Direct Energy, 2018a

14 LIE ANALYSIS 319
and
y → y(1 + ε) (14.30)
The above transformation says that if we scale any solution of a linear
equation, y(t), by a constant (1 + ε), the result will also be a solution of
the equation. By denition, a linear equation obeys exactly this property.
By knowing a solution of the wave equation and this symmetry, we can nd
a whole family of related solutions, and this family of solutions is illustrated
in Figure 14.2.
The wave equation also contains the symmetry transformation described
by the innitesimal generator
U = sin(ω 0 t)∂ y . (14.31)
The operators ξ and η can be identied directly from the innitesimal
generator.
ξ =0 (14.32)
η = sin(ω 0 t) (14.33)
Again we can nd the corresponding nite transformations using Eq. 14.14.
t → e εU t = t (14.34)
(
y → e εU y = 1+ε sin (ω 0 t) ∂ y + 1 )
2! (ε sin (ω 0t) ∂ y ) 2 + ... y (14.35)
y → y + ε sin (ω 0 t) (14.36)
If we know a solution y(t) to the wave equation, this transformation tells us
that y(t)+ε sin (ω 0 t) is also a solution. Since ε can be any innitesimal or
nite constant, we have found another family of solutions using symmetry
concepts, and these solutions are illustrated in Figure 14.3.
In this section, we have discussed three of the symmetries of the wave
equation. The wave equation actually contains eight continuous symmetry
transformations. Deriving these transformations is left as a homework
problem.