Plz Help with the following math problem ! I really appreciate it ! :
Two pure notes of the same amplitude and of frequencies 15 Hz. and 17 Hz. sounded together to produce beats. Each note is modeled as a sine wave.
Quests :
a - What values should the constants a and b take in the function Y1 = sin(at) + sin(bt) if it is to model the sound of the two notes playing together ?
b - What values should the constants c and d take in the function Y2 = 2 sin(ct) cos(dt) if it is to model the sound of the two notes playing together ?
c - Plot both functions Y1 and Y2 on your calculator,choosing a suitable window to show the beats over a period of one second , and then sketch what is on the screen , labeling the axes.
Thanksss in advance !
We propose that
sin(at) + sin(bt) = 2sin(ct)cos(dt)
for some given values of a, b, c, and d and for all t.
Now,
sin(ct)cos(dt) = (1/2)*[sin((c + d)t) + sin((c - d)t)]
Thus we require:
sin(at) + sin(bt) = sin((c + d)t) + sin((c - d)t)
So let
a = c + d
b = c - d
Thus
c = (1/2)(a + b) = (1/2)(30(pi) + 34(pi)) = 32(pi) rad/s
d = (1/2)(a - b) = (1/2)(30(pi) - 34(pi)) = -2(pi) rad/s
-Dan