CBSE Class 12-science Answered
Find graphically the resultant E(t) of the following wave disturbances.
E1 =E0 sint
E2 =E0 sin(t+15o)
E3 =E0 sin(t+30o) E4 =E0 sin(t+45o)
Asked by Pgpgpgpgn | 02 Jan, 2021, 03:09: AM
Expert Answer
Let us consider frequency f of all waves as f = 50 Hz.
Hence angular frequency , ω = ( 2π f ) = 100π rad/s.
Let each wave has amplitude Eo = 10 ( arbitrary units )
Excel chart tabulated for displacements of all the waves are shown below
| t | E1 | E2 | E3 | E4 | E |
| 0 | 0.000 | 2.588 | 5.000 | 7.071 | 14.659 |
| 2 | 5.878 | 7.771 | 9.135 | 9.877 | 32.662 |
| 4 | 9.511 | 9.986 | 9.781 | 8.910 | 38.188 |
| 6 | 9.511 | 8.387 | 6.691 | 4.540 | 29.128 |
| 8 | 5.878 | 3.584 | 1.045 | -1.564 | 8.942 |
| 10 | 0.000 | -2.588 | -5.000 | -7.071 | -14.659 |
| 12 | -5.878 | -7.771 | -9.135 | -9.877 | -32.662 |
| 14 | -9.511 | -9.986 | -9.781 | -8.910 | -38.188 |
| 16 | -9.511 | -8.387 | -6.691 | -4.540 | -29.128 |
| 18 | -5.878 | -3.584 | -1.045 | 1.564 | -8.942 |
| 20 | 0.000 | 2.588 | 5.000 | 7.071 | 14.659 |
| 22 | 5.878 | 7.771 | 9.135 | 9.877 | 32.662 |
| 24 | 9.511 | 9.986 | 9.781 | 8.910 | 38.188 |
| 26 | 9.511 | 8.387 | 6.691 | 4.540 | 29.128 |
| 28 | 5.878 | 3.584 | 1.045 | -1.564 | 8.942 |
| 30 | 0.000 | -2.588 | -5.000 | -7.071 | -14.659 |
Last column in above table is sum of all displacements of waves
Graph of resultant E = E1 + E2 + E3 + E4 plotted as a function of time is given below
Let Resultant wave be of the form , ER = ERo sin ( ω t + φ )
Amplitude ERo is the maximum value of resultanat, we get ERo = 38.2
Angular frequency ω is determined from period T of the resultant wave .
We get period T = 20 ms , hence f = 50 Hz and ω = 2πf = 100π rad/s
Initial phase φ is obtained from initial displacement at t = 0 and amplitude ERo
at t= 0, total displacement = 14.659 . Hence φ = sin-1 ( 14.659 / 38.188 ) = 22.6o
From graph and tabulated values , we get the resultant , ER = 38.2 sin ( 100π t + 22.6o )
Answered by Thiyagarajan K | 02 Jan, 2021, 10:12: AM
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