Question

The equation of a wave travelling on a string is $y=\left(0.10mm\right)sin\left[\right(31.4m^{-1})x+(314s^{-1}\left)t\right].$ (a) In which direction does the wave travel ? (b) Find the wave speed, the wavelength and the frequency of the wave. (c) What is the maximum displacement and the maximum speed of a portion of the string ?

Answer 1

The equation of the wave is given by

$y=\left[\right(0.1mm\left)sin\right(31.4m^{-1})x+(31.4s^{-1}\left)t\right]$

$y=rsin\left\{\right(\frac{2\pi x}{\lambda })+\omega t\}$

(a) Negative x-direction.

(b) $k=31.4m^{-1}$

$\Rightarrow \frac{2\pi }{\lambda }=31.4$

$\Rightarrow \lambda =\frac{2\pi }{31.4}=0.2m=20cm$

Again, $w=314s^{-1}$

$\Rightarrow 2\pi f=314$

$\Rightarrow f=\frac{314}{2\pi }$

$=\frac{314}{2\times 3.14}$

$=50se{c}^{-1}=50Hz$

wave speed,

$v=\lambda f=20\times 50$

$=1000cm/s$

(c) Max. displacement = 0.10 mm

Max. velocity $=aw=0.1\times {10}^{-1}\times 314$

$=3.14cm/sec$