Chapter 15: Stationary Waves

Stationary / Standing Waves
When two similar waves travel in opposite directions they can superpose to form stationary wave.  This usually happens when one wave is a reflection of the other. Stationary waves occur by resonance only at the natural frequencies of vibration of a medium.

Formation of stationary wave


Here,

At time t =0, the progressive waves travelling to the left and right are in phase. The waves combine constructively, giving amplitude twice that of each wave.

After a time equal to one quarter of a period (t = T/4). Consequently, the two waves are in antiphase. The waves combine destructively, giving zero displacement.

After time equal to one half of a period (t=T/2), the two waves are back in phase again and combine constructively.

After a time equal to three quarters of a period (t=3T/4), the waves are in antiphase.

After time equal to one whole period (t=T), the waves combine constructively.

The separation between two adjacent nodes or two adjacent antinodes = λ/2

Separation between adjacent node and antinode = λ/4

Wavelength and speed of sound

Harmonics

As we increase the frequency of the vibration generator we will see standing waves being set up.

The first will occur when the generator is vibrating at the fundamental frequency, f0, of the string.

First Harmonics

It has 2 nodes and 1 antinode.

f = f0            λ= 2L

Second Harmonic

It has 3 nodes and 2 antinodes.

f = 2f0        λ= L

Third Harmonic

It has 4 nodes and 3 antinodes.

f = 3f0      λ = 2/3 L

Fourth Harmonic

It has 5 nodes and 4 antinodes.

f = 4f0     λ = ½ L