Rombes Physics 1B: PSet 2 PROBLEM I Problem 1 Orthogonal Functions 8: Fourier Decomposition In lecture, we mentioned that sinusoidal travelling waves provide \"building blocks\" out of which all waves can be built. Let's examine why. (8) Recall that a general vector L7 in three dimensions can be \"built\" out of the Cartesian unit vectors 5:, g}, and 2: V = V1i+ Vyg+ m. V1, V9, and Vz tell us \"how much\" of the vector lies in the :c-, y, or z-direction, respectively. Using the fact that the Cartesian unit vectors form an orthonormal set, i.e. H) H) H H: \"a\": II \"a\": \"a\": ll \":3; H) {w N> II II H O H D O H N) N: H show that we can \"select out\" the component of the vector l7 along a particular direction by dotting V with the relevant unit vector. (d) It turns out that any (well-behaved) function f(x) on the interval [-L, L] can be represented as a sum of sine and cosine functions of different periods: + Elan cos ("IF) + by sin ("I" ) ] n=1 This is sometimes called its Fourier decomposition. Show that, just like in parts (a) and (b), we can "select out" the component of f(x) corresponding to cos (maz ) or sin (maz ) by integrating f(x) against cos (maz) or cos ( mNZ ) "." ), respectively. That is, show that am = I f(x) cos ( max) om = 1 / " s(x ) sin (maz ) de.Rombes Physics 13: PSet 2 PROBLEM I (C) We were able to decompose vectors in this way clue to the orthonormality of the basis vectors. It turns out that we can do this with functions too, as well as vectors. Here we will consider functions of a single variable 1' that are dened on the interval a: E [L, L]. The \"unit functions\" that take the place of unit vectors are sine and cosine functions, sin ($), cos ($), for n = 0,1,2, . . .. For functions, integrals take the place of the dot product (they are both examples of a more general notion of \"inner product\" on a vector space). Prove the following orthonormality relations between these functions: imnewek 2:2\" i/Lsmcaewer r. L 1 mm: max 1 n=m90 E _L5( L )S( L )= 0 \"9"\" 2 n=m=0 You may nd trig identities involving the product of sines and cosines to be useful here. (b) In part (a), we considered a vector in three dimensions. Now, imagine we have a vector [7 in N dimensions, whose unit vectors are labelled 6,. and comprise an orthonormal set, i.e. A A {1 n = m an - em = This vector can be written as N U = Z Una\". n=1 Again, Show that we can \"select out" the component of the vector [7 along a particular direction by dotting U with the relevant unit vector