Wave Superposition, Interference, Phase Difference and Coherence

This topic is part of the HSC Physics course under the section Wave Behaviour.

HSC Physics Syllabus

• explain the behaviour of waves in a variety of situations by investigating the phenomena of:

– reflection
– refraction
– diffraction
– wave superposition (ACSPH071, ACSPH072)

What is Wave Superposition?

Wave superposition is the phenomenon where two or more waves combine to form a resultant wave. The resultant wave's amplitude at any point is the sum of the amplitudes of the individual waves at that point.

Constructive interference occurs when waves meet in phase, their amplitudes add together, creating a wave with a larger amplitude.

Destructive interference occurs when waves meet out of phase, their amplitudes cancel out, creating a wave with a smaller or even zero amplitude.

It is important to beware and understand that waves often undergo a combination of constructive and destructive interference.

In-Phase and Out-of-Phase Explained

Understanding the concepts of in-phase and out-of-phase is vital to grasp the phenomenon of wave superposition. These terms describe the relative positions of the crests and troughs of waves as they interact.

In-Phase

Two or more waves are said to be in-phase when their crests and troughs align perfectly. In other words, the waves reach their maximum and minimum points simultaneously. When in-phase waves meet, they undergo constructive interference, where their amplitudes add together. This creates a resultant wave with a larger amplitude.

The resultant wave of constructive interference has an increased amplitude.

Waves are in-phase when their phase difference is an integral multiple of wavelengths.

This is commonly expressed as:

$$\text{path difference} = n \lambda \, \text{where n = 1, 2, 3...}$$

Example: If two identical sound waves are in-phase, they will combine to produce a sound wave that is louder at the points where they meet.

Out-of-Phase

Out-of-phase waves are waves where the crests of one wave align with the troughs of another. They are essentially “out of step” with each other. When out-of-phase waves meet, they undergo destructive interference, where their amplitudes cancel each other out. This can result in a resultant wave with a smaller amplitude or even complete cancellation if the waves are perfectly out of phase.

The resultant wave of destructive interference has a reduced amplitude.

When two waves of the same amplitude and wavelength are of out of phase, the resultant wave has zero amplitude.

Waves are out of phase when their phase difference is 0.5, 1.5, 2.5... wavelengths. This is commonly expressed as:

$$\text{path difference} = (n + \frac{1}{2}) \lambda \, \text{where n = 0, 1, 2, 3...}$$

Example: Noise-cancelling headphones utilise the principle of out-of-phase waves. They produce sound waves that are out-of-phase with ambient noise, thereby cancelling it out.

What are Coherent Waves?

Two or more waves are described as coherent when they have the same frequency and their phase differences are constant. Incoherent waves have different frequency and/or variable phase differences.

While both coherent and incoherent can undergo wave superposition i.e. constructive and destructive interference, only coherent waves produce a fixed interference pattern. The examples of interference shown above are all between coherent waves (same frequency and constant phase difference if any).

In the example above, the red and blue waves are coherent as they have the same frequency, and separated by a constant phase difference of 1/4 \lambda. This can be clearly identified by looking at the difference between the waves' crests, and their troughs – both are 2 units apart. The resultant wave (black) also has the same frequency.

In the example above, the red and blue waves are incoherent as they have different frequencies, and separated by a variable phase difference. At one instance, the crests of both waves meet, undergoing constructive interference, but at another instance, the crest of the blue wave meets the trough of the red wave, undergoing destructive interference. As a result, the resultant wave (black) has a variable interference pattern (as seen by the changing amplitude).

Examples of Coherent Waves:

1. Lasers: Perhaps the most well-known source of coherent light. Lasers emit light waves with a consistent phase relationship over long distances, making them ideal for various applications like precision measurements, optical data transmission, and medical procedures.

2. Radio Antennae in Arrays: When radio antennae are synchronised to emit signals at the same frequency and phase, they produce coherent radio waves. This principle is used in phased array radar systems and radio astronomy.

Examples of Incoherent Waves:

1. Incandescent Bulbs: These bulbs emit light due to the heating of a filament. The emitted light consists of a wide range of frequencies and random phases, making it incoherent.

2. Sunlight: While sunlight contains a broad spectrum of wavelengths (visible light, ultraviolet light, and infrared), the waves from the sun are incoherent due to the myriad of atomic transitions happening in the sun.

3. Random Noise in Electronic Devices: Thermal noise or "white noise" in electronic circuits is inherently incoherent, originating from the random motion of electrons.

4. Sound from Uncoordinated Sources: If you were in a busy marketplace or a crowded room, the cacophony of individual, uncoordinated voices and noises produces incoherent sound waves.