A wave is represented by the equation $y = A \sin \left(10\pi x + 15\pi t + \frac{\pi}{3}\right)$

A wave is represented by the equation $y = A \sin \left(10\pi x + 15\pi t + \frac{\pi}{3}\right)$
  1. a wave travelling in positive $x$-direction with a velocity $1.5\text{ ms}^{-1}$
  2. a wave travelling in negative $x$-direction with a velocity $1.5\text{ ms}^{-1}$
  3. a wave travelling in the negative $x$-direction having a wavelength $0.4\text{ m}$
  4. a wave travelling in positive $x$-direction of wavelength $0.1\text{ m}$

Solution

Here, $\omega = 15\pi\text{ rad s}^{-1}$, $k = 10\pi\text{ rad m}^{-1}$ $v = \frac{\omega}{k} = 1.5\text{ ms}^{-1} \Rightarrow \lambda = \frac{2\pi}{k} = \frac{2\pi}{10\pi} = 0.2\text{ m}$ Positive sign between $kx$ and $\omega t$ means wave is travelling in negative $x$-direction.

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