Two pulses in a stretched string, whose centres are initially 8 cm apart, are moving towards each ot

Q: If the string is completely flat at $t = 2\text{ s}$, why isn't the total energy zero?

While the displacement of the string is zero everywhere, the particles of the string are still in motion. The energy of a wave consists of both potential energy (from deformation) and kinetic energy (from particle motion). At this instant, the potential energy is zero, but the kinetic energy is at its maximum, keeping the total energy conserved.

Q: How do we know the particles are moving if the string looks flat?

The particles have velocity because they are transitioning from positive to negative displacement (or vice versa). Mathematically, particle velocity is $v_p = -v_{\text{wave}} \frac{\partial y}{\partial x}$. Even though the net displacement $y$ is zero at this instant, the rate of change of displacement with respect to time is at its maximum.

Q: Do the waves destroy each other permanently after $2\text{ s}$?

No, they do not. Waves propagate independently of one another. After $t = 2\text{ s}$, the two pulses will pass through each other and emerge completely unchanged, continuing their journeys in opposite directions.

Q: What would happen if both pulses were crests?

If both pulses were crests, they would undergo constructive interference at $t = 2\text{ s}$. The displacement would double, and the energy at that instant would be a mix of both potential and kinetic energy, with potential energy reaching its maximum.

Q: Is this phenomenon unique to transverse waves on a string?

No, this is a fundamental property of all waves obeying the linear wave equation, including sound waves and electromagnetic waves. Destructive interference always involves a temporary redistribution of energy, never its destruction.

Question

Two pulses in a stretched string, whose centres are initially $8	ext{ cm}$ apart, are moving towards each other as shown in the figure. The speed of each pulse is $2	ext{ cm/s}$. After $2	ext{ s}$ the total energy of the pulses will be

Options:

(A) zero

(B) purely kinetic

(C) purely potential

(D) partly kinetic and partly potential

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine a stretched string, a perfect medium for wave propagation.

At time $t = 0$, we introduce two identical but opposite wave pulses.

One is a positive **crest** traveling to the right, and the other is a negative **trough** traveling to the left.

Their centers are initially separated by a distance of $8	ext{ cm}$, and both move with a constant speed of $2	ext{ cm/s}$ towards each other.

$$	ext{Initial Separation } (d) = 8	ext{ cm}$$
$$	ext{Speed of each pulse } (v) = 2	ext{ cm/s}$$

### The Midpoint Encounter

Let us calculate the distance traveled by each pulse in $2	ext{ s}$:

$$	ext{Distance } (s) = v 	imes t = 2	ext{ cm/s} 	imes 2	ext{ s} = 4	ext{ cm}$$

Since both pulses travel $4	ext{ cm}$ towards each other, they cover a combined distance of $8	ext{ cm}$.

Their centers now coincide exactly at the midpoint of their initial separation.

At this precise instant, the two pulses completely overlap in space.

### The Principle of Superposition

When waves overlap, they obey the **Principle of Superposition**.

The resultant displacement $y(x, t)$ at any point is the algebraic sum of the individual displacements $y_1(x, t)$ and $y_2(x, t)$:

$$y(x, t) = y_1(x, t) + y_2(x, t)$$

Since the two pulses are identical in shape but opposite in sign (one is a crest, the other a trough), they undergo **complete destructive interference** at $t = 2	ext{ s}$.

Therefore, the resultant displacement of the string is exactly zero everywhere:

$$y(x, t=2	ext{ s}) = 0 \quad \forall \ x$$

The string appears completely flat and horizontal, as if no wave is present.

### Where Did the Energy Go?

The total mechanical energy of a wave on a string is the sum of its **potential energy** ($U$) and **kinetic energy** ($K$):

$$E_{	ext{total}} = U + K$$

Potential energy is stored in the elastic deformation of the string, which depends on its slope:

$$U \propto \int \left(\frac{\partial y}{\partial x}ight)^2 dx$$

Since the string is completely flat at $t = 2	ext{ s}$, there is no deformation and no slope.

Thus, the potential energy of the string is exactly **zero** ($U = 0$).

However, by the law of **conservation of energy**, the total energy cannot simply vanish.

If we look closely at the particles of the string, they are not at rest.

The particles of the left-moving pulse and the right-moving pulse have velocities that add up constructively.

At the instant of complete overlap, the particle velocities are at their maximum, meaning the **kinetic energy** of the string is at its maximum.

Since $U = 0$, the entire mechanical energy of the system is stored purely as kinetic energy:

$$E_{	ext{total}} = K_{	ext{max}}$$

Thus, the total energy of the pulses after $2	ext{ s}$ is **purely kinetic**, which corresponds to **option (b)**.

Animated Solution for Physics - Waves: Two pulses in a stretched string, whose centres are initially $8 cm$ apart, are moving towards each other as shown in the figure. The speed of each pulse is $2 cm/s$ . After $2 s$ the total energy of the pulses will be

Select Answer:

Visualized Solution

Visualizing the Initial State at $t = 0$

Calculating Distance Traveled in $2 s$

Determining the Positions at $t = 2 s$

Applying the Principle of Superposition

Resultant Displacement of the String

Analyzing Potential Energy

Analyzing Particle Velocity and Kinetic Energy

Total Energy is Purely Kinetic

Conclusion and Correct Option

The Way Forward: Post-Overlap Propagation

The Sigma Insight: Interference of Waves

Analyzing the Setup

The Midpoint Encounter

The Principle of Superposition

Where Did the Energy Go?

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