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A wave travels along the positive x-direction with a speed of 20 m/s. The amplitude of the wave is 0.20 cm and the wavelength 2.0 cm. (a) Write a suitable wave equation which describes this wave, (b) What is the displacement and velocity of the particle at x = 2.0 cm at time t = 0 according to the wave equation written? Can you get different values of this quantity if the wave equation is written in a different fashion?
Wave speed, v = 20 m/s A = 0.20 cm λ = 2 cm a) Equation of wave along the x-axis y = A sin (kx – wt) ∴ k = 2π/λ = 2π/2 = π cm–1 T = λ/v = 2/2000 = 1/1000 sec = 10–3 sec ⇒ ω = 2π/T = 2π * 10–3 sec–1 So, the wave equation is, ∴ y = (0.2 cm)sin[(π cm–1)x – (2π * 103 sec–1)t] b) At x = 2 cm, and t = 0, y = (0.2 cm) sin (/2) = 0 ∴ v = rω cos πx = 0.2 * 2000π * cos 2π = 400 π = 400 * (3.14) = 1256 cm/s = 400 π cm/s = 4π m/s
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