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?
[Physics] Stress and Strain in a standing wave
newtonian-mechanicsstress-strainwaves
Related Solutions
The easiest way is to join both the formulas in definition of sound intensity:
$$ i = pv $$
where $p$ and $v$ are acoustical pressure and velocity respectively. In a standing wave there is a $\pi/2$ phase shift between them and hence the intensity maxima out the nodes and antinodes of pressure and velocity.
However, usually you use and measure not the immediate intensity $i$, but it's mean value in time $I$ which is:
$$ I = \frac{p^2_{acoustic \ RMS}}{\rho_0 c_0} $$
There is nothing wrong with the definition in terms of displacement because half a period later the graph looks like this.
So perhaps it would be better to have it as
- the position where maximum displacement occurs in a standing wave system
You could also use the terms "maximum amplitude" or "maximum magnitude of the displacement".
Best Answer
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.