Wave Motion

We will calculate the frequencies of the normal modes for a system of two equal masses connected by three springs with spring constants , , and .

Step 1: Writing the Equations of Motion

Let the displacements of the two masses be denoted by and . The forces acting on the masses using Hooke's law are:

For mass 1:

For mass 2:

Step 2: Normal Mode Solutions

Assume solutions of the form , . Substituting, we get:

Step 3: Solving the Eigenvalue Problem

We now solve the determinant equation:

Using determinant properties, this simplifies to:

Solving this quadratic gives:

Step 4: Substituting Values

Given , , and :

Final Answer: The correct frequencies are .

Therefore, the correct option is (A).