Question

A wave on a string has an amplitude of 0.567 mm and is propagating with a speed of 15.0 m/s in the positive x direction. If the frequency of the wave is 42.0 Hz, what is the speed of the string at x = 1.25 cm and t= 0.0625 s? (Hint: not looking for a wave speed)

Answer 1

Let the wavelength of the wave is $\lambda$ .

The velocity of the wave is v=15.0 m/s.

The frequency of the wave is n=42.0 Hz.

We know the wavelength of the wave

  $\lambda =\frac{v}{n}$

Let the equation of the wave for the direction of the positive x-axis is

  $y=a\sin \frac{2\pi }{\lambda }\left ( vt-x \right )$

Here a is the amplitude of the wave, y is the displacement of the positive y-axis of the string.

Let the speed of the string is V.

$\therefore \, \, \, V=\frac{\mathrm{d} y}{\mathrm{d} t}$

  $or \, \, \, V=\frac{2\pi v}{\lambda }a\cos \frac{2\pi }{\lambda }\left ( vt-x \right )$

  $or \, \, \, V=2\pi na\cos \frac{2\pi }{\lambda }\left ( n\lambda t-x \right )$

  $or \, \, \, V=2\pi \times 42.0\times 0.567\times 10^{-3}\times \cos\left ( 2\pi nt-\frac{2\pi n}{v} x\right )$$or \, \, \, V=2\pi \times 42.0\times 0.567\times 10^{-3}\times \cos\left ( 2\pi \times 42.0\times 0.0625-\frac{2\pi \times 42.0\times 1.25\times 10^{-2}}{15.0} \right )\: \: \: m/s$

  $\therefore \, \, \, V=-0.126\: \: \: m/s$

$\therefore$ The speed of the string is -0.126 m/s or speed is 0.126 m/s and direction is negative y-axis.