Initial and Boundary Conditions

Oct 13, 2025 by windailys.com 10 views

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Introduction

The wave equation is a fundamental partial differential equation (PDE) that describes the propagation of waves in various physical systems, such as strings, membranes, and acoustic waves. In this article, we will discuss the initial value problem (IVP) of the wave equation with a non-homogeneous term, specifically the equation $u_{tt} = 4u_{xx} + \sin(ct)\cos(x)$ , where $-1 < x < 1$ and $t > 0$ . This equation is a classic example of a PDE with a non-trivial solution, and its study has important implications in various fields, including physics, engineering, and mathematics.

Background

The wave equation is a second-order linear PDE that describes the propagation of waves in a medium. It is given by the equation $u_{tt} = c^2u_{xx}$ , where $u = u(x,t)$ is the wave function, $x$ is the spatial coordinate, $t$ is time, and $c$ is the wave speed. In this case, we have a modified wave equation with a non-homogeneous term, $\sin(ct)\cos(x)$ , which represents an external force or a source term.

Initial and Boundary Conditions

The initial value problem (IVP) of the wave equation is defined by the following initial and boundary conditions:

Initial conditions: $u(0,x) = \sin(x)$ and $u_t(0,x) = 0$ for $x \geq 0$
Boundary conditions: $u(0,x) = 0$ and $u_t(0,x) = 0$ for $x < 0$

These conditions specify the initial displacement and velocity of the wave, as well as the boundary conditions at $x = 0$ .

Method of Undetermined Coefficients

To solve the IVP of the wave equation, we will use the method of undetermined coefficients. This method involves assuming a solution of the form $u(x,t) = X(x)T(t)$ , where $X(x)$ is a function of $x$ alone and $T(t)$ is a function of $t$ alone. Substituting this assumption into the wave equation, we get:

X(x)T''(t) = 4X''(x)T(t) + \sin(ct)\cos(x)

Separation of Variables

We can separate the variables by dividing both sides of the equation by $X(x)T(t)$ :

\frac{T''(t)}{T(t)} = 4\frac{X''(x)}{X(x)} + \frac{\sin(ct)\cos(x)}{X(x)T(t)}

Since the left-hand side depends only on $t$ and the right-hand side depends only on $x$ , both sides must be equal to a constant, say $k$ . This gives us two ordinary differential equations (ODEs):

T''(t) - kT(t) = 0

4X''(x) - kX(x) = -\sin(ct)\cos(x)

Solution of the ODEs

We can solve the ODEs using standard techniques. The first ODE is a second-order linear homogeneous ODE with constant coefficients, and its solution is given by:

T(t) = Ae^{kt} + Be^{-kt}

where $A$ and $B$ are arbitrary constants.

The second ODE is a second-order linear non-homogeneous ODE with constant coefficients, and its solution is given by:

X(x) = C\cos(\sqrt{\frac{k}{4}}x) + D\sin(\sqrt{\frac{k}{4}}x) + \frac{1}{4}\cos(x)

where $C$ and $D$ are arbitrary constants.

Application of the Boundary Conditions

We can apply the boundary conditions to the solution $u(x,t) = X(x)T(t)$ to determine the values of the constants $A$ , $B$ , $C$ , and $D$ . The boundary condition $u(0,x) = 0$ for $x < 0$ implies that $X(x) = 0$ for $x < 0$ , which means that $C = 0$ . The boundary condition $u_t(0,x) = 0$ for $x < 0$ implies that $T'(0) = 0$ , which means that $B = 0$ .

Final Solution

The final solution of the IVP of the wave equation is given by:

u(x,t) = \frac{1}{4}\cos(x)\sin(ct)\cos(\sqrt{4}t)

This solution satisfies the initial and boundary conditions, and it represents a wave propagating in the positive $x$ -direction with speed $c$ .

Conclusion

In this article, we have discussed the initial value problem (IVP) of the wave equation with a non-homogeneous term, specifically the equation $u_{tt} = 4u_{xx} + \sin(ct)\cos(x)$ , where $-1 < x < 1$ and $t > 0$ . We have used the method of undetermined coefficients and separation of variables to solve the IVP, and we have applied the boundary conditions to determine the values of the constants. The final solution represents a wave propagating in the positive $x$ -direction with speed $c$ . This solution has important implications in various fields, including physics, engineering, and mathematics.

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Q&A

Q: What is the wave equation and why is it important?

A: The wave equation is a fundamental partial differential equation (PDE) that describes the propagation of waves in various physical systems, such as strings, membranes, and acoustic waves. It is a second-order linear PDE that is used to model a wide range of phenomena, including sound waves, light waves, and water waves.

Q: What is the significance of the non-homogeneous term in the wave equation?

A: The non-homogeneous term in the wave equation represents an external force or a source term that affects the wave propagation. In this case, the term $\sin(ct)\cos(x)$ represents a periodic force that is applied to the wave.

Q: How do you solve the IVP of the wave equation using the method of undetermined coefficients?

A: To solve the IVP of the wave equation using the method of undetermined coefficients, you assume a solution of the form $u(x,t) = X(x)T(t)$ , where $X(x)$ is a function of $x$ alone and $T(t)$ is a function of $t$ alone. You then substitute this assumption into the wave equation and separate the variables to obtain two ordinary differential equations (ODEs).

Q: What are the boundary conditions for the IVP of the wave equation?

A: The boundary conditions for the IVP of the wave equation are given by $u(0,x) = 0$ for $x < 0$ and $u_t(0,x) = 0$ for $x < 0$ . These conditions specify the initial displacement and velocity of the wave at the boundary.

Q: How do you apply the boundary conditions to determine the values of the constants?

A: To apply the boundary conditions, you substitute the solution $u(x,t) = X(x)T(t)$ into the boundary conditions and solve for the values of the constants. In this case, the boundary condition $u(0,x) = 0$ for $x < 0$ implies that $C = 0$ , and the boundary condition $u_t(0,x) = 0$ for $x < 0$ implies that $B = 0$ .

Q: What is the final solution of the IVP of the wave equation?

A: The final solution of the IVP of the wave equation is given by:

u(x,t) = \frac{1}{4}\cos(x)\sin(ct)\cos(\sqrt{4}t)

This solution satisfies the initial and boundary conditions, and it represents a wave propagating in the positive $x$ -direction with speed $c$ .

Q: What are the implications of the solution in various fields?

A: The solution of the IVP of the wave equation has important implications in various fields, including physics, engineering, and mathematics. It can be used to model a wide range of phenomena, including sound waves, light waves, and water waves.

Additional Resources

Conclusion

In this Q&A article, we have discussed the IVP of the wave equation with a non-homogeneous term, specifically the equation $u_{tt} = 4u_{xx} + \sin(ct)\cos(x)$ , where $-1 < x < 1$ and $t > 0$ . We have answered various questions related to the wave equation, its significance, and its solution. The final solution represents a wave propagating in the positive $x$ -direction with speed $c$ , and it has important implications in various fields.