Standing Waves (HSC SSCE Physics): Revision Notes

Standing Waves

What are standing waves?

Standing waves are wave patterns that appear to remain stationary in one location rather than travelling through a medium. While the pattern stays fixed, the particles within the medium continue to vibrate in their normal transverse or longitudinal motion. The wave pattern itself, however, does not move along the string, spring, or air column.

When you create waves in a string that is fixed at one end, the forward-moving waves and the reflected waves interfere with each other through superposition. Most of the time, this interference creates a complex, chaotic pattern. However, at specific frequencies, something special happens - a stable pattern of maximum and minimum displacements forms. This is a standing wave.

How standing waves form

A standing wave forms when two conditions are met:

• Two waves with identical frequency and amplitude travel in opposite directions through the same medium
• The length of the string, spring, or air column has a specific mathematical relationship with the wavelength

When these conditions are satisfied, the waves interfere with each other in a way that creates fixed points of zero displacement (nodes) and fixed points of maximum displacement (antinodes).

Nodes - points of zero displacement

Nodes are locations along the medium where destructive interference occurs continuously. At a node, the displacement is always zero, regardless of what time you observe it.

This happens because at any moment, the displacement caused by one wave is exactly equal in magnitude but opposite in direction to the displacement caused by the other wave.

The result is that particles located at nodes remain perfectly still, even though waves are continuously travelling through those points. This might seem strange, but it's a direct consequence of destructive interference.

Antinodes - points of maximum displacement

Antinodes are locations where particles oscillate with the maximum possible amplitude. At these points, constructive interference occurs continuously.

The maximum displacement happens when:

• Two crests meet at the same point simultaneously, or
• Two troughs meet at the same point simultaneously

When this occurs, the displacements add together, giving a total displacement equal to the sum of the individual wave amplitudes (which are equal, so the antinode displacement is twice the amplitude of each individual wave).

Key characteristics of standing waves

Standing wave patterns always show an alternating pattern of nodes and antinodes along the medium. This pattern has specific mathematical properties:

• The distance between two consecutive nodes equals $\frac{\lambda}{2}$ (half a wavelength)
• The distance between two consecutive antinodes also equals $\frac{\lambda}{2}$
• The distance from a node to the nearest antinode equals $\frac{\lambda}{4}$ (quarter wavelength)

The solid line in the diagram represents the string's position at one instant in time, while the dotted line shows its position half a period ($\frac{T}{2}$) later. Notice how nodes remain at zero displacement while antinodes reach their maximum displacement in the opposite direction.

Investigation: Using standing waves to calculate wave speed in a spring

This practical investigation demonstrates how to use standing wave patterns to determine the speed of waves travelling through a spring.

Aim

To observe how stretching a spring affects wave speed by measuring standing wave properties.

Hypothesis

Stretching a spring further increases the speed of waves travelling through the spring.

Materials

• Slinky springs
• Metre ruler
• Stopwatch
• Optional: digital camera or recording device

Method overview

The investigation involves creating standing waves in a slinky spring and measuring key properties:

• Extend the slinky along the ground with one end fixed and create travelling waves by moving the free end side to side
• Adjust the movement frequency until standing waves form (nodes and antinodes become visible and stationary)
• Time 10 complete side-to-side movements to determine the period
• Measure the distance between two consecutive nodes while maintaining the standing wave
• Record the investigation with video if possible
• Repeat the entire process with the spring stretched to a different length

Analysis of results

From your measurements, calculate the wave properties:

Period: Calculate the period $T$ by dividing the time for 10 movements by 10.

Frequency: Use the relationship $f = \frac{1}{T}$ to find the wave frequency.

Wavelength: Since the distance between consecutive nodes equals $\frac{\lambda}{2}$, the wavelength is:

$\lambda = 2 \times \text{(distance between nodes)}$

Wave speed: Finally, calculate the wave speed using the wave equation:

$v = f\lambda$

Repeat these calculations for both spring lengths to compare how stretching affects wave speed.

Remember!