Waves in two dimensions

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Introduction

Lessons

  1. Waves in two dimensions
  2. Introduction to reflection of two-dimensional waves
  3. Introduction to refraction of two-dimensional waves
  4. Introduction to diffraction of two-dimensional waves

Examples

Lessons

  1. A train of water waves reflected at a boundary. The angle between the incident ray and normal in 30°. What would be the angle between normal and reflected wave?
    1. Sound wave are reaching a rigid boundary at a speed of 330 m/s. What would be the speed of the reflected wave bouncing of the same medium?
      1. Refraction of water waves occurred from deep to shallow water. Explain what happens to the wavelength, speed and the frequency of the refracted wave.
        1. Light waves reflect off a boundary with frequency of 30Hz. The angle between the reflected wave and incident wave is given to be 60°. What are the values for angle of reflection and angle of incident?
          1. What is the difference between the diffraction of light waves through small gaps and large gaps?

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            Topic Basics

            In this lesson, we will learn:

            • Reflection of two-dimensional waves
            • Recreation of two-dimensional waves
            • Diffraction of two-dimensional waves

            Notes:

            • Waves moving along rope or spring are mechanical waves that move only in one dimension.
            • Waves on the surface of water or sound waves move in two dimensions.
            • Similar to one one-dimensional waves, two-dimensional waves get both reflected and refracted.
            • As waves from one medium to another, partially reflect and partially refract.
            • Properties of two-dimensional waves can be shown using a ripple tank which contains a thin layer of water. Vibrating boards produce wave pulses with constant frequency. The lamp above the tank produces shadows below the tank to show the exact location of crest and troughs.
            • Pulses travelling towards a rigid barrier get “Reflected”.
            • Pulses reaching a boundary might change direction or get “Refracted”.
            • Pulses passing through a small hole, spread or get “Diffracted”.

            Reflection

            • A ray diagram model is used to study the reflection of two-dimensional waves.
            • A ray is a line drawn at right angles to the crests of the waves.

            Key terms using a ray model to study reflection:

            Incident Ray: An arrow pointing towards the boundary

            Point of incidence: Point where incident ray meets the boundary

            Reflected Ray: An arrow pointing away from the boundary

            A straight line: showing the direction of the barrier

            Normal: A line drawn perpendicularly to the barrier at point of incidence

            Angle on incidence (<i < i ): The angle between incident ray and NORMAL

            Angle of reflection (<r < r ): The angle between reflected ray and NORMAL

            Barrier

            Waves in two dimensions

            • According to the law of reflection; angle of incidence is equal to angle of reflection ( θi=θr \angle \theta_{i} = \angle \theta_{r} )
            • Upon reflection Speed, Wavelength, and Frequency do NOT change.

            Refraction


            Key terms using ray model to study refraction:

            Incident Ray: An arrow pointing towards the boundary

            Point of incidence: Point where incident ray meets the boundary

            Refracted Ray: An arrow pointing away from the boundary in the other medium

            A straight line: showing the direction of the barrier

            Normal: A line drawn perpendicularly to the barrier at point of incidence

            Angle on incidence (θi \angle \theta_{i}): The angle between incident ray and NORMAL

            Angle of refraction (θr \angle \theta_{r}): The angle between refracted ray and NORMAL

            Waves in two dimensions


            • As waves move from one medium to another with different densities (shallow or deep), they get refracted, change direction at the boundary.
            • Shallow medium: Lower Velocity, Shorter Wavelength, Same Frequency
            • Deep medium: Higher Velocity, Longer Wavelength, Same Frequency

            Diffraction of waves

            • The process of diffraction is the spreading of waves through a small gap, waves circulating out of the gap.
            • Size of the gap affects the shape of the diffracted wave;


            Waves in two dimensions

            Interference of diffracted waves
            • Diffracted waves interfere with each other producing constructive and destructive interface.
            • The result of light interference through small gaps can be collected on a screen, light and dark fringes are seen on the screen. Points where waves interference constructively

            Waves in two dimensions