Example

Question:

A wave travelling along a string is described by
\( y(x, t) = 0.005 \sin (80.0\,x - 3.0\,t) \),
where all quantities are in SI units.
(a) Calculate the amplitude, (b) the wavelength, (c) the period and frequency of the wave. Also, calculate the displacement \( y \) of the wave at \( x = 30.0\,\mathrm{cm} \) and \( t = 20\,\mathrm{s} \).

Solution:

(a) Amplitude: \( a = 0.005\, \mathrm{m} = 5\,\mathrm{mm} \)
(b) Angular wave number: \( k = 80.0\,\mathrm{m^{-1}} \), angular frequency \( \omega = 3.0\,\mathrm{s^{-1}} \)
Wavelength: \[ \lambda = \frac{2\pi}{k} = \frac{2\pi}{80.0} = 0.0785\,\mathrm{m} = 7.85\,\mathrm{cm} \] (c) Period: \[ T = \frac{2\pi}{\omega} = \frac{2\pi}{3.0} = 2.09\,\mathrm{s} \] Frequency: \[ f = \frac{1}{T} = 0.48\,\mathrm{Hz} \] Displacement at \( x = 0.3\,\mathrm{m},\, t = 20\,\mathrm{s} \): \[ y = 0.005 \sin [80.0 \times 0.3 - 3.0 \times 20] = 0.005 \sin (24.0 - 60.0) = 0.005 \sin (-36.0) = 0.005 \sin (-36 + 12\pi) = 0.005 \sin (1.699) \approx 0.005 \times 0.99 = 0.00495\,\mathrm{m} \approx 5\,\mathrm{mm} \]