Notes: One Dimensional Wave Equations

Lecture Introduction

• Instructor: Anuj Kumar, Assistant Professor, Department of Mathematics
• Subject: One Dimensional Wave Equations
• In the previous lecture, the solution of one-dimensional wave equations was discussed.

Important Topic: Question Session

• In this lecture, important questions from an exam perspective will be discussed.
• These are often asked in Delhi University and CCS University.

Focus on Conditions

• The basis for solving all questions will be the boundary and initial conditions given in the previous video.
• Boundary conditions can change.

Special Question: Stretched and Fastened String

• A string instrument is stretched between 22 points.
• Initial position: Lifting and releasing the string.
• Initial displacement: u(x, 0) = A * sin(kx)

Standard Solution Process

1. Apply the one-dimensional wave equation:

   • u_tt = c^2 * u_xx
   • where c is the speed of wave propagation.

2. Boundary conditions:

   • u(0,t) = 0
   • u(L,t) = 0
   • Here, L is the length of the string.

Analysis

• Initial conditions:

  • u(x, 0) = f(x)
  • u_t(x, 0) = g(x)

• These conditions give us information about the initial state of the wave.

Process to Find Solution

1. Use binary partition:

   • u(x,t) = (X(x))(T(t))

2. Total Solution:

   • Different cases: Positive, Negative, and Zero
   • Based on different initial and boundary conditions.

Results

• All solutions are presented in generalized form.

• Final solution:

  • u(x,t) = A * sin(kx) * cos(ωt)

• It is necessary to pay attention to correctly applying the sine and cosine system in the proper context for the general solution.

Conclusion

• The next lecture will discuss two-dimensional wave equations.

• In conclusion, all students are requested to carefully study all the important points related to this topic.

• Thank you!