1-D Wave Equation (String Vibrations) (continued)

Strategy: A general approach to solving the 1-D wave equation

∂²u/∂t² = a² ∂²u/∂x²

is to assume separation of variables. i.e. Assume:

u(x,t) = X(x) T(t)

Substitution of u(x,t) = X(x) T(t) into the wave equation gives

X(x) dT²/dt² = a² d²X/dx²

Then by division (d²X/dx²)/X = (dT²/dt²)/a²T = ˗ λ = separation constant

So d²X/dx² + λ X = 0 and dT²/dt² + λa²T = 0

Note: To solve the 1-D wave equation you need to solve two eigenvalue problems.

Further note that the separation constant could be zero, negative, or positive.

Examine each case separately.

Solution of eigenvalue problem. Strategy: Start with the eigenvalue problem for X(x).

d²X/dx² + λ X = 0 subject to the boundary conditions

X(0) = X(L) = 0

And consider each case for λ separately.

The cases are λ =0, λ < 0, and λ > 0.

The result of this calculation yields the eigenvalues λ_n and eigenvectors, X_n(x).

Strategy: Substitute the eigenvalues, λ_n , into the equation for T(t) to obtain

dT²/dt² + λ_na²T = 0

Solution of this equation normally yields T_n(t) = C_n cos a√λ_nt + D_n sin a√λ_nt

Then combine with X_n(x) with T_n(t) to obtain the product solution T_n(t) X_n(x).

u_n(x,t) = [C_n cos a√λ_nt + D_n sin a√λ_nt ] X_n(x).

Click here to continue discussion on vibrations of a string.

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