boundary condition

Examples of boundary condition in the following topics:

• Mathematical Represenation of a Traveling Wave

  • The wave function is further determined by taking additional information, usually given as boundary conditions and some others.
  • A solution of the wave equation in two dimensions with a zero-displacement boundary condition along the entire outer edge.

• Exercises

  • We have two boundary conditions that we can apply.
  • The second boundary condition is that as $r$, the potential must approach that of the original, unperturbed E-field:
  • If we apply the orthogonality condition to the $r=a$ boundary condition, we can see that
  • In the limit of large $r$, our boundary condition only constrains terms involving positive power of $r$, since the negative powers of go to zero.
  • It is clear from this that we can satisfy the boundary condition at infinity only if all the of $A$ coefficients are zero expect the $\ell = 1$ term.

• Wave Nature of Matter Causes Quantization

  • Once the string becomes a "bound system" with specific boundary restrictions, it allows waves with only a discrete set of frequencies.
  • However, once an electron is "bound" by a Coulomb potential of a nucleus, it can no longer have an arbitrary wavelength as the wave needs to satisfy a certain boundary condition.
  • We now realize this as a condition for constructive interference of an electron in a (bound) circular orbit.

• Reflection and Transmission

  • We will impose additional restriction on the waves by applying "boundary conditions" at x=0.
  • At the boundary x=0, the wave must be continuous and there should be no kinks in it.

• Winds

  • The crucial difference is that the boundary conditions for a wind differ from those for accretion.

• Introduction to Spherical and Cylindrical Harmonics

  • When solving boundary value problems for differential equations like Laplace's equation, it is extremely handy if the boundary on which you want to specify the boundary con- ditions can be represented by holding one of the coordinates constant.
  • On the other hand, if we tried to use Cartesian coordinates to solve a boundary value problem on a spherical domain, we couldn't represent this as a fixed value of any of the coordinates.
  • Obviously this would be much simpler if we used spherical coordinates, since then we could specify boundary conditions on, for example, the surface $x = r \cos \phi \sin \theta$ constant.

• 1-D Separation of Variables: Summary of the Argument

  • The clamped end boundary conditions imply that $X(0) = X(l) = 0$ .
  • This solution obviously will not satisfy general initial conditions.
  • Now we have some hope of satisfying the initial conditions.
  • The symbol to the left indicates that on the WWW page you will find a Mathematica notebook; in this case one that solves the 1D problem for initial conditions corresponding to the string being pulled up in the middle and released at $t = 0$.

• Total Internal Reflection and Fiber Optics

  • Total internal reflection happens when a propagating wave strikes a medium boundary at an angle larger than a particular critical angle.
  • Total internal reflection is a phenomenon that happens when a propagating wave strikes a medium boundary at an angle larger than a particular critical angle with respect to the normal to the surface .
  • If the refractive index is lower on the other side of the boundary and the incident angle is greater than the critical angle, the wave cannot pass through and is entirely reflected.
  • The angle of incidence is measured with respect to the normal at the refractive boundary (see diagram illustrating Snell's law).

• Thin Film Interference

  • This is a phenomenon that occurs when incident rays reflected by the upper and lower boundaries of a thin film interfere with one another and form a new wave.
  • Additionally, a phase shift of 180° or $\pi$radians may be introduced upon reflection at a boundary depending on the refractive indices of the materials on either side said boundary.
  • However, this condition may change if phase shifts occur upon reflection.
  • Demonstration of the optical path length difference for light reflected from the upper and lower boundaries.

• Phase Changes and Energy Conservation

  • During a phase transition, certain properties of the medium change, often discontinuously, as a result of some external condition.
  • During a phase transition of a given medium certain properties of the medium change, often discontinuously, as a result of some external condition, such as temperature or pressure.
  • The measurement of the external conditions at which the transformation occurs is termed the phase transition.
  • The solid lines—boundaries between phases—indicate temperatures and pressures at which the phases coexist (that is, they exist together in ratios, depending on pressure and temperature).