# Is a bending wave transverse or longitudinal?

A bending wave in a metal bar or string is simply called a transverse wave because the macroscopic oscillation is transverse. However, paradoxically, in a frame co-moving with the atoms, the atoms are oscillating longitudinally. The outer boundary of a curved section is being stretched, while the inner boundary is being compressed, somewhat like a bimetal.

Is the longitudinal motion in the co-moving frame really irrelevant to the term 'transverse wave'? Or is there a less paradoxical terminology?

• There’s also transverse motion that leads to the stretching. For all reasonably-strong materials, isn’t that larger? Jan 8, 2020 at 15:26

We normally distinguish between a shear wave and a compression wave.

A bending wave of the type you describe is a pure shear wave in the limit of zero amplitude, but as you say it also involves longitudinal motion when the amplitude of wave is large. However in many (most?) cases the amplitudes aren't large enough for this to be an issue and the interpretation of the wave is straightforward.

• Wouldn't a wave in a metal string be shear free, at least a standing wave? For example in a node of the standing wave, in the co-moving frame, the atom lattice rotates in accordance with the local direction of the string. My intuition says that rotation prevents shear, although I admit it isn't proper mathematics. Sep 26, 2017 at 16:59

I don't think that it will be a transverse wave. A wave is considered transverse if the displacement vector is orthogonal to the wave vector. If we consider the vertical ($$z$$) direction to be the direction for deflection and the horizontal ($$x$$) direction the direction for extension, we have the following for Euler-Bernoulli Beams.

\begin{align} &u_x = u_0 - x\frac{d w}{d x}\, ,\\ &u_z = w(x)\, , \end{align}

where $$u_0$$ is the extension of the mid-axis.

So, we can see that the displacement in the horizontal direction depends on the deflection, and the displacement won't be completely transverse.

Transverse and longitudinal waves are the general solutions of the elastodynamic equations in an elastic (isotropic) medium. If we let out the theoretical case of an infinite (unbounded) 3D medium, most (if not all) solutions to elastic wave problems will involve a superposition of these two solutions (or more fundamentally, a solution deriving from the superposition a curl-free and divergence-free vector field). Bending waves can be written as a superposition of a transverve wave and a longitudinal wave satisfying the traction-free boundary conditions of the bounded medium experiencing bending (e.g a beam,a plate), they are neither a pure transverse nor a pure longitudinal wave.