3.2 Solution of the Vibrating String Problem

3.2 Solution of the Vibrating String Problem

  • Check the dimensions of each term.

  • If the string is located in a medium that resists movement, then it's there.
    • The resistance is expressed as a force in either direction or magnitude.
    • It only affects Eq.
    • Proceed to derive a new equation.
  • The displacement of the vibrating string is described by the initial value-boundary value problem.
    • The method of separation of variables may be applied.

  • Both members of the equation need to be constant for the equality to hold.

  • a.

  • There are some standing waves on the CD.

  • a.

  • A given function is to be expanded in a series of sines if there are two initial conditions.

  • a.

  • There is an animated version of this solution on the CD.
  • Although the solution in the example can be considered valid, it is difficult to see what shape the string will take at different times.

  • The string returns to its initial position through the positions shown during the second half-period.
    • The string has a nonzero velocity in the horizontal portion.

  • In later sections, we will generalize.

  • The vibrating string problem is solved in the fifth exercise.

  • Find the eigenvalues and eigenfunctions associated with the wave equation for each set of boundary conditions.

  • For a thin beam and a string, write out formulas for the first four frequencies.

  • The antenna is free to move.

  • This comes from the boundary conditions and the partial differential equation.
  • Take the first two eigenfunctions and compare them to the figure.
  • Find a solution by separation of variables.

  • There are 2 constants.

  • The series is in Eq.

The only function that can satisfy the boundary conditions is 0

  • In some cases, we could express the solution of the wave equation directly in terms of the initial data.
    • We change variables to see what the wave equation looks like.
  • It is possible to find the general solution of this equation.

  • a.

  • a.

  • a.

  • The wave equation and initial conditions are satisfied.

  • It can be adapted to other cases.

  • The graphs were graphically averaged in the two preceding steps.
    • The boundary conditions should be satisfied.
  • The units are reflected in the horizontal axis.
    • The boundary conditions should be satisfied.

  • The Eq is being used.

  • The vibrating string problem can be solved with a sketch.

  • The wave equation can be changed by using the change of variables at the beginning of the section.