# Calculate intial angle of a refracted wave

• BOYLANATOR
In summary, the problem is that two waves of different frequencies are traveling towards each other and the waves cancel out. The solution is to use Snell's Law to calculate the distance between the two waves and then use that distance to calculate the angle between the waves.
BOYLANATOR
Please see attached image for problem and brief description of attempted solution.

Can you be more clear on what exactly is known?

BOYLANATOR said:
Please see attached image for problem and brief description of attempted solution.

[ ATTACH=full]84469[/ATTACH]
Yes this is solvable.

I'm looking for a general solution when you know the velocity contrast of the layers (to give the ratio for snells law) and you know the thickness of the two layers as well as the separation of the source and receiver.

BOYLANATOR said:
I'm looking for a general solution when you know the velocity contrast of the layers (to give the ratio for snells law) and you know the thickness of the two layers as well as the separation of the source and receiver.
Do you know how the velocities are related to index of refraction?

Yes v2/v1 =n1/n2. But whether I use the ratio of refractive indices or the ratio of velocities doesn't really matter. Either way they are just two known variables that carry through.

Without writing out my attempt at a derivation in full, what I did was re-write Θ2 in terms of Θ1. Then you get a nasty term of the form tan(arcsin(x)) which can be replaced by $x/(sqrt(1-x^2))$. I get rid of the other tan(theta) in terms of opp/adj in an attempt to first work them out (getting theta back will be easy at the end) but then the equations are hard to solve.

Sorry I don't have my workings here. I can write out some maths tomorrow.

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Lets say that D/2 is the sum of the horizontal distances traveled in the two layers (only considering the downgoing wave):

$D/2 = (d_1+d_2=) z_1tan\theta_1+z_2tan\theta_2$

From Snell's Law:

$\theta_2=sin^{-1}(\frac{v_2}{v_1}sin\theta_1$)

$D/2 = z_1tan\theta_1+z_2tan(sin^{-1}(\frac{v_2}{v_1}sin\theta_1))$

There is a substitution of the form:

$tan(sin^{-1}(x))=\frac{x}{\sqrt{1-x^2}}$

Therefore,

$D/2 = z_1tan\theta_1+z_2\frac{\frac{v_2}{v_1}sin\theta_1}{\sqrt{1-(\frac{v_2}{v_1}sin\theta_1)^2}}$

From here, I wasn't sure what direction to go in. I tried swapping the angles for known distances using basic trig but the algebra became very long. The goal is really just to rearrange for either $\theta_1$ or if it's easier to swap out the angles then I would want to solve for either $d_1$ where $d_1$ is the horizontal distance traveled in the first layer.

Last edited:
I needed the solution for this to run a computer model. I have since just solved the problem iteratively in MATLAB but I am interested to know if the maths leads to a reasonably neat solution. Any ideas?

## What is the initial angle of a refracted wave?

The initial angle of a refracted wave is the angle at which the wave enters a new medium from a different medium. It is measured between the incident ray and the normal line, or the perpendicular line, drawn at the point of incidence.

## How is the initial angle of a refracted wave calculated?

The initial angle of a refracted wave can be calculated using Snell's law, which states that the ratio of the sine of the angle of incidence to the sine of the angle of refraction is equal to the ratio of the velocities of the wave in the two different mediums. This can be expressed as: sin(θ1) / sin(θ2) = v1 / v2, where θ1 is the angle of incidence, θ2 is the angle of refraction, and v1 and v2 are the velocities of the wave in the two mediums.

## What factors can affect the initial angle of a refracted wave?

The initial angle of a refracted wave can be affected by the change in the speed of the wave as it travels from one medium to another, the density of the two mediums, and the angle at which the wave enters the new medium. The refractive index of the two mediums also plays a role in determining the initial angle of the refracted wave.

## Why is it important to calculate the initial angle of a refracted wave?

Calculating the initial angle of a refracted wave is important because it helps us understand the behavior of waves when they travel from one medium to another. It also allows us to predict the direction of the refracted wave and how it will interact with the new medium.

## Can the initial angle of a refracted wave be greater than 90 degrees?

No, the initial angle of a refracted wave cannot be greater than 90 degrees. This is because the angle of incidence and the angle of refraction cannot be greater than 90 degrees according to Snell's law. If the angle of incidence is greater than 90 degrees, total internal reflection will occur instead of refraction.

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