Elementary travelling wave problem

[patrickmoloney]

Homework Statement

A sinusoidal wave of frequency 50 Hz travels along a string at velocity of 28 m/s. At a given instant the displacement and velocity of a certain point in the string are 24 mm and 1.2 m/s respectively. Taking the certain point and given instant to be x=0, t=0, derive the traveling wave equation which gives the displacement of any point on the string as a function of position x, and time t.

A point in the string has an acceleration of 1800 m/s², at a time 3.0 ms before the instant specified above. What is the minimum distance possible between this and the point x=0

Homework Equations

λ=v/f , k=2π/λ , y(x,t)= Asin(kx-ωt+$\varphi$)

I assume ∂²x/∂t²= -Aω²Sin(kx-ωt+$\varphi$)

The Attempt at a Solution

I would really appreciate the second part explained to me.

I got the first part:

y= Asin(kx-ωt+$\varphi$)

λ= v/f = 28/50 = 0.56 m and k= 2π/λ = 2π/0.56 = 11.2 rad/m and ω= 2πf = 100π

the velocity of the displaced point is obtained using a differential equation:

dy/dt = -Aωcos(kx-ωt+$\varphi$)

y= 0.024 = Asin($\varphi$) at x=0, t=0

dy/dy= 1.2= -Acos($\varphi$)

solving the simultaneous equations we have $\varphi$ = -1.41 rad and A = -0.024 m

∴Equation of traveling wave is y= -0.024sin(11.2x-314t-1.41).

no idea about the next part. Can someone do it for me?

 

[voko]

The second part is done much like the first one. Differentiate the velocity function; that will give the acceleration, whose value you are given. You are further given the time, so the only unknown is the distance. Find it from the acceleration equation.