in wave equations sq rt of -1 appears. could you kindly explain why.
I helps to simplify the manipulation and solution of equations.
d'Alembert's solution to the wave equation would be,
u(x,t) = F(x-ct) + G(x+ct)
which can be re-written as
u(x,t) = f(x,t) + g(x,t)
and we can write
f(x,t) = A.exp[i(kx - wt)]
g(x,t) = B.exp[i(kx + wt)]
w = kc.
Although the deformation u(x,t) will clearly have to be a real number for all values of x and t, it turns out to be useful to consider complex soutions to the wave equation as well. The real and imaginary parts of a complex solution will individually satisfy the wave equation, so a complex solution encodes two real solutions.
Where a solution involves trigonometric functions, e.g.
u(x,t) = A.cos(x - ct) + B.cos(x + ct)
rather than a complex exponential function, then the solution and algebraic manipulation of the latter is often much easier than the former.
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