Question

b) Calculate the resultant amplitude of N-coherent wave superposition with phases of successive waves differing by a constant amount? ​

Answer 1

Explanation:

Ankit sold two jeans for 990 each. On one he gains 10% and on the other he lost

10%. Find his gain or loss per cent in the whole transaction.

Answer 2

Answer:

Resultant amplitude = $a.\frac{sin{\frac{N\theta}{2}}}{sin\frac{\theta}{2}}$

Explanation:

Let the wave be represented by,  $x_1=acos(wt)$

Complex representation of: $x_1= a.e^{i(wt)}$

similarly,

$x_2=a.cos(wt+\theta) = a.e^{i(wt+\theta)}$

$x_3=a.cos(wt+2\theta) = a.e^{i(wt+2\theta)}$

.................................

$x_N=a.cos(wt+(N-1)\theta)=a.e^{i(wt+(N-1)\theta)}$

Resultant = $X=x_1+x_2+x_3+......+x_N$

$X= a.e^{i(wt)}+a.e^{i(wt+\theta)}+a.e^{i(wt+2\theta)}+...........+a.e^{i(wt+(N-1)\theta)}$

$X= a.e^{i(wt)}[1+e^{\theta}+e^{2\theta}+...........e^{(N-1)\theta}$

$X= a.\frac{1-e^{N\theta}}{1-e^{\theta}} e^{iwt}$

$X= a.e^{iwt}.\frac{e^{N\theta /2}}{e^{\theta /2}}\frac{e^{N\theta /2}-e^{-N\theta /2}}{e^{\theta /2}-e^{-\theta /2}}$

$X= a.e^{iwt + (N-1)\theta}.\frac{e^{N\theta /2}-e^{-N\theta /2}}{e^{\theta /2}-e^{-\theta /2}}$

$X= a.\frac{sin{\frac{N\theta}{2}}}{sin\frac{\theta}{2}}.exp[i[wt+(N-1)\frac{\theta}{2}]]$

Resultant amplitude = $a.\frac{sin{\frac{N\theta}{2}}}{sin\frac{\theta}{2}}$

[Resultant amplitude can also be derived using graphical method.]

#SPJ3