Stress and Strain in a standing wave

Question

Where are stress and strain maximum in a Standing wave? what about a transverse wave? My book says "strain is maximum at the nodes in a standing wave". But since Antinodes are the position of maximum displacement, +strain should be maximum at the antinodes. Can someone explain this intuitively?

Answer 1

The stress and strain referred in a wave, are the stress and strain developed in the medium in which the wave is propagating.

Intuitively, in a standing wave on a string, the strain will be maximum at the point around which stretching is the most (this is by basic definition of strain). The nodes are the stationary points, due to which any movement of the string will lead to more stretching than a point which can move. Since the antinode is the point with the maximum displacement, the local streching around it is the least, in most cases we even say it to be zero.

For a more mathematical reasoning you can refer to the image, which mathematically shows the strain in transverse and longitudinal waves.

Here also you can see that the slope in a standing wave is maximum at the node, hence having maximum strain.