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.

• Defining Boundaries

  • Social groups are defined by boundaries.
  • One important factor in how symbolic boundaries function is how widely they are accepted as valid.
  • Symbolic boundaries are a "necessary but insufficient" condition for social change.
  • According to sociologists, it is "only when symbolic boundaries are widely agreed upon can they take on a constraining character… and become social boundaries. " Thus, rituals and traditions to define boundaries are extremely influential in determining how groups interact.
  • He saw the symbolic boundary between the sacred and the profane as the most profound of all social facts, and the one from which lesser symbolic boundaries were derived.

• 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.

• Major Features of a Phase Diagram

  • Phase diagrams are useful because they allow us to understand in what state matter exists under certain conditions.
  • A phase diagram is a graph which shows under what conditions of temperature and pressure distinct phases of matter occur.
  • The major features of a phase diagram are phase boundaries and the triple point.
  • Phase boundaries, or lines of equilibrium, are boundaries that indicate the conditions under which two phases of matter can coexist at equilibrium.
  • Along the blue phase boundary, water exists as both a vapor and a liquid.

• Interpreting Phase Diagrams

  • The lines that separate these single phase regions are known as phase boundaries.
  • Along the phase boundaries, the matter being evaluated exists simultaneously in equilibrium between the two states that border the phase boundary.
  • By examining the phase boundaries and the triple point, researchers can use phase diagrams to understand under which conditions a pure sample of matter exists in two or three state equilibrium.
  • General observations from the diagram reveal that certain conditions of temperature and pressure favor certain phases of matter.
  • The dotted green line refers to the solid-liquid phase boundary for water.

• Faith in the Face of Suffering

  • During that period the boundaries of the Empire shifted, but generally included what is known as Germany, Austria, and parts of Denmark, Northern Italy, France and Central and Eastern Europe .
  • Towns were especially hard-hit because of the crowded conditions.
  • Conditions were further unsettled by the return of the plague throughout the rest of the 14th century.
  • The boundaries of the Holy Roman Empire shifted during its reign but generally included what is known as Germany, Austria, and parts of Denmark, Northern Italy, France and Central and Eastern Europe

• 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.