Flexural waves in beam
EXPERIMENTAL METHODOLOGY
Wave propagation through Euler Bernoulli beam:
The governing equation of the Euler-Bernoulli (EB) beam for free vibration can be written as follows:
where ρ and E are density and Young's Modulus of the EB beam respectively; A is the cross-sectional area and I is the second moment of area.
stand for the transverse displacement, axis distance, and time.
It can be assumed that 
where X(z) is the mode shape function of the function of the beam at z, which is a complex value. Thus, Eq.(1) can be expressed as
Mode shape function Xn (z) can also be assumed:
Substitute Eq.(3) into Eq.(2), we can obtain wave number (λn):
Phase velocity "Vp" defined as Eq. (5).
Hence,
The relation between excitation frequency 
and wave number λn is called dispersion relation. As here the phase velocity increase with increase in excitation frequency as per Eq. (6), the beam can be seen as dispersive medium.