Question

The displacement of a string is given by $y(x, t)=0.06 \sin (2 \pi x / 3) \cos (120 \pi t)$ where $x$ and $y$ are in $m$ and $t$ in $s$. The length of the string is $1.5\, m$ and its mass is $3.0 \times 10^{-2} kg$.

• A. It represents a progressive wave of frequency 60 Hz
• B. It represents a stationary wave of frequency 60 Hz
• C. It is the result of superposition of two waves of wavelength 3 m, frequency 60 Hz each travelling with a speed of 180 m / s in opposite direction
• D. Amplitude of this wave is constant

Answer 1

Answer: (B), (C)

(a) The given equation is $y(x, t)=0.06 \sin \left(\frac{2 \pi x}{3}\right) \cos (120 \pi t)$
(b) As terms involving $x$ and $t$ are independent of each other, the given equation represents a stationary wave.
(c) Compare the given equation with the standard form of equation of stationary wave
$y(x, t) =2 r \sin k x \cos \omega t $
$k =\frac{2 \pi}{\lambda}=\frac{2 \pi}{3}$
$\therefore \lambda =3\, m$
$\omega=120\, \pi$
$\therefore v=\frac{\omega}{2 \pi}=\frac{120 \pi}{2 \pi}=60\, Hz$
and $v=v \lambda=60 \times 3=180\, m / s$
Hence the given stationary wave is the result of superposition of two waves of wavelength $3 \,m$ and frequency $60\, Hz$ each, travelling with a velocity of $180\, m / s$ in opposite directions.