ABSTRACT
Figure 15.1. Arbitrary disturbance at two successive instants of time Consider an arbitrary scalar disturbance Ψ which propagates with velocity v along the x-axis, in the laboratory frame S, as illustrated in Figure 15.1. In its rest frame
whose Cartesian system ,Σ ξηζΘ coincides with that in S, Oxyz, at time we have ,0=t x vtξ = − (15.1)
Since the disturbance has a constant position and profile with respect to it must depend on the time t through the coordinate
,Σ ,ξ as expressed by some function
( ) ( ) ( )vtxfftx −==Ψ ξ, (15.2) where ( )f ξ defines the profile or the waveform. For any profile f, Eq.(15.2) gives
01 =Ψ⎟⎠ ⎞⎜⎝
⎛ ∂ ∂+∂
∂ tvx
(15.3)
which describes a travelling wave along the x-axis. If v were negative, the wave would propagate in the opposite direction, obeying the equation
01 =Ψ⎟⎠ ⎞⎜⎝
⎛ ∂ ∂−∂
∂ tvx
(15.4)
and this is satisfied by a function of arbitrary profile g, which is not required to be related with ,f that is ( ) ( )vtxgtx +=Ψ , (15.5) By applying the differential operator (15.4), Eq.(15.3) becomes
( ) ( ) 0111 2
2 =−⎟⎟⎠
⎞ ⎜⎜⎝ ⎛
∂ ∂−∂
∂=−⎟⎠ ⎞⎜⎝
⎛ ∂ ∂+∂
∂⎟⎠ ⎞⎜⎝
⎛ ∂ ∂−∂
∂ vtxf tvx
vtxf tvxtvx
whereas by applying the operator (15.3), Eq.(15.4) reads
( ) ( ) 0111 2 2
2 =+⎟⎟⎠
⎞ ⎜⎜⎝ ⎛
∂ ∂−∂
∂=+⎟⎠ ⎞⎜⎝
⎛ ∂ ∂−∂
∂⎟⎠ ⎞⎜⎝
⎛ ∂ ∂+∂
∂ vtxg tvx
vtxg tvxtvx
It follows that the equation
21 x v t x v t
⎛ ⎞∂ ∂ ∂ Ψ ∂ Ψ− Ψ = =⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠ (15.6)
is satisfied by a linear superposition of the two wave functions (15.2) and (15.5), of arbitrary form and unrelated to each other. Equation (15.6) is called the equation of wave motion in one dimension and permits waves of given profile to progress with velocity v in either direction along the x-axis. Since Eq.(15.6) is valid for functions of arbitrary profile, which is determined by various sources, it predicts the same wave motion irrespective of nature of the wave. The wave equation is linear, so that each of the wave functions (15.2) and (15.5) can in turn be written as a linear combination of any number
Ψ
of other functions of .vtx ± This is known as the principle of superposition of waves, which states that any solution to the wave equation can be subdivided into simpler partial waves, whose linear combination constitutes the actual wave. In two dimensions, a wave travelling along the x-axis can be defined by the wave function ( ) ( ) ( ) ( ) ( )vtergvterfvtxgvtxftr xx +⋅+−⋅=++−=Ψ GGGGG, where rG is the position vector in the xy plane, represented in Figure 15.2. We will assume that on any line perpendicular to the x-axis, Ψ is constant and will maintain the same constant value if the line moves along the x-axis with constant velocity .v± The moving line is called a wavefront. A line wave function in the direction of an arbitrary unit vector ϕϕ sincos yxv eee GGG += has the form ( ) ( ) ( )vtergvterftr xx +⋅+−⋅=Ψ GGGGG, ( ) ( )cos sin cos sinf x y vt g x yϕ ϕ ϕ ϕ= + − + + vt−
(15.7) where the arbitrary functions f and g obey
01 =⎟⎟⎠ ⎞
⎜⎜⎝ ⎛
∂ ∂+∂
∂+∂ ∂ f
tv e
y e
x e vyx
GGG (15.8)
for a wave travelling in the positive ve
G -direction, and
01 =⎟⎟⎠ ⎞
⎜⎜⎝ ⎛
∂ ∂−∂
∂+∂ ∂ g
tv e
y e
x e vyx
GGG (15.9)
if the wave propagates in the opposite direction.
