Wave Behaviour (HSC SSCE Physics): Revision Notes

Wave Superposition

What is superposition?

When waves of similar type (both transverse or both longitudinal) travel through the same medium, they can pass through each other. During this interaction, something remarkable happens: the displacement values from each wave combine mathematically at every point where they overlap. After passing through each other, the waves continue travelling with their initial characteristics completely unchanged. This phenomenon is called superposition.

The mathematical combination typically creates a complex pattern. At any given point, the interaction may produce either constructive interference (resulting in larger amplitude) or destructive interference (resulting in reduced amplitude), depending on how the two waves align.

Constructive interference

Constructive interference occurs when two waves align perfectly so that their crests coincide with each other and their troughs also coincide. When this happens, the waves reinforce each other.

When two identical waves meet with this perfect alignment, they create a resultant wave with twice the amplitude of either original wave. The wave appears to grow larger because the displacements from both waves add together positively.

Destructive interference

Destructive interference occurs when two waves meet with opposite alignment—a crest from one wave coinciding with a trough from the other wave. In this situation, the waves' displacements work against each other.

When two identical waves arrive at the same location completely out of phase (crest meeting trough), their displacements cancel completely. The result is zero displacement—the waves effectively disappear at that point! This complete cancellation demonstrates the remarkable nature of wave superposition.

Waves with different wavelengths

When two waves with different wavelengths overlap, the pattern becomes more interesting. The resultant wave shows alternating regions where constructive and destructive interference occur.

In the diagram above, you can see three waves:

• The blue wave (higher frequency, shorter wavelength)
• The green wave (lower frequency, longer wavelength)
• The red wave (the resultant of adding the blue and green waves)

Real-world application: acoustics and the Sydney Opera House

The principle of superposition is crucial in acoustics—the study of sound behaviour. Acoustic engineers use their understanding of wave superposition to design buildings with excellent sound qualities for concerts, speeches, and theatrical performances.

The Sydney Opera House Concert Hall provides a fascinating case study. When the Concert Hall first opened in the 1970s, the acoustics were poor—the orchestra sounded muffled and lacked clarity. The problem was destructive interference: sound waves reflected off various surfaces were arriving out of phase with each other, causing cancellation that reduced the sound quality.

Investigation 8.4: Destructive and constructive interference in the classroom

Aim: To experience how destructive and constructive interference can occur with sound in a classroom

Materials:

• Two signal generators (smartphones using tone generator apps with suitable speakers work well)
• Optional: smartphone app that measures sound level intensity

Method:

1. Position the two speakers at least $3 \text{ m}$ apart at the front of the room. Set both to play the same sine wave sound at exactly the same frequency (approximately $600 \text{ Hz}$ works well).
2. Start at a position equidistant from both speakers. Move slowly sideways across the room, listening carefully for changes in the sound loudness.
3. Measure the distance you need to move between two positions where constructive interference occurs (where the sound is loudest). A sound level intensity app can help identify these positions precisely.

Results:

Record your observations about how the loudness varies as you move. Create a scale map of the classroom showing where you observed constructive and destructive interference along your path.

Analysis of results:

Investigation 8.5: The principle of superposition

Aim: To demonstrate the principle of superposition using electronic equipment

Materials:

• Cathode ray oscilloscope (CRO) with two inputs
• Two microphones
• Signal generators or sound sources (musical instruments like guitars also work)
• Device to record video (camera, phone, laptop or tablet)

Safety considerations:

Risk	Management
Electrical equipment near water	Keep all devices plugged into $240 \text{ V}$ mains power well away from water

Method:

1. Connect both microphones to separate inputs on the CRO. Set up two sound sources (signal generators or musical instruments) producing sounds at slightly different frequencies.
2. Adjust the CRO's vertical scale so that each microphone's signal has approximately the same size when displayed separately.
3. Change the CRO display mode to 'add'. This makes the oscilloscope add the signals from both microphones to display one combined waveform.
4. Observe what happens both on the screen and with your ears. You should see and hear alternating periods of constructive and destructive interference.

Results:

Record your observations by taking video of the changing CRO display or by capturing snapshots at different times. Notice how the waveform changes shape and amplitude over time.

Investigation 8.6: Constructing a resultant wave using the principle of superposition

Aim: To apply the principle of superposition to produce a resultant waveform from two component waves

Materials:

• Grid paper (preferably $10 \text{ mm} \times 10 \text{ mm}$ grids)
• Pencil
• Ruler

Method:

1. Draw a graph of wave A starting at the origin with wavelength $\lambda = 8 \text{ cm}$ and amplitude $A = 3 \text{ cm}$.
2. On the same axes, draw wave B starting at the origin with $\lambda = 12 \text{ cm}$ and $A = 2 \text{ cm}$.
3. At each vertical grid line, measure the displacement of wave A and wave B from the horizontal axis. Add these two values together and plot the sum.
4. Connect your plotted points with a smooth curved dashed line. This is the resultant wave (A + B).

Results:

The curved dashed line represents the resultant wave formed by superposition of waves A and B.

Analysis of results:

1. Measure the amplitude of the resultant wave A + B. Notice that it varies along the wave—it's not constant like the original waves.
2. Identify and label regions where:
   • Constructive interference occurred (where the resultant amplitude is largest)
   • Destructive interference occurred (where the resultant amplitude is smallest)

Musical instruments and characteristic sound

Every musical instrument produces a unique sound, even when playing the same note as another instrument. For example, a saxophone playing middle C sounds noticeably different from a clarinet playing middle C, despite both instruments using a vibrating reed to generate sound.

This distinctive quality, called timbre, results from superposition. Each instrument doesn't produce just one pure frequency. Instead, it generates multiple sound waves at different frequencies simultaneously. These component waves superpose to create the complex resultant wave that we hear. The unique combination of frequencies and their relative amplitudes gives each instrument its characteristic sound.

Key Takeaways