Question

If you take snapshots of a standing wave on a string, are there certain instants when the string is totally flat?

1) True
2) False

Answer 1

Yes, a standing wave on a string can appear flat at certain instants when all points coincide with nodes due to destructive interference. Standing waves result from two identical waves moving in opposite directions and can exhibit higher amplitudes at antinodes through constructive interference.

Step-by-step explanation:

If you take snapshots of a standing wave on a string at certain instants, it is true that the string may appear to be totally flat. This occurs at a specific point in time during the wave cycle known as the node, where the amplitude is zero. In a standing wave, the points known as nodes are points of no displacement, and when all points on the string coincide with nodes, the string can appear flat for an instant. This happens at the moment of complete destructive interference between the two waves that create the standing wave.

Standing waves are formed by the superposition of two identical waves moving in opposite directions, not in the same direction as might be assumed. When the waves have equal amplitude and frequency and they align precisely, at some points known as antinodes they amplify each other's effect leading to higher amplitude due to constructive interference, while at nodes, they cancel each other out resulting in no movement.

In scenarios where waves overlap on a string that is snapped from both ends, we observe the creation of a standing wave assuming the waves are in phase and of the same frequency. If the string is snapped in opposite directions at both ends, it creates opposite phases which would still contribute to the formation of a standing wave with nodes and antinodes, but the initial shape would be different.

Answer 2

Standing waves on a string never result in the string being totally flat at any instant due to the dynamic interplay of nodes and antinodes in the wave pattern. The interference of incident and reflected waves ensures that there is always some degree of displacement along the string.Thus,the correct option is 2) False

Step-by-step explanation:

Standing waves on a string are formed by the interference of incident and reflected waves, resulting in a pattern of nodes (points of zero displacement) and antinodes (points of maximum displacement). The string is never completely flat at any instant; there are always points on the string experiencing some degree of displacement.

This phenomenon is a consequence of the superposition of waves, and it can be mathematically explained through the principle of interference.

When two waves with the same frequency and amplitude travel in opposite directions on a string and interfere constructively, they create standing waves. The nodes occur at points where the two waves always interfere destructively, causing the string to remain at its equilibrium position.

However, the antinodes, which experience maximum displacement, are never completely flat. The amplitudes of the incident and reflected waves at these points combine to create the standing wave pattern.

In the mathematical framework, if
`\(y_1\) and \(y_2\)` represent the displacements of the incident and reflected waves, respectively, the resultant displacement (y) is given by
`\(y = y_1 + y_2\).` At the nodes,
`\(y_1 = -y_2\)`, resulting in (y = 0), but at the antinodes,
`\(y_1 = y_2\)`, leading to (y) being nonzero. Therefore, the string never becomes entirely flat, highlighting the dynamic nature of standing waves.

Thus,the correct option is 2) False