Question

The displacement of a string is given \[{\rm{y}}({\rm{x}},{\rm{t}}) = 0.06\sin \dfrac{{2\pi {\rm{x}}}}{3}\cos 120\pi {\rm{t}}\] where x and y are in m and t in s. The length of the string is $1.5\,m$ and its mass is $3 \times {10^{-2}}\,kg$.
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 the superposition of two waves of wavelength 3 m, frequency 60 Hz each travelling with a speed of 180 m/s in the opposite direction.
D. Amplitude of this wave is constant.

Answer 1

Hint: A displacement is a vector in geometry and mechanics whose length is the shortest distance between the original and final positions of a moving point $P$. It measures the distance and direction of net or absolute motion in a straight line from the point trajectory's initial location to its final position. The translation that maps the original position to the final position can be used to identify a displacement.

Complete step by step answer:
Here the given equation is,
\[{\rm{y}}({\rm{x}},{\rm{t}}) = 0.06\sin \dfrac{{2\pi {\rm{x}}}}{3}\cos 120\pi {\rm{t}}\]
A standing wave, also known as a stationary wave, is a wave that oscillates in time but does not travel in space due to its peak amplitude profile. The wave oscillations' peak amplitude is constant with time at any point in space, and the oscillations at different points in the wave are in phase.

The given equation describes a stationary wave since the terms containing $x$ and $t$ are independent of one another. When we compare the given equation to the standard form of the stationary wave equation, we get
\[y(x,t) = 2r\sin kx\cos \omega t\]
Substituting the values we get
\[{\rm{ }}k = \dfrac{{2\pi }}{\lambda } \\
\Rightarrow {\rm{ }}k= \dfrac{{2\pi }}{3}\]
\[\Rightarrow \lambda = 3\,m\]
Also \[\omega = 120\pi \]
\[\Rightarrow \nu = \dfrac{\omega }{{2\pi }} \\
\Rightarrow \nu= \dfrac{{120\pi }}{{2\pi }} \\
\Rightarrow \nu= 60\,Hz\]
We know that
\[v = \nu \times \lambda \\
\Rightarrow v= 60 \times 3 \\
\therefore v= 180\,m/s\]

Hence option B and C are correct.

Note: Michael Faraday was the first to note standing waves in 1831. Standing waves on the surface of a liquid in a vibrating container are detected by Faraday. Around 1860, Franz Melde invented the word "standing wave" and explained the effect of vibrating strings in his classic experiment.