Oscillation question

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#1
Sound is a pressure oscillation that may be the sum of more than one harmonic oscillation. For example, a sound you hear is made of two oscillations of same pressure amplitude, but slight different frequencies: A cos(w1t) and A cos(w2t). Assume the frequencies are similar, w2>w1 but delta w=w2-w1<<w1.
A) What overall pressure displacement do you have at t=0?
B)Give an equation, in terms of w1 and delta w, that describes after how many oscillations, n, of the first oscillator, the first oscillator will have displacement A, and the second oscillator will have displacement -A.
C)After how many more seconds will you get the same overall displacement as t=0 again?
D)Describe how you can make a tunable acoustic oscillator have exactly the same frequency as a reference oscillator using only your ear, without measurement of frequency.

The answer i got for A is 0, as there would be no displacement (I think). But I am stuck on part B.Any help would be appreciated.
0
1 year ago
#2
(Original post by Wqyds)
Sound is a pressure oscillation that may be the sum of more than one harmonic oscillation. For example, a sound you hear is made of two oscillations of same pressure amplitude, but slight different frequencies: A cos(w1t) and A cos(w2t). Assume the frequencies are similar, w2>w1 but delta w=w2-w1<<w1.
A) What overall pressure displacement do you have at t=0?
B)Give an equation, in terms of w1 and delta w, that describes after how many oscillations, n, of the first oscillator, the first oscillator will have displacement A, and the second oscillator will have displacement -A.
C)After how many more seconds will you get the same overall displacement as t=0 again?
D)Describe how you can make a tunable acoustic oscillator have exactly the same frequency as a reference oscillator using only your ear, without measurement of frequency.

The answer i got for A is 0, as there would be no displacement (I think). But I am stuck on part B.Any help would be appreciated.
So, your question is just about the B part? does the letter w stands for omega? If so, you have to consider the mathematical relationships for this quantity.

The cosine oscillations have their repetitions and thus their periods. The period of an oscillation is the inverse of the frequency: T = 1/f.

The frequence f is defined as the quotient of the circuit frequency w and the full rotation (2*pi): f = w/(2*pi)

From this it follows that T = (2*pi)/w. This converted to the ciruit frequency, you get w = (2*pi)/T

As it is asked about the number of oscillations n (repetitive periods!), you must add this to your equation of the Period: n*T = (2*pi)/w

What leads to w = (2*pi)/(n*T). For w1 you get: w1 = (2*pi)/(n*T1).

And for delta w = w2-w1, you get: delta w = [(2*pi)/n*T2)] - [(2*pi)/n*T1)]

The equations would look like these:

f(t) = A*cos(w1*t) = A*cos((2*pi)/(n*T1)*t)

f(t) = A*cos((w2-w1)*t) = A*cos([(2*pi)/n*T2)] - [(2*pi)/n*T1)]*t)

Those are the equations for displacement A, for -A you just have to change the sign behind A from positive to negative.

K-Man_PhysCheM I need your help to find possible mistakes again. Are there any errors in reasoning and thus in approaches and conclusions?
0
1 year ago
#3
btw, the answer to the first part is not zero.
This is a cos wave and you are adding two of them with amplitude A, at t=0, where the value of cos is 1.
So the answer should be amplitude is 2A at t=0
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