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Question

The harmonic wave yi = (2.0 × 103) cos π (2.0 x 50t) travels along a string toward a boundary at x = 0 with a second string. The wave speed on the second string is 50 m/s. What are the expressions for reflected and transmitted waves. Assume SI units


A

6.67 x 10-4 cos π (2.0x + 50t), 2.67 x 10-3 cos π (x-50t)

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B

6.67 x 10-4 cos π (2.0x - 50t), 2.67 x 10-3 cos π (x-50t)

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C

6.67 x 10-4 cos π (2.0x + 50t+π), 2.67 x 10-3 cos π(x-50t)

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D

None of these

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Solution

The correct option is A

6.67 x 10-4 cos π (2.0x + 50t), 2.67 x 10-3 cos π (x-50t)


In such situation a part of the wave gets reflected while some gets transmitted. The amplitude of reflected wave is given by

Ar = (v2 v1v1 + v2)Ai where v2 and v1 are velocities in second and first medium respectively.

Here v1 = ωk1 = 50π2π = 25ms1 from the given equation.

yi = (2.0 × 103)cos[π(2.0x 50t)]

v2 = 50ms1

Ar = (50 2550 + 25)(2 × 103

= 6.67 × 104 m

So the equation of reflected wave would be

yr = 6.67 × 104 cos[π(2.0x + 50t)]

as the wave is going from higher density medium to lower density medium, so there is no phase change

In the second medium,

k2 = ωv2[frequency does not change]

= 50π50 = πm1

Also. At = (2v2v1 + v2)Ai

= (2 × 5025 + 50) × 2 × 103

= 2.67 × 103 m

So the wave equation is yr = 2.67 × 103 cos[π(x 50t)]


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