8.1.2 Stationary Waves

• Stationary waves, or standing waves, are produced by the superposition of two waves of the same frequency and amplitude travelling in opposite directions
• This is usually achieved by a travelling wave and its reflection. The superposition produces a wave pattern where the peaks and troughs do not move

Formation of a stationary wave on a stretched spring fixed at one end

Stretched strings

• Vibrations caused by stationary waves on a stretched string produce sound
  • This is how stringed instruments, such as guitars or violins, work

• This can be demonstrated by a length of string under tension fixed at one end and vibrations made by an oscillator:

Stationary wave on a stretched string

• As the frequency of the oscillator changes, standing waves with different numbers of minima (nodes) and maxima (antinodes) form

Microwaves

• A microwave source is placed in line with a reflecting plate and a small detector between the two
• The reflector can be moved to and from the source to vary the stationary wave pattern formed
• By moving the detector, it can pick up the minima (nodes) and maxima (antinodes) of the stationary wave pattern

Using microwaves to demonstrate stationary waves

Air Columns

• The formation of stationary waves inside an air column can be produced by sound waves
  • This is how musical instruments, such as clarinets and organs, work

• This can be demonstrated by placing a fine powder inside the air column and a loudspeaker at the open end
• At certain frequencies, the powder forms evenly spaced heaps along the tube, showing where there is zero disturbance as a result of the nodes of the stationary wave

Stationary wave in an air column

• In order to produce a stationary wave, there must be a minima (node) at one end and a maxima (antinode) at the end with the loudspeaker

• A stationary wave is made up of nodes and antinodes
  • Nodes are where there is no vibration
  • Antinodes are where the vibrations are at their maximum amplitude

• The nodes and antinodes do not move along the string. Nodes are fixed and antinodes only move in the vertical direction
• Between nodes, all points on the stationary are in phase
• The image below shows the nodes and antinodes on a snapshot of a stationary wave at a point in time

• L is the length of the string
• 1 wavelength λ is only a portion of the length of the string

ANSWER: C

• Stationary waves have different wave patterns depending on the frequency of the vibration and the situation in which they are created

Two fixed ends

• When a stationary wave, such as a vibrating string, is fixed at both ends, the simplest wave pattern is a single loop made up of two nodes and an antinode
• This is called the fundamental mode of vibration or the first harmonic
• The particular frequencies (i.e. resonant frequencies) of standing waves possible in the string depend on its length L and its speed v
• As you increase the frequency, the higher harmonics begin to appear
• The frequencies can be calculated from the string length and wave equation

Diagram showing the first three modes of vibration of a stretched string with corresponding frequencies

• The nth harmonic has n antinodes and n + 1 nodes

One or two open ends in air column

• When a stationary wave is formed in an air column with one or two open ends, we see slightly different wave patterns in each

Diagram showing modes of vibration in pipes with one end closed and the other open or both ends open

• Image 1 shows stationary waves in a column which is closed at one end
  • At the closed end, a node forms
  • At the open end, an antinode forms

• Therefore, the fundamental mode is made up of a quarter wavelength with one node and one antinode
  • Every harmonic after that adds on an extra node or antinode
  • Hence, only odd harmonics form

• Image 2 shows stationary waves in a column which is open at both ends
  • An antinode forms at each open end

• Therefore, the fundamental mode is made up of a half wavelength with one node and two antinodes
  • Every harmonic after that adds on an extra node and an antinode
  • Hence, odd and even harmonics can form

• In summary, a column length L for a wave with wavelength λ and resonant frequency f for stationary waves to appear is as follows:

Air Column Length & Frequencies Summary Table

Step 1: Determine the positions of the nodes and antinodes

• One end of the column is closed, and the loudspeaker represents an open end
• Hence, an antinode forms at the loudspeaker (open end) and a node forms at the closed end
• The fundamental frequency represents the lowest note - this would be 1 node and 1 antinode
• So, the second lowest note must have 2 nodes and 2 antinodes

Step 2: Write an expression for the length of the sound wave in the column

• In the column, there is a quarter wavelength and a half wavelength, or 
• Therefore the length of the column is:

• Note: for a column with an open and closed end, , this would represent the third harmonic (n = 3)

Step 3: Determine the wavelength of the second lowest note

Step 4: Calculate the frequency using the wave equation

• Note: you could combine steps 3 and 4 by using the expression