There are two harmonic waves on a string: y1=Aexp(i(kx-wt)) y2=Bexp(i(-kx-wt)) (1) y(x=0)=0 (2) y(=L) = 0 The total wave is (3) y=exp(-iwt)[Aexp(ikx) + Bexp(-ikx)] (1) gives A = -B, so that (3) can be written y = exp(-iwt)[Aexp(ikL) - Aexp(-ikL)] = 0 For (2) to be true, we must have exp(ikL) - exp(-ikL) = 0, that is cos(kL) + i sin(kL) - cos(-kL) - i sin (-kL) = 0 ---> 2sin(kL) = 0 ---> kL = n*pi Is my book wrong when it says kL = 2n*pi?
The book is not wrong but they ignore integer multiples of angle (pi), where sin equals zero, and they take only integer multiples of the angle 2*pi. I dont know why they do it, i am not a mechanical engineer!
I'd say that the book is wrong if they omit an infinite number of solutions! The correct answer is kL = n*pi, where n is an integer. It's easy to show that kL = pi is a solution to the equation, a solution not included in the book's answer.
My intuition says the book is wrong but most times, after a careful study, we are wrong and the books are correct for some reason. However, i am not saying that books are always correct!
I taught college math for 18 years, using a pretty fair number of textbooks. That was long enough for me to realize that there frequently are errors in the answers. The way it works is that the textbook writers make up a bunch of problems and hire grad students at minimum wage (or slightly higher) to work them. I'm not saying that the books were full of errors in the answers, but it would be unusual to find a textbook, especially in calculus and above, for which the answers were error-free.