A: Wave PhenomenaA.2 Standing (Stationary) Waves

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

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.

Nodes and Antinodes

Points oscillating with the biggest amplitude in a stationary wave are called antinodes.

Points undergoing zero oscillation are called nodes.

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.

λ = 2L

Frequency = f

λ = L

Frequency = 2f

λ = 2/3 L

Frequency = 3f

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

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

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 withstrings.

Must have antinodes at both ends.

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!

Click Here for a Standing Wave Simulation

Notice the top ones are similar to closed pipesThe middle ones are similar to open pipesThe bottom ones are for strings.

Click HERE for a Standing Waves Visual Simulation

Question

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