27: Higher dimenstional waves

for 2D wave, height is defined as some function ___

z(x,y,t)

Wave equation in 2D

A² z/Ax² + A² z/Ay² = 1/v² A²z/At²

Solution to equation of motion in 2D

z(x,y,t) = A cos (kx +ky -wt)

velocity of 2D wave

v = w/ sqrt(kx² + ky²)

wavenumber k of 2D wave

k = sqrt(kx² + ky²)

wave vector k definition

direction of wave propogation

angle (a) relative to x axis

tan a = ky/kx

what is angle a

angle between propogation direction and x axis

3D wave equation

A²psi/Ax² + A²psi/Ay² + A²psi/Az² = 1/v² A² psi/At²

3D wave equation solution for periodic travelling wave

psi (x y z t) = A cos (kx + ky + kz - wt)

3D wave velocity

v = w / sqrt (kx² + ky² + kz²)

3D wave number k

k = sqrt (kx² + ky² + kz²)

del operator

del = A/Ax, A/Ay etc , scalar → vector

Gradient

del f(xyz) = A/Ax f(xyz), A/Ay f(xyz) etc

taking the gradient ___

turns scalar to vector function

taking the gradient, taking each point in our vector field now tells us about ___

rate/direction of change in f(xyz) at point (xyz)

Gradient - larger vector arrows = ____ slopes, _____ rates of change

steeper, greater

center of 2D gradient field:_____ change, _____ points

little to no change, max/min

Divergence fomula:

del * vector field (xyz) = A/Ax v(xyz) + Ay v (xyz) etc

Divergence turns

scalar to vector field

Divergence measures ______, depending on ______ in ___ directions around a point

net flux of the vectors in or out of a given point (x,y,z), how

the vectors change with respect to position, all

Divergence gives us an indication of ____

net flow rate of the vector quantity at all points

Curl formula:

del x v(xyz)

taking the curl turns

vector field → vector field

curl definition:

rotational change of a vector field at each

point measure of how much the vector fiel is “curling” at any point

laplacian =

A²/ Ax² + A²/ Ay² + A²/ Az²

taking the laplacian of a ____, returns a ____, and

scalar, scalar, vice versa

d’Alembertian (wave operator) definition

time derivative of wave equation - laplaction = 1/v² A²/At² - laplacian

Wave equation in 3D in terms of wave operator

square psi = 0

Wave equation in 3D in terms of laplacian

laplacian psi = 1/v² A²psi/At²