Standing Wave in superposition equation problem

[beebo021]

A string with tension F Newtons, mass m kilograms, and length L meters is clamped at each end (y=0 at x=0 and x=L). At time t=0, the displacement in the y-direction for each point x on the string is defined as:

y(x, t = 0) = 2 sin ( 2πx/L) + 3 sin (πx/L)
= y1(x, t = 0) + y2(x, t = 0)

(a) (1 pts) What is the value of the amplitudes A1 and A2?
(b) (2 pts) What is the phase velocity vp?
(c) (1 pts) What is the value of k1 and k2?
(d) (1 pts) What is the value of ω1 and ω2
(e) (2 pts) What is the displacement y for each point x at any time t? (Write out y(x,t) as a function of y1 and y2. You do not need to simplify.)

Period (s) T = 2π/ω
Angular Velocity (radians/s) ω = vpk = 2πf
Wavenumber (radians/m) k = 2π λ
y(x,t) = Ʃyi(x,t)
y(x, t) = Acos (kx − ωt)

For part a A1=2 and A2=3, For part e y(0,0)=0 for x=0, and y(L,0)=2sin(2π)+3sin(π).

Attempt at k1: L=(2)(λ/2) due to the fact that there are 2 nodes. k=2π/λ, so k1=2π/L

Attempt at k2: k2=(2/1)(π/L)

attempt at w1 and w2: both would be zero as time is at the zero mark, so the period will be zero. If this is true the phase velocity will be zero as well.

 

[jbsteele]

(a) A1=2 and A2=3(b) vp=0(c) k1=2π/L and k2=2π/L(d) ω1=0 and ω2=0(e) y(x, t) = y1(x, t) + y2(x, t) = 2sin(2πx/L - 0t) + 3sin(πx/L - 0t)