# Optics: Finding the wave equation given position and amplitude information

A harmonic wave traveling in +x-direction has, at t = 0, a displacement of 13 units at x = 0 and a displacement of -7.5 units at x = 3λ/4. Write the equation for the wave at t = 0.

## Homework Equations

The equation for a harmonic wave is

r = asin(kx-vt+θ)

a being the amplitude
k being the wave number k=2π/λ
v being the velocity of the wave
θ being the initial phase angle

## The Attempt at a Solution

I set up the wave equations at both positions because we have two unknowns so we need two equations

13 = asin(θ) & -7.5= asin((2π/λ)(3λ/4) + θ)
13/sin(θ ) = a -7.5=asin(3π/2 + θ)

Now I plugged in 13/sin(θ ) for a in the other equation and I ended up with

-7.5=(13/sin(θ ))sin(3π/2 + θ)
-7.5sin(θ ) = 13sin(3π/2 + θ)

This is where I got stuck. Am I on the right track? I imagine there is a trig identity that will help me solve for θ and then I can easily solve for the amplitude. The answer according to the book is:

15sin(kx+π/3)

## Answers and Replies

I solved it, but thanks to those who may have read it. For those that are curious you use the trig identity sin(u+v) = sin(u)cos(v) + sin(v)cos(u) and from there it is simple algebra