The Mathematics of Wave Motion

목차

1. 1차원 파동 (One-Dimensional Waves)
1) 파동 함수 (Wave function)
2) 파동 방정식 (사실은 Maxwell 방정식으로부터)

2. 조화파 (Harmonic Waves)
1) 파장의 정의 : wavelength (λ)
2) 주파수의 정의 : frequency
3) 다른 parameter의 정의
4) another representation

3. 위상과 위상 속도 (phase and phase velocity)
1) phase ( ) : the entire argument of the sine function for a harmonic wave
2) derivation of phase velocity

4. The superposition Principle

5. The Complex Representation

6. 평면파 (Plane waves)
1) Mathematical expression of a plane ( ⊥ )
2) the wave function of plane wave

7. The Three-Dimensional Differential Wave Equation

8. 구면파 (Spherical waves)

9. Cylindrical Waves (주면파)

본문내용

pression of a plane ( ⊥
vec k
)
vec r_o
: position vector of one particular point on the plane
If
vec r
exists on the plane,
(vec r ``-`` vecr_o ) · vec k ~=~ 0
vec k · vec r ~=~ {rm const} ~=~ a
* Every planes defined by this eq. are
parallel to each other.
2) the wave function of plane wave
Since the plane wave has constant disturbance over every plane defined by
vec k · vec r ~=~ {rm const}
,
psi( vec r ) ~=~ A ``e^{i ``vec k · vec r }
, (
vec k
= propagation vector )
By introducing the time dependence in an analogous fashion to that of the 1-dim. wave
psi( vec r , t ) ~=~ A ``e^{i (vec k · vec r ``-`` ωt `)}
== The phase velocity of the plane wave is parallel to the propagation vector.
파면 (wavefront or wave surface)
: the surface joining all points of equal phase at any given time.
Any three-dimensional wave can be expressed as a combination of plane waves.
2.7 The Three-Dimensional Differential Wave Equation
∇^2 psi ~=~ 1 over v^2 {partial^2 psi} over {partial t^2}
psi (boldr , t) ``=`` C_1 f `( boldk · boldr
2.8 구면파 (Spherical waves)
general wave function
psi(r,t) ~=~ C_1 {f(r-vt)} over r ``+`` C_2 {g(r+vt)} over r
, C1 and C2 are constants.
(∵ from the conservation of energy and spherical symmetry)
wave function of harmonic spherical wave
psi(r,t) ~=~ A over r ``cos k(r `±` vt)
psi(r,t) ~=~ A over r ``e^{i``k(r `±` v``t)}
, A is the source strength.
The surface of constant phase is given by kr = constant.
2.9 Cylindrical Waves (주면파)
wave function of harmonic cylindrical wave
psi(r,t) ~approx~ A over sqrt{r} ``cos k(r `±` vt)
for r 》 0
psi(r,t) ~approx~ A over sqrt{r} ``e^{i``k(r `±` v``t)}
[참고] Scalar and Vector waves
transverse wave :
vector wave ( in order to describe the direction along which the disturbance occurs.)
longitudinal wave :
scalar wave

키워드

Mathematics,   Wave,   Motion,   파동,   방정식