Final answer:
The characteristic angular frequencies, ωₙ, of the vibration of a string in an elastic medium can be found by solving the Klein-Gordon equation. The solution involves solving separate equations for the space and time variables. The solution for the space variable will depend on the boundary conditions at the ends of the string.
Step-by-step explanation:
In this equation, u is the displacement of the membrane along the y direction, v is the wave speed, and λ is a constant determined by the elasticity of the medium.
To find the characteristic angular frequencies, we need to solve the differential equation by assuming a solution of the form u(x,t) = X(x)T(t). By substituting this into the equation, we get two separate equations: one for X(x) and one for T(t). Solving these equations will give us the characteristic angular frequencies ωₙ.
The solution for X(x) will depend on the type of boundary conditions at the ends of the string. For a clamped string, the solution can be written as X(x) = A sin(nπx/a), where n is an integer representing the mode of vibration and A is the amplitude of the vibration. The corresponding solution for T(t) is T(t) = B cos(ωₙ t), where B is the amplitude of the time variation and ωₙ is the characteristic angular frequency.
Your question is incomplete, but most probably the full question was:
String in an elastic medium. The vibration of a string with length a (0<x<a) in an elastic medium is described by Klein-Gordon equation: ∇²u(x,t)= (1/v²) ∂² u/∂t² + λ² u, where u is the displacement of the membrane along y direction, v is the wave speed and λ is a constant determined by the elasticity of the medium.
Assume the string is clamped at both ends.
Find the characteristic angular frequencies ωₙ