How Do You Derive the Equation for a Traveling Sinusoidal Wave?

[moonkey]

Homework Statement

A sinusoidal wave of frequency 50Hz travels along a string at a velocity of 24m/s. At a given instant the displacement and velocity of a certain point in the string are 22mm and 1.6m/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 2000m/s², 3ms before the instant specified above. What is the minimum distance possible between this point and the point x=0?

Homework Equations

From the information given what I know is

f=50Hz, v=24m/s, λ=0.48m, ω=314.16/s, k=13.09/m

Equation for sinusoidal wave:

y(x,t)=ASin(ωt-kx+φ)

The Attempt at a Solution

I managed to derive the traveling wave equation as

y=-0.02258Sin(314.16t-13.09x-1.343)

where A=-0.02258, φ=-1.343

The answer in the back of the book is

y=-0.02258Sin(13.09x-314.16t-1.343)

Am I right in thinking that both those equations are the same thing?The biggest problem I'm having is the second part of the question.

Here's what I've done (the answer in the back of the book is -55mm)

y=ASin(ωt-kx+φ)

dy/dx=-ωACos(ωt-kx+φ)

d²y/dy²=-ω²ASin(ωt-kx+φ),

where d²y/dy²=acceleration = 2000m/s² and t=3ms=0.003s

so subbing in my knowns and rearranging I get:

arcSin[2000/(-(314.16²)*(-0.02258)) - 314.16*(-0.003) + 1.343]/-13.09 = x

x=-0.2597Help please

 

[moonkey]

Never mind. Figured it out. My differentiation was wrong