1. A.2 Standing (Stationary) Waves
A: Wave Phenomena
A.2 Standing (Stationary) Waves

2. Standing (Stationary) Waves
When two identical travelling waves pass in the opposite direction through a medium, a stationary wave may be produced. In this situation there is zero net transfer of energy through the medium.
Demo: Stationary waves on a rubber cord:
Vibration generator
Pulley

4. Formation of Standing Waves
Standing waves form as a result of constructive and destructive superposition of two waves moving in opposite directions with equal amplitude and frequency.
If at an instant when the waves overlap any points are in-phase, constructive superposition occurs.
If each wave moves ¼ of a cycle in opposite directions, the waves become ½ a cycle out of phase and destructive interference occurs.

5. Nodes and Antinodes
Points oscillating with the biggest amplitude in a stationary wave are called antinodes.
Points undergoing zero oscillation are called nodes.

6. For any stationary wave the node to node (loop) distance = ½ λ
Harmonic Frequencies
For any stationary wave the node to node (loop) distance = ½ λ
So for waves on a string, a stationary wave will only occur when the length is a whole number of half wavelengths. Thus, because the speed of the wave in the string is fixed (depending on the string and its tension), a stationary wave pattern can only be formed at certain frequencies – ‘harmonics’—thank you resonance.

7. λ = 2L
Frequency = f
λ = L
Frequency = 2f
λ = 2/3 L
Frequency = 3f

8. Harmonic frequency = no. of loops x natural frequency
The lowest resonant frequency of the system is called the fundamental frequency (or 1st harmonic frequency). The next are called the 2nd harmonic, 3rd harmonic etc.
For a stationary wave in a string:
Harmonic frequency = no. of loops x natural frequency

9. Progressive Waves Standing Waves
Comparing Standing Waves to Progressive Waves
Progressive Waves
Standing Waves
Wave profile progresses (moves)
Wave profile doesn’t progress (stays still)
All points oscillate with the same amplitude
Amplitude varies from zero (node) to maximum (antinode)
All points are out of phase with each other
All points between any two nodes oscillate in phase with each other

10. Standing Waves in Open Pipes
Wind instruments often use different columns of air to produced different frequencies.
Open (at both ends) – Similar to the math with strings.
Must have antinodes at both ends.

11. Standing Waves in Closed Pipes
The second type of pipe is a closed pipe—however, one end is closed and the other is open (air has to enter somehow).
Nodes are at the closed end.
Antinodes are at the open end.
Notice the harmonics!

12. Click Here for a Standing Wave Simulation
Click HERE for a Standing Waves Visual Simulation
Click Here for a Standing Wave Simulation
Notice the top ones are similar to closed pipes
The middle ones are similar to open pipes
The bottom ones are for strings.

13. Question
Find the fundamental frequency of an open pipe of length 85 cm. The speed of sound in air is 340 m/s.