IGNOU PHE 2 2018 - Two waves, travelling along the same direction, are given byy1(x, t)

Question

Two waves, travelling along the same direction, are given by
y1(x, t) = asin (ω₁t − k1x)
and y2 (x, t) = asin (ω₂t − k2x)
Suppose that the values of ω₁ and k₁ are respectively slightly greater than ω₂ and
k₂ . i) Obtain an expression for the resultant wave due to their superposition. ii)
Explain the formation of wave packet.

25 Jan 2018

Answer :

Word Count : 498

To solve this problem step by step, let's go over the two parts: ### Part i: Expression for the Resultant Wave We are given two waves traveling in the same direction: 1. \( y_1(x, t) = a \sin(\omega_1 t - k_1 x) \) 2. \( y_2(x, t) = a \sin(\omega_2 t - k_2 x) \) The resultant wave is the superposition of both waves, which means adding the two waves together: \[ y_{\text{resultant}}(x, t) = y_1(x, t) + y_2(x, t) \] Using the trigonometric identity for the sum of sines: \[ \sin(A) + \sin(B) = 2 \sin\left(\frac{A + B}{2}\right) \cos\left(\frac{A - B}{2}\right) \] Substitute \( A = \omega_1 t - k_1 x \) and \( B = \omega_2 t - k_2 x \) into the identity: \[ y_{\text{resultant}}(x, t) = a \sin(\omega_1 t - k_1 x) + a \sin(\omega_2 t - k_2 x) \] \[ y_{\text{resultant}}(x, t) = 2a \sin\left(\frac{(\omega_1 t - k_1 x) + (\omega_2 t - k_2

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