Reading...
Play button
Play button
Use LEFT and RIGHT arrow keys to navigate between flashcards;
Use UP and DOWN arrow keys to flip the card;
H to show hint;
A reads text to speech;
27 Cards in this Set
- Front
- Back
|
The divergence can be thought of as
|
the flux per unit volume.
|
|
|
Take the div of a ____, the gradient of a ____
|
Vector field, scalar field
|
|
|
The divergence of the gradient is the _____.
|
Laplacian, which takes a function and makes another function
|
|
|
Solutions to the wave equation of the form
q(r, t) = Aei(k·r−wt), w = |k|v are called_________. |
Plane wave solutions.
|
|
|
Every solution to the 3D wave equation can be obtained by _________.
|
superimposing real plane wave solutions.
|
|
|
The gradient of a function is always perpendicular to __________.
|
the surfaces upon which the function is constant.
|
|
|
What kind of differential equations will "separate"?
|
Linear, homogeneous differential equations.
|
|
|
Bessel's equation is what type of DE?
|
Linear, homogeneous, ordinary...
|
|
|
Cylindrical symmetry implies that the solution ...
|
Is independent of phi and z, which suggest that n=0 and k=0.
|
|
|
The solution to the spherical polar wave equation via separable solution consists of
|
two harmonically oscillating pieces (T and phi), the associated Legendre equation (for theta), and a spherical Bessel function.
|
|
|
Spherically symmetric solutions to the wave equation imply that
|
l=0, m=0, no theta or phi dependence, no Neumann component (for finite).
|
|
|
Azimuthal (axial) symmetry in a solution implies
|
no dependence on phi, l=0, m=0.
|
|
|
The energy current density is
|
a vector field representing the flow of energy from point to point.
|
|
|
What is a continuity equation?
|
The manifestation of the conservation of energy in the dynamics of a continuous medium.
|
|
|
The continuity equation is satisfied whenever
|
q(r,t) satisfies the wave equation.
|
|
|
The energy density of a system is given by
|
p(x,t)= (u/2)(diff(q,t))^2 + (t/2)(diff(q,x))^2. The first term is kinetic energy density, the second is potential energy density.
|
|
|
The energy current density, or energy flux, is given by
|
j(x,t)= -(t)(diff(q,t))*(diff(q,x))
|
|
|
To get total energy, you just
|
integrate the appropriate energy density on the boundary of the surface.
|
|
|
The total energy is constant for transverse waves on a string because
|
normal mode solutions do not interact, that is, they are independent of each other. They therefore can be added without interacting.
|
|
|
Energy density =
Current density = |
scalar field
vector field |
|
|
Continuity equation equates
|
time derivatives of energy density p with space derivatives of energy current density j - so 3D is diff(p,t)= - div(j)
|
|
|
Azimuthal symmetry means what for the Legendre functions?
|
They are the Legendre polynomials! m=0!
|
|
|
What is Rodriquez's formula?
|
A nice way to calculate the Legendre polynomials.
|
|
|
P, Q, J, Y, H?
|
Leg. funcs of 1st kind, Leg. funcs of 2nd kind, Bessel of first, Bessel of second, Hankel (Bessel of Third)
|
|
|
What kind of wave would emerge from a pulsating sphere at the origin?
|
A spherically symmetric traveling wave.
|
|
|
For the spherical traveling wave, the radiative part is what? the non-radiative part is what?
|
The radiative part goes like 1/r^2, the non- like 1/r^3. The idea is that the non-radiative part looks like a standing wave - energy just goes back and forth. As a result of these terms, which are called local fields, the current goes like 1/r.
|
|
|
The divergence theorem states that the
|
integral of the divergence of V over some volume is equal to the flux of V through the area bounding that surface (which is closed = has no boundary).
|
