Question

(1) A wave is traveling in the positive-y direction with wavelength λ f Hz. At t is 6m 12 m and frequency 0, there is no displacement of the medium at the origin, but its amplitude a) Write the equation for the displacement of this wave as a function of position and (b) Draw a quantitatively accurate history graph (at y 3m) and snapshot graph (at t 2s) (c) What is the speed of the wave?

Answer 1

​​​​​​

The wave equation is given by w = A sin (k Vector middot r vector - wt) where w = 2 pi f = 2 pi times 1/6 = pi/3 magnitude of wave vectorVertical-bar k Vector Vertical-bar = 2 pi/lambda = 2 pi/12 = pi/6 and direction is + y^hat for the wave propagating along the positive y - direction Given amplitude A = 6 cm The position of r Vector = x x^hat + y y^hat + z z^hat = > k Vector middot r Vector = (pi/6 y^hat) middot (x x^hat + y y^hat + z z^hat) = pi/6 y^hat) middot (x x^hat + y y^hat + z z^hat) = pi/6 x y^hat middot x^hat + pi/6 y y^hat middot y^hat + pi/6 z y^hat middot z^hat k Vector middot r vector = 0 + pi/6 y + 0 [y^hat middot x^hat = 0, y^hat middot z^hat = 0 y^hat middot y^hat = 1]

(c) The speed of the wave is given by

$v=\lambda f = 12m \times \frac{1}{6}Hz = 2m/s$