The transverse displacement of a wire (clamped at both its ends) is described as : $y(x, t)=0.06 \sin \left(\frac{2 \pi}{3} x\right) \cos (120 \pi t)$ The mass of the wire is $6 \times 10^{-2} \mathrm{~kg}$ and its length is $3 \mathrm{~m}$. Provide answers to the following questions:(i) Is the function describing a stationary wave or a travelling wave?(ii) Interpret the wave as a superposition of two waves travelling in opposite directions. Find the speed, wavelength and frequency of each wave.

Home » The transverse displacement of a wire (clamped at both its ends) is described as : The mass of the wire is and its length is . Provide answers to the following questions:(i) Is the function describing a stationary wave or a travelling wave?(ii) Interpret the wave as a superposition of two waves travelling in opposite directions. Find the speed, wavelength and frequency of each wave.

The transverse displacement of a wire (clamped at both its ends) is described as : $y(x, t)=0.06 \sin \left(\frac{2 \pi}{3} x\right) \cos (120 \pi t)$ The mass of the wire is $6 \times 10^{-2} \mathrm{~kg}$ and its length is $3 \mathrm{~m}$ .
Provide answers to the following questions:
(i) Is the function describing a stationary wave or a travelling wave?
(ii) Interpret the wave as a superposition of two waves travelling in opposite directions. Find the speed, wavelength and frequency of each wave.

CBSE | Class 11 | NCERT | Physics | Waves

The transverse displacement of a wire (clamped at both its ends) is described as : $y(x, t)=0.06 \sin \left(\frac{2 \pi}{3} x\right) \cos (120 \pi t)$ The mass of the wire is $6 \times 10^{-2} \mathrm{~kg}$ and its length is $3 \mathrm{~m}$ .
Provide answers to the following questions:
(i) Is the function describing a stationary wave or a travelling wave?
(ii) Interpret the wave as a superposition of two waves travelling in opposite directions. Find the speed, wavelength and frequency of each wave.

As we know,

The standard equation of a stationary wave is known as,

$y(x, t)=2 a \sin k x \cos w t$

Given equation is,

$y(x, t)=0.06 \sin \left(\frac{2 \pi}{3} x\right) \cos (120 \pi t)$

It is similar to the general equation .

i) Thus, the given function describes a stationary wave.

ii) We know, a wave travelling in the positive $x$ -direction can be represented as :

$y_{1}=a \sin (\omega t-k x)$

And,

A wave travelling in the negative $x$ -direction is represented as :
$y_{2}=a \sin (w t+k x)$

On Super positioning these two waves, we get,

$y=y_{1}+y_{2}$

$=a \sin (\omega t-k x)-a \sin (w t+k x)$

$=\operatorname{asin}(\omega t) \cos (k x)-\operatorname{asin}(k x)$
$\cos (\omega t)-\operatorname{asin}(\omega t) \cos (k x)-a \sin (k x) \cos (\omega t)$
$=-2 \operatorname{asin}(\mathrm{kx}) \cos (\omega \mathrm{t})$
$=-2 \operatorname{asin}\left(\frac{2 \pi}{\lambda} x\right) \cos (2 \pi v t) \ldots \ldots \ldots \ldots(1)$