Asked by Hana
When the waves Y1=asinwt and Y2=acoswt are superimposed, find the resultant amplitude?
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The resultant amplitude can be found using the formula:
A = √(A1^2 + A2^2 + 2A1A2cosφ)
where A1 and A2 are the amplitudes of Y1 and Y2 respectively, and φ is the phase difference between them.
In this case, A1 = a and A2 = a, since Y1 = asinwt and Y2 = acoswt have the same amplitude.
To find φ, we need to express Y1 and Y2 in terms of a common reference. Let's use cos:
Y1 = a sin wt = a cos (π/2 - wt)
Y2 = a cos wt
Now we can see that φ = π/2 - wt.
Substituting these values into the formula, we get:
A = √(a^2 + a^2 + 2a*a cos(π/2-wt))
Simplifying,
A = √(2a^2(1-cos(π/2-wt)))
A = √(2a^2(1+sinwt))
A = a√(2(1+sinwt))
Therefore, the resultant amplitude is a multiplied by the square root of 2 times (1+sinwt).
A = √(A1^2 + A2^2 + 2A1A2cosφ)
where A1 and A2 are the amplitudes of Y1 and Y2 respectively, and φ is the phase difference between them.
In this case, A1 = a and A2 = a, since Y1 = asinwt and Y2 = acoswt have the same amplitude.
To find φ, we need to express Y1 and Y2 in terms of a common reference. Let's use cos:
Y1 = a sin wt = a cos (π/2 - wt)
Y2 = a cos wt
Now we can see that φ = π/2 - wt.
Substituting these values into the formula, we get:
A = √(a^2 + a^2 + 2a*a cos(π/2-wt))
Simplifying,
A = √(2a^2(1-cos(π/2-wt)))
A = √(2a^2(1+sinwt))
A = a√(2(1+sinwt))
Therefore, the resultant amplitude is a multiplied by the square root of 2 times (1+sinwt).
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