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

• azolotor
In summary, the equation for the harmonic wave at t = 0 is r = 15sin(kx+π/3). This can be found by setting up two equations for the wave at two different positions and solving for the amplitude using trigonometric identities and algebra. The final equation can be simplified using the trig identity sin(u+v) = sin(u)cos(v) + sin(v)cos(u).
azolotor
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)

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

## What is the wave equation and why is it important in optics?

The wave equation is a mathematical formula that describes the behavior of waves, including light waves in optics. It is important because it allows us to predict and understand the properties of waves, such as their amplitude, frequency, and wavelength.

## How do you find the wave equation given position and amplitude information?

To find the wave equation, you will need to know the amplitude of the wave and its position or wavelength. Once you have this information, you can use the formula A*sin(kx-wt), where A is the amplitude, k is the wave number, x is the position, w is the angular frequency, and t is time.

## What is the relationship between wavelength and frequency in the wave equation?

The wavelength and frequency of a wave are inversely proportional, meaning that as one increases, the other decreases. This relationship is described by the formula v = f*λ, where v is the speed of the wave, f is the frequency, and λ is the wavelength.

## How does the wave equation differ for different types of waves in optics?

The wave equation may differ for different types of waves in optics, depending on their properties. For example, the wave equation for electromagnetic waves, including light waves, is different from the equation for mechanical waves, such as sound waves.

## What are some real-life applications of the wave equation in the field of optics?

The wave equation has many practical applications in optics, including predicting the behavior of light in different materials, designing and optimizing optical devices such as lenses and mirrors, and understanding phenomena such as diffraction and interference. It is also used in fields such as telecommunications, imaging, and fiber optics.

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