Saved Bookmarks
| 1. |
Explain how the interference of waves is formed. |
Answer» Solution :Interference is a phenomenon in which two waves superimpose to form a resultant wave of GREATER,lowe or the sameamplitude. Let us consider two harmonic waves having identical frequencies constant phase difference `varphi` and same dwave form (can be trated as coherent source), but having amplitudes `A_(1) and A_(2)` then `y_(1)=A_(1)s in(kx-OMEGAT) ""....(1)` `y_(1)=A_(1)s in(kx-omegat+varphi) ""....(2)` Suppose they move simultaneously in a particular direction, then interference occur (i.e., overlap of these two waves). Mathematically `y=y_(1)+y_(2) ""...(3)` Hennce by substituting equation (1) and equation (2) in equation (3) we get `y={{:(A_(1)sin(kx-omegat)),(+A_(2)sin(kx- omegat+varphi)):}` USING trigonometric IDENTITY `sin (alpha+beta)=(sin alpha COS beta+ cos alpha sin beta)`, we get `y={{:(A_(1)sin(kx-omegat)),(+A_(2)sin(kx- omegat+varphi)),(+cos(kx-omega t) sin varphi"]"):}` `{{:(sin (kx- omega t)(A_(1)+A_(2)cos varphi)),(+A_(2)sin varphi cos(kx-omegat)""....(4)):}` Let us re-define `Acos theta=(A_(1)+A_(2)cos varphi) ""...(5)` and `A sin theta=A_(2) sin varphi ""....(6)` Then equation (4) can be rewritten as `{{:(A sin (kx-omegat)cos theta),(+A cos (kx- omegat) sin theta):}` `{{:(A "("sin (kx-omegat)cos theta),(+sin theta cos (kx- omegat) sin theta):}` `A=A sin (kx- omegat+theta) ""...(7)` By squaring and adding equation (5) and (6), we get `A^(2)=A_(1)^(2)+A_(2)^(2)+2A_(1)A_(2)cos varphi...(8)` Since intensity is square of the amplitude `(I=A^(2))`, we get `I=I_(1)+I_(2)+ 2sqrt(I_(1)I_(2)) cos varphi "" ...(9)` This means the resultant intensity at any point depends on the phase difference at that point. |
|