Solving the Wave Equation with Separation of Variables and Fourier Series

The wave equation is a second-order partial differential equation that plays a pivotal role in various fields of physics and engineering, including acoustics and electromagnetism. This article explores how to solve the wave equation using the method of separation of variables and Fourier series. We will provide a step-by-step guide to finding the solution, complete with boundary and initial conditions.

Understanding the Wave Equation

The wave equation we are considering is:

(U_{tt} c^2 U_{xx})

where (c) is the wave speed. This equation is subject to the following boundary and initial conditions:

(U_x(0,t) 0) (U_x(pi,t) 0) (U(x,0) 0) (U_t(x,0) cos^2{x})

Step-by-Step Solution

Let's go through the solution step-by-step.

Step 1: Separation of Variables

Assume a solution of the form:

(U(x,t) X(x)T(t))

Substituting this into the wave equation:

(X(x)T''(t) c^2 X''(x)T(t))

We can separate the variables by dividing both sides by (c^2 X(x)T(t)), which gives:

(frac{T''(t)}{c^2 T(t)} frac{X''(x)}{X(x)} -lambda)

where (lambda) is a separation constant.

Step 2: Solve the Spatial Equation

The spatial equation is:

(frac{X''(x)}{X(x)} -lambda)

The general solution for this equation is:

(X(x) A cos(sqrt{lambda} x) B sin(sqrt{lambda} x))

Applying the boundary conditions:

(X(0) 0) (X(pi) 0)

We get:

(A cos(0) B sin(0) 0) implies (A 0) (-sqrt{lambda} A sin(sqrt{lambda} pi) 0) implies (sin(sqrt{lambda} pi) 0), so (sqrt{lambda} n) for (n 0, 1, 2, ldots)

Thus, the eigenvalues are:

(lambda_n n^2)

and the eigenfunctions are:

(X_n(x) A_n cos(nx))

Step 3: Solve the Temporal Equation

The temporal equation is:

(frac{T''(t)}{c^2 T(t)} -lambda_n -n^2)

The general solution for this equation is:

(T_n(t) C_n cos(c n t) D_n sin(c n t))

Step 4: Construct the General Solution

The general solution is a sum of the eigenfunctions:

(U(x,t) sum_{n0}^{infty} left(C_n cos(c n t) D_n sin(c n t)right) cos(nx))

Step 5: Apply Initial Conditions

Applying the initial condition (U(x,0) 0):

(U(x,0) sum_{n0}^{infty} C_n cos(nx) 0)

This implies:

(C_n 0)

for all (n). Therefore, the solution simplifies to:

(U(x,t) sum_{n0}^{infty} D_n sin(c n t) cos(nx))

Applying the initial condition (U_t(x,0) cos^2{x}):

(U_t(x,0) sum_{n0}^{infty} D_n c n cos(nx) cos^2{x})

Using the identity:

(cos^2{x} frac{1 cos(2x)}{2})

This implies that:

The term (frac{1}{2}) corresponds to (n 0) The term (frac{1}{2} cos(2x)) corresponds to (n 2)

Therefore:

(D_0 0) (D_2 frac{1}{2c})

Thus, the final solution is:

(U(x,t) frac{1}{2c} sin(2 c t) cos(2 x))

Conclusion

The above method allows us to solve the wave equation with the given boundary and initial conditions. The solution (U(x,t) frac{1}{2c} sin(2 c t) cos(2 x)) satisfies the wave equation, the boundary conditions, and the initial conditions. This approach is a powerful technique in analytical methods for solving partial differential equations in physics and engineering.