametric plot of the functions
x = a sinh(t), y = a cosh(t).
The result is a hyperbola and is the reason why sinh and
cosh are referred to as hyperbolic functions.
Fn = 83 to 85 shows other cases of Fn = 82.
One of the consequences of the circular property of
sinusoids is that a wide variety circle based patterns can
easily be deduced from elementary arithmetic operations with
sinusoids.
Fn = 86 to 538 shows a tiny subset of the possibilities.
The following gif images show some of the circle based patterns.
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COPYRIGHT © 2012 Cuthbert Nyack.