The displacement of a wave is given by y = 0.001 sin (100t + x) where x and y are in metre and t is insecond. This represents a wave(a) of wavelength one metre.(b) travelling with a velocity of 100 m/s in the negative x-direction.(c) of frequency (100/π) Hz.(d) travelling with a velocity of 50 π m/s in the positive x-direction
Question
The displacement of a wave is given by y = 0.001 sin (100t + x) where x and y are in metre and t is insecond. This represents a wave(a) of wavelength one metre.(b) travelling with a velocity of 100 m/s in the negative x-direction.(c) of frequency (100/π) Hz.(d) travelling with a velocity of 50 π m/s in the positive x-direction
Solution
The given wave equation is y = 0.001 sin (100t + x). This is a standard form of a wave equation y = A sin(wt + kx + φ), where A is the amplitude, w is the angular frequency, k is the wave number, and φ is the phase constant.
(a) The wavelength (λ) of a wave is given by 2π/k. Here, k = 1, so the wavelength is 2π/1 = 2π meters. Therefore, the statement is incorrect.
(b) The velocity (v) of a wave is given by w/k. Here, w = 100 and k = 1, so the velocity is 100/1 = 100 m/s. However, since the wave equation has a positive x, the wave is moving in the negative x-direction. Therefore, the statement is correct.
(c) The frequency (f) of a wave is given by w/2π. Here, w = 100, so the frequency is 100/2π = 50/π Hz. Therefore, the statement is incorrect.
(d) As calculated in part (b), the velocity of the wave is 100 m/s. Therefore, the statement is incorrect. The wave is not travelling with a velocity of 50 π m/s in the positive x-direction.
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