Question

let L be the total length of the rod, Y its Young's modulus, and p its density. Express the normal mode frequencies in terms of L, Y and p( density). and find the allowed wavelengths.

Answer 1

Part 1 -> Forming the differential eq

Let the density of the rod be: ,

axial force acting on the rod: F,

cross-sectional area of the rod: A

Length of rod: L

displacement of rod (longitudinal): y(x,t)

Youngs modulus: Y

For an infinitesimal deformation in length of rod:

while strain (being the measure of deformation) on rod:

stress (defined as force applied per unit area): , and the stress-strain relationship is given by Hooke's law:

substituting values of stress and strain, the relationship becomes:

Differentiating eq(1),

From Newtons second law(change in momentum is equal to force applied): (p: linear momentum)

thus,

Comparing eq(2) and eq(3),

From general wave equation: , thus above found equation can be rearranged as:

Therefore, velocity can be written as:

Part 2 -> Solving the differential eq

the above found differential eq could be solved using Separation of Variables i.e

on differentiating w.r.t x,

and now differentiating w.r.t t:

Putting into general wave eq:

for oscillatory motion: , therefore,

Thus the equation becomes: and from eq(6):

The general solutions of eq(6) and (7): and

Now, y(0,t) = 0, thus C₂ = 0

and at y(L,t), the axial force vanishes,

therefore,

which is the normal mode frequency of the rod.