The harmonic wave yi = (2.0 × 10−3) 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
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 = 25ms−1 from the given equation.
yi = (2.0 × 10−3)cos[π(2.0x −50t)]
v2 = 50ms−1
Ar = (50 − 2550 + 25)(2 × 10−3
= 6.67 × 10−4 m
So the equation of reflected wave would be
yr = 6.67 × 10−4 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 = πm−1
Also. At = (2v2v1 + v2)Ai
= (2 × 5025 + 50) × 2 × 10−3
= 2.67 × 10−3 m
So the wave equation is yr = 2.67 × 10−3 cos[π(x − 50t)]