Q.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 1.5m and its mass is $3.0 \times 10^{-2}$ kg. (Note: more than one of the given options may be correct.)

Concept: Standing waves from superposition; wave parameters

The given displacement $y(x,t) = 0.06\sin\left(\frac{2\pi x}{3}\right)\cos(120\pi t)$ has the form of a standing (stationary) wave because it factors into a purely spatial part and a purely temporal part. A progressive wave would have the form $\sin(kx \pm \omega t)$.

Step 1: Extract wave parameters

Compare with the standard standing-wave form $y = A\sin(kx)\cos(\omega t)$:

• Wave number: $k = \frac{2\pi}{3}$ rad/m, so wavelength $\lambda = \frac{2\pi}{k} = 3$ m
• Angular frequency: $\omega = 120\pi$ rad/s, so frequency $f = \frac{\omega}{2\pi} = 60$ Hz
• Wave speed: $v = f\lambda = 60 \times 3 = 180$ m/s

Step 2: Check each option

(A) False – the $\sin(kx)\cos(\omega t)$ form is characteristic of a standing wave, not a progressive wave.

(B) True – it is a stationary wave with frequency 60 Hz. …

The given displacement has the form $y(x,t) = A\sin(kx)\cos(\omega t)$, which is the signature of a standing (stationary) wave. A standing wave arises when two identical progressive waves traveling in opposite directions interfere. Unlike a progressive wave where the disturbance travels, in a standing wave certain points (nodes) remain permanently at rest while others (antinodes) oscillate with maximum amplitude. The key is recognizing the product structure: one factor depends only on position, the other only on time.

Let me extract the wave parameters and check each option systematically.

Identifying the wave parameters

The given equation is:

Comparing with the standard standing-wave form $y(x,t) = A\sin(kx)\cos(\omega t)$:

1. Wave number and wavelength:

Since $k = \frac{2\pi}{\lambda}$:

2. Angular frequency and frequency:

Since $\omega = 2\pi f$:

3. Wave speed:

The relationship between wave speed, frequency, and wavelength is:

We can verify this makes physical sense. The linear mass density is:

For $v = 180$ m/s, the required tension would be $T = \mu v^2 = 0.02 \times 180^2 = 648$ N, which is reasonable for a string.

Analyzing each option

Option (A): Progressive wave of frequency 60 Hz?

A progressive wave has the form $y(x,t) = A\sin(kx \pm \omega t)$ or $A\cos(kx \pm \omega t)$, where the phase $(kx \pm \omega t)$ mixes position and time. Our equation separates into $\sin(kx) \times \cos(\omega t)$, which is not a progressive wave. The disturbance does not travel; instead, each point oscillates in place with an amplitude that depends on its position.

Option (A) is incorrect.

Option (B): Stationary wave of frequency 60 Hz?

The product form $\sin(kx)\cos(\omega t)$ is exactly the definition of a standing (stationary) wave. The frequency is indeed 60 Hz.

Option (B) is correct.

Option (C): Superposition of two waves traveling in opposite directions?

A standing wave is formed by the superposition of two identical waves traveling in opposite directions. Using the trigonometric identity:

This shows our wave is the sum of: …

Step 1: $y(x,t) = 0.06\sin(2\pi x/3)\cos(120\pi t)$ has the product form $\sin(kx)\cos(\omega t)$ — the signature of a standing wave (position and time factors are separate), not a progressive wave $\sin(kx\pm\omega t)$. Rules out (a).

Step 2: $k = 2\pi/3$ rad/m $\Rightarrow \lambda = 3$ m. $\omega = 120\pi$ rad/s $\Rightarrow f = 60$ Hz. Confirms (b).

Step 3: Wave speed $v = f\lambda = 60 \times 3 = 180$ m/s. …