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Waves — Physics STD 11 Science — Question

Rajasthan BoardEnglish MediumSTD 11 SciencePhysicsWaves5 Marks
Question
What do you mean by interference of waves? Distinguish between constructive and destructive interference.
Standing waves are produced by the superposition of two waves
$\text{y}_1=0.05\sin(3\pi\text{t}-2\text{x})$ and $\text{y}_2=0.05\sin(3\pi\text{t}+2\text{x})$
where y and x are measured in metres and t in seconds. Find the amplitude of a particle at x = 0.5m.
✓

Answer

Interference of waves is the phenomenon of redistribution of energy in space on account of superposition of two waves of same nature, same frequency and equal or comparable amplitudes and travelling in the given medium in the same direction.
Constructive interference takes place when the two superposing waves are in same phase i.e., crest of one wave (in transverse waves) coincides with crest of another wave and vice-versa. As a result, the resultant amplitude and hence intensity of the resultant wave is maximum. Thus, for constructive interference, the phase difference between the superposing waves $\Delta\phi=0$ or $2\text{n}\pi,$ where n is an integer i.e., n = 1, 2, 3.....
Destructive interference takes place when two superposing waves are in mutually opposite phase i.e., in superposing of two transverse waves crest of one wave exactly coincides with trough of another wave. As a result, the resultant amplitude and hence intensity of the resultant wave is minimum. For destructive interference, the phase difference $\Delta\phi=(2\text{n}-1)\pi,$ where n = 1, 2, 3....
Numerical:
The resultant displacement is given by
$\text{y}=\text{y}_1+\text{y}_2=0.05\{\sin(3\pi\text{t}-2\text{x})+\sin(3\pi\text{t}+2\text{x})\}$
Using trigonometric relation
$\sin(\alpha+\beta)+\sin(\alpha-\beta)=2\sin\alpha\cos\beta,$ we have
$\text{y}=0.1\cos2\text{x}\sin3\pi\text{t}$ or $\text{y}=\text{A}\sin3\pi\text{t}$
where A, the amplitude of standing waves, is
given by $\text{A}=0.1\cos2\text{x}$ with
$\text{x}=0.5\text{m}$
$\cos2\text{x}=\cos(2\times0.5\text{ rad})$
$=\cos(1\text{ rad})=\cos\Big(\frac{\pi}{3.142}\Big)=\cos57.3^\circ$
$=0.54$
Amplitude A at (x = 0.5) = 0.1 × 0.54
= 0.054m.

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