Solving Wave Equations with a Wall - Reflection and Amplitude

[inferno_gogo]

Homework Statement

We have waves propagating in the air. First part asks for a traveling wave solution in the x direction , given a wave length 2pi and amplitude 3m. I have obtained u(x,t) = 3 cos ( x - 345t). The question then asks what happens if a wall is placed at x=0, what is the form of u(x,t) including the reflect wave and incident wave. Also to find the amplitude and phase of the reflect wave given u(0,t) = 0.

Homework Equations

z(x,t)=Asin(ks-vt)

The Attempt at a Solution

I have hit a wall. No idea how to proceed with the form of the equation. Once I get that, I am pretty sure I will manage to find the phase shift and amplitude.

 

[tiny-tim]

I have hit a wall.

Then you should reflect! ​

Hint: draw a diagram!

Draw the wave going through the wall as if it wasn't there.

Then turn one half of the wave round …

does it have the same frequency? the same amplitude? the same velocity? the same phase?

 

[Moetasim]

I understand the frustration of hitting a roadblock in problem solving. Let's break down the problem and see if we can find a solution together.

First, let's review the given information. We have a traveling wave solution in the x direction with a wavelength of 2pi and an amplitude of 3m. This can be represented as u(x,t) = 3 cos (x - 345t).

Now, let's consider what happens when a wall is placed at x=0. This wall will act as a boundary, causing the wave to reflect back in the opposite direction. This means that the reflected wave will have the same wavelength and amplitude, but will be traveling in the opposite direction.

To represent this in our equation, we can add a negative sign to the x term, indicating the change in direction. This would give us u(x,t) = 3 cos (-x - 345t).

Next, we need to consider the incident wave, which is the wave that is approaching the wall before it reflects. This can be represented as u(x,t) = 3 cos (x - 345t).

Now, to find the form of u(x,t) including the reflected and incident waves, we simply add the two equations together. This would give us u(x,t) = 3 cos (x - 345t) + 3 cos (-x - 345t).

To find the amplitude and phase of the reflected wave, we can use the given information that u(0,t) = 0. This means that at x=0, the total wave will be zero, indicating that the reflected wave must have the same amplitude and opposite phase as the incident wave.

Therefore, the amplitude of the reflected wave will also be 3m, and the phase will be shifted by pi. This can be represented as u(x,t) = 3 cos (x - 345t) - 3 cos (-x - 345t + pi).

I hope this helps you move forward with your solution. Remember to always break down the problem into smaller parts and use the given information to guide your steps. Good luck!