Q.The transverse displacement of a string (clamped at its both ends) is given by $y(x,t) = 0.06\sin(2\pi x/3)\cos(120\pi t)$. All the points on the string between two consecutive nodes vibrate with (Note: more than one of the given options may be correct.)

Concept: Standing Wave Properties

The given displacement describes a standing wave. Rewrite it in the standard form:

Between two consecutive nodes (where $A(x) = 0$), every point oscillates as $y = A(x)\cos(120\pi t)$.

Analysis of each property:

• (A) Frequency: The time dependence $\cos(120\pi t)$ is identical for all points. Every point oscillates with angular frequency $\omega = 120\pi \text{ rad/s}$, hence the same frequency $f = 60 \text{ Hz}$. ✓

• (B) Phase: All points have the factor $\cos(120\pi t)$ with no additional phase shift; they reach maximum displacement simultaneously. Between nodes, $\sin(2\pi x/3)$ keeps the same sign, so all points move in phase. ✓ …

The given displacement describes a standing wave on a string. Standing waves form when two identical traveling waves moving in opposite directions superpose. The key insight is that unlike a traveling wave where the disturbance propagates, in a standing wave each point oscillates in place with an amplitude that depends on its position.

The general form is $y(x,t) = A(x) \cos(\omega t + \phi)$, where $A(x) = 0.06\sin(2\pi x/3)$ is the position-dependent amplitude and $\omega = 120\pi$ rad/s governs the time variation.

Nodes occur where $A(x) = 0$, i.e., where $\sin(2\pi x/3) = 0$, giving $x = 0, \frac{3}{2}, 3, \frac{9}{2}, \ldots$ Between any two consecutive nodes, the sine function is either entirely positive or entirely negative, which determines the behavior of all points in that segment.

Let me examine each option for points between consecutive nodes:

1. Frequency (Option A)

Every point on the string oscillates with the time dependence $\cos(120\pi t)$. The angular frequency is $\omega = 120\pi$ rad/s, corresponding to a frequency:

This is independent of position $x$. All points, whether near a node or an antinode, complete 60 oscillations per second.

Option (A) is correct.

2. Phase (Option B)

The displacement at any point $x$ between two consecutive nodes is:

Between consecutive nodes, $\sin(2\pi x/3)$ has a fixed sign (all positive or all negative). This means:

• If $\sin(2\pi x/3) > 0$ in the segment, all points have displacement proportional to $+\cos(120\pi t)$
• If $\sin(2\pi x/3) < 0$ in the segment, all points have displacement proportional to $-\cos(120\pi t)$

In either case, when $\cos(120\pi t) = +1$, all points in that segment reach their maximum displacement simultaneously (in the same direction). When $\cos(120\pi t) = 0$, all points pass through equilibrium together. They move in phase with each other.

Option (B) is correct.

3. Energy (Option C) …

Step 1: Write the standing wave as $y(x,t) = A(x)\cos(\omega t)$ with $A(x) = 0.06\sin(2\pi x/3)$; nodes are where $A(x)=0$.

Step 2: The time factor $\cos(120\pi t)$ is identical for every point on the string, so every point oscillates with the same frequency, $f=60$ Hz. Confirms (a).

Step 3: Between two consecutive nodes, $\sin(2\pi x/3)$ keeps one fixed sign, so all points there reach maximum displacement, and pass through zero, at exactly the same instants — they move in phase. Confirms (b). …