Homework for Waves (Fourier series)

Analysis of normal modes of a string (start in class): A string (mass per unit length $\mu$, under tension $T$) is anchored at $x$ = 0 and $x = L$. It is then displaced so that it has the following profile at $t$ = 0, and the transverse velocity at all points is zero at $t$ = 0. Give a complete mathematical description of the motion of the string, including a Mathematica animation.

Here is a guide to help you:

(a) Write this wave form very generally as a superposition of the allowed standing wave modes of vibration of this string (Remember the group activity in class? Use that and generalize to many wavelengths).

(b) Make sure that the time dependence satisfies what you know about the velocity at $t$ = 0, and that the space dependence satisfies what you know about where the string is anchored.

(c) Addressing the spatial dependence only, find out what contribution each “normal mode” makes at $t$ = 0. (Fourier series will be useful here even though this is not a periodic function – why?)

(d) Now return to the full wave function, including the time dependence of each normal mode. Plot the function (you need to include enough terms to form a reasonable approximation to the function), and animate it. Describe what you see.

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