Oscillations and Waves: Winter-2026
HW 1a: Due W1 D3

1. Harmonic Integrals

   Show that: \begin{equation*} \frac{2}{T}\int_0^T\sin(n\omega t)\sin(m\omega t)dt=\delta_{m,n} \end{equation*} Here the period \(T=2\pi/\omega\), and \(n\) and \(m\) are integers greater than zero. Recall that \(\delta_{m,n}\) (the "Kronecker delta") is given by \[\delta_{m,n}= \begin{cases} 1 &m=n\\ 0 &m\ne n \end{cases} \] You will have to treat the two cases separately. Do not choose specific values of \(m\) and \(n\), prove this relationship in general for ANY integer \(m\) and \(n\).

   Hints: Since it is easy to integrate exponentials, even if the exponent is a complex number, use Euler's formula to change the sines into exponentials: \begin{equation*} \sin(n\omega t)=\frac{e^{i n\omega t}-e^{-i n\omega t}}{2i} \end{equation*} Beware of zero in the denominator of fractions!

   Please evaluate all integrals analytically by hand.

2. Change the Period in Fourier Series The integrals that show the Fourier series basis functions are orthonormal on the interval \(x=0\dots 2\pi\) are: \begin{align*} \int_{0}^{2\pi} \cos nx \cos mx\, dx &=\pi \delta_{m,n}&n,m=1,\dots,\infty\\ \int_{0}^{2\pi} \cos nx \cos mx\, dx &=2\pi \delta_{m,n}&n=0\\ \int_{0}^{2\pi} \sin nx \sin mx\, dx &=\pi \delta_{m,n}&n,m=1,\dots,\infty\\ \int_{0}^{2\pi} \sin nx \cos mx\, dx &=0&n,m=1,\dots,\infty \end{align*} Notice that in the equations above, the variable \(x\) is dimensionless. In practical physics problems, you often want to work with a function which is periodic on the range \(y=0\dots L\), where \(y\) is a variable with dimensions of length. Use a simple change of variables to find equations analogous to the ones above if the interval is \(y=0\dots L\). Do the change of variables by hand, not with technology.