ABSTRACT
A.1 Waves EquationIf we have a stationary wave (non-sinusoidal form), as shown in Fig. A.1.1, as y¢ = f(x¢) at t = 0 (A.1.1)
we can have a traveling wave as y = f(x ± vt) at t = t (A.1.2)where x¢ = x ± vt, ∂ ∂
= x x ¢ 1 and ∂
∂ = ±
x t
v ¢
Also when y = f(x + vt), the wave moves in the left direction. When y = f(x – vt), the wave moves in the right direction. From the space derivative as∂
∂ =
∂ ∂
◊ ∂ ∂
= ∂ ∂
y x
f x
x x
f x¢
¢ ¢
(A.1.3) ∂
∂ =
∂ ∂
∂ ∂
= ∂ ∂ ∂
∂ ◊ ∂ ∂
= ∂ ∂
∂ ∂
= ∂ ∂
y
x x y x
y x x
x x x
f x
f
x
( / )
¢ ¢
¢ ¢ ¢ (A.1.4)and the time derivative as
∂∂ = ∂∂ ◊ ∂∂ = ± ∂∂yt fx xt v fx¢ ¢ ¢ (A.1.5)∂ ∂
= ∂ ∂
∂ ∂
= ∂ ∂ ∂ ∂
◊ ∂ ∂
= ∂ ∂
± ∂ ∂
Ê ËÁ
ˆ ¯˜ ◊± = ∂
t t y t
y t x
x t x
v f x
v v ( / )
¢ ¢
¢ ¢ f
x∂ ¢2 (A.1.6) Equations A.1.4 and A.1.6 yield∂ ∂
= ◊ ∂ ∂
1y
x v
y
t (A.1.7) On the other hand, if we have the wave function of a traveling wave (sinusoidal waveform) as y = A sin k (x ± vt) (A.1.8)the solution to the partial differential equation will be as follows:
Space derivative ∂ ∂
= ± y x
kA k x vtcos ( ) (A.1.9) ∂
∂ = - ±
x k A k x vtsin ( ) (A.1.10)
Time derivative ∂ ∂
= ± ± y t
kvA k x vtcos ( ) (A.1.11)
∂ ∂
= - ± 2
t k v A k x vtsin ( ) (A.1.12) Equations A.1.10 and A.1.12 also yield∂
∂ = ◊
∂ ∂
1y
x v
y
t (A.1.13) On the other hand, a harmonic wave (sinusoidal form) can be written as y A k x vt= ±
sin ( ) (A.1.14)where A and k are the amplitude and the propagation constant of the periodic wave form/character, respectively. Since the difference between sine and cosine functions is a relative translation of 0.5p radian, it is also enough to treat only one of these wave functions. For a sine wave of amplitude, as shown in Fig. A.1.2., at constant time t
Figure A.1.2 Harmonic waves. y A k x vt A k x vt= + + = + +sin ( ) sin ( )l p2 (A.1.15) A sin (kx + kl + kvt) = A sin(kx + kvt + 2p) (A.1.16)Clearly, kl = 2p (A.1.17) At a constant point x, y = A sin k(x + v[t + T]) = A sin k(x + vt) + 2p (A.1.18) y = A sin (kx + kvt + kvT) = A sin (kx + kvt + 2p) (A.1.19)Clearly, KvT = 2p (A.1.20)
Equations A.1.17 and A.1.20 yield v T
= l and v f= l (A.1.21)
If we define the angular frequency w p
= 2 T
, a more compact form of the harmonic wave in Eq. A.1.14 is y A x t T
= ±Ê ËÁ
ˆ ¯˜cos
l (A.1.22) y A kx t= ±
sin ( )w (A.1.23) Furthermore, at a constant point x, the phase angle j is constant or j = k(x ± vt) (A.1.24) dj = 0 = k(dx ± vdt) (A.1.25)dx
dt v= ∓ (A.1.26) From Eq. A.1.14, the general form of the harmonic wave at initial phase j0, as shown in Fig. A.1.3, can be modified as y A k x vt= ± + cos
sin [ ( ) ]f
Figure A.1.3 Harmonic waves showing the initial phase. If we define the initial conditions for harmonic wave at x = 0, t = 0, and y = y0, we get y A
1 0= Ê ËÁ
ˆ ¯˜
y A
using sine function (A.1.28)
The initial phase j0 is generally set equal to zero for simplicity. Let a complex number be defined asz a ib= + (A.1.29)where a = Re(z), b = lm(z), and i2 = −1. z can be shown in the polar plot as seen in Fig. A.1.4.
