Fundamental Principles of Wave Physics and Interference

What is a Wave?

A wave is the transmission of energy via oscillations from one location to another without the net overall transfer of matter. Particles only oscillate about a fixed point; it is energy that is transmitted by the wave.

This energy transfer occurs because as each particle vibrates, it pushes its neighbor, transferring energy to it, and this process continues down the line. An individual particle's speed within the wave does not necessarily affect the speed of the wave itself, as the wave's speed is determined by the medium.

Transverse and Longitudinal Waves

• Transverse Waves: A wave in which the oscillations are perpendicular to the direction of wave travel and energy transmission.
• Longitudinal Waves: A wave in which the oscillations are parallel to the direction of wave travel and energy transmission, moving back and forth at a single point. Wavelength is measured between compressions.

Interference and Superposition

What is Interference?

Interference is the superposition of coherent waves where their displacements algebraically combine. This leads to regions of reinforcement (constructive interference) or cancellation (destructive interference), depending on their path difference.

What is Superposition?

Superposition is the addition of overlapping waves in the same medium. A limitation to note is that the medium determines the speed. As frequency remains constant when a wave moves between media, its wavelength changes. Variability would cause a continuous phase shift in waves.

Constant Interference Patterns

Superposition results in a constant interference pattern when using coherent sources. These are sources that emit waves with the same frequency (or wavelength) and maintain a constant phase difference.

Proof of Wave Model

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Wave Properties and Formulas

• Wavelength: The distance between two consecutive identical points on a wave.
• Amplitude: The maximum displacement of a particle from its equilibrium position.
• Period: The time taken for one complete oscillation of a wave or particle.
• Frequency: The number of complete oscillations or cycles per unit of time.

Wave Formulas: If measurements are in cm, multiply by 10^-2 to convert to meters.

Displacement-Distance Graphs

The amplitude and wavelength can be determined from these graphs. They are better for transverse waves as they displace particles in the same direction as the graph, whereas longitudinal waves displace particles in a different direction to the graph.

Path Difference and Interference

Path Difference

Path difference is the difference in distance traveled by two waves from coherent sources to a particular point. Note: The 3rd node means n = 2 because you start counting from 0.

Nodes

A node is a point where destructive interference consistently occurs.

• Amplitude is always zero.
• The waves meet completely out of phase (phase difference of 180°).
• Path difference at a node is an odd number of half-wavelengths.

Antinodes

An antinode is a point where constructive interference consistently occurs; the waves constantly reinforce each other.

• Point of maximum amplitude.
• The waves meet in phase.
• Path difference at an antinode is a whole number multiple of the wavelength.

Standing Waves

What is a Standing Wave?

Unlike progressive waves, a stationary wave does not travel and therefore has no speed. It does not transfer energy between two points. A standing wave consists of two traveling waves of the same frequency and amplitude traveling in opposite directions (180° inversion) superimposing to create stationary regions of maximum and minimum displacement.

How is a Standing Wave Formed?

A standing wave forms when a traveling wave reflects off a fixed end of a medium, inverting by 180 degrees, and superposes with an identical incoming wave. This interference creates a stationary pattern.

Nodes and Antinodes in Standing Waves

Nodes are stationary regions of minimum (or zero) displacement where destructive interference constantly occurs. For example, the fixed ends of a string must be nodes. Antinodes are stationary regions of maximum displacement where constructive interference occurs. A particle at an antinode moves between maximum displacement and zero.

Conditions for Standing Waves

Standing waves form in a medium with two fixed ends under these conditions:

• There is a node at each end.
• There is at least one antinode in the middle; the wavelength is double the length of the string for the fundamental frequency.
• The number of half-wavelengths that fit along the string must be an integer.
• The frequency is an integer multiple of the fundamental frequency.

A standing wave will form if n is an integer.

Diffraction

Diffraction is the spreading out of a wave as it passes through a narrow opening or around an obstacle.

Factors Influencing Diffraction

The extent of diffraction depends on the ratio of the wavelength (λ) to the width of the opening or obstacle (w):

• If λ/w ≈ 1, significant diffraction occurs.
• If λ/w ≳ 1, increasing the ratio results in a greater extent of diffraction.
• If λ/w ≪ 1, very little to no diffraction occurs.