How Does Time Affect the Displacement of a Plucked Violin String?

[danmel413]

Homework Statement

A violin string is plucked to the shape of a triangle with initial displacement:

y(x,0) = { 0.04x if 0 < x < L/4
(0.04/3)(L-x) if L/4 < x < L

Find the displacement of the string at later times. Plot your result up to the n = 10 term, for t = L/10v, L/5v, and L/2v, where v = p T /µ is the speed of the wave. Are all harmonics excited?

Homework Equations

The normal Fourier Series equations (the ones I use are here)

The Attempt at a Solution

The solution is supposed to be of the form y(x,t) = Σa_nsin(nπx/L)cos(nπvt/L)

My only issue is conceptually. I understand how to go through the math to get a_n. Why is it that the vt is within the cosine and not the sine? How am I even supposed to know to add the vt in? I feel like this is something unbelievably basic that I'm missing.

 

[Orodruin]

Why would you think it should be a sine? The string is released from rest. How would you solve the problem?

 

[olivermsun]

The solution is supposed to be of the form y(x,t) = Σa_nsin(nπx/L)cos(nπvt/L)

My only issue is conceptually. I understand how to go through the math to get a_n. Why is it that the vt is within the cosine and not the sine? How am I even supposed to know to add the vt in? I feel like this is something unbelievably basic that I'm missing.

Notice first that the solutions terms are a product of sin(function of x only) and cos(function of t only). What do you think these two "separated" pieces represent, conceptually?