The orthogonality of sine functions is a key concept in understanding Fourier series. When we talk about orthogonal functions, we're referring to functions that are "independent" of each other in a specific sense. For sine functions within a Fourier series, this means they do not overlap when integrated over a specific interval.Consider two sine functions, \( \sin\left(\frac{m\pi x}{L}\right) \) and \( \sin\left(\frac{n\pi x}{L}\right) \). When you integrate the product of these functions over the interval from 0 to \( L \), something interesting happens:
- If \( m eq n \), the integral equals zero.
- If \( m = n \), the integral equals \( \frac{L}{2} \).
This orthogonality property is fundamental because it simplifies the Fourier series. It allows us to isolate each term in the series when finding the coefficients. This is useful in many applications, such as signal processing and solving differential equations.