Thread: Plz Help , stuck on final exercise !

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 !

Originally Posted by speed01

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 ?

a and b represent angular frequencies, so:
f = 15 Hz = a/(2*(pi)) ==> a = 2*(pi)*15 = 30(pi) rad/s.
f = 17 Hz = b/(2*(pi)) ==> b = 2*(pi)*17 = 34(pi) rad/s.

-Dan

Originally Posted by speed01

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 :

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 ?

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