Two wave pulses identical in shape but inverted with respect to each other are produced at the two ends of a stretched string. At an instant when the pulses reach the middle, the string becomes completely straight. What happens to the energy of the two pulses ?

Explanation

To solve the problem, we need to analyze the situation of two identical wave pulses that are inverted with respect to each other. Here’s a step-by-step breakdown of what happens to the energy of the two pulses when they meet at the center of the string:

### Step 1: Understanding the Wave Pulses
We have two wave pulses that are identical in shape but inverted. This means that one pulse is a crest (upward displacement) and the other is a trough (downward displacement).

### Step 2: Movement Towards Each Other
The two pulses are moving towards each other from opposite ends of the stretched string. As they approach the center, they will eventually overlap.

### Step 3: Meeting at the Center
At the instant when the two pulses meet in the middle of the string, they will overlap completely. Since one pulse is a crest and the other is a trough, they will interfere with each other.

### Step 4: Destructive Interference
When the crest of one pulse meets the trough of the other, they will undergo destructive interference. This means that the displacements of the two waves will cancel each other out. Mathematically, if the amplitude of the crest is +A and the amplitude of the trough is -A, the resultant displacement at the point of overlap will be:
\[ y = A + (-A) = 0 \]
This results in a completely straight string at that instant.

### Step 5: Energy Consideration
In wave mechanics, energy is associated with the amplitude of the wave. When the two pulses interfere destructively, the total mechanical energy of the system does not disappear; rather, it is transformed. At the point of complete cancellation, the kinetic energy of the waves is converted into potential energy in the medium (the string).

### Step 6: Conclusion
After the pulses have passed through each other, they will continue to propagate in their respective directions. The energy that was present in the two pulses is not lost; it is conserved and is temporarily stored as potential energy when the string is straight. Eventually, the pulses will continue to travel away from the center, regaining their original shape and energy.

### Summary
When two identical wave pulses that are inverted with respect to each other meet, they undergo destructive interference, resulting in a straight string momentarily. The energy of the two pulses is conserved and converted into potential energy during this interaction.

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