In transverse waves on a string, we neglect the longitudinal displacement of the particles, the reason given in most books is that the slope and the amplitude of the string is very small.
Can someone please prove it mathematically that it's reasonable to neglect the longitudinal motion of the element of the string under these conditions?
( The angle $\phi$ is exaggerated in figure)
A very small displacement, d, would create a very small angle, ϕ, with the strings original position. The transverse displacement, T, would be proportional to d∗cos(ϕ) and the longitudinal displacement,L, would be proportional to d∗sin(ϕ). So as ϕ tends to zero the transverse displacement would tend to d and the longitudinal displacement would tend to zero.
