Stationary waves in the lab: strings, microwaves and air columns

The main symbols used in this lesson:

A stationary wave needs two identical waves travelling in opposite directions. In practice you make the second wave by reflecting the first.

• On a string, the wave reflects at the fixed end (or at the pulley).
• For microwaves, a metal plate reflects the beam straight back.
• In a tube, the sound reflects at the water surface.

In each case the incident and reflected waves have the same frequency and travel in opposite directions, so they superpose into a stationary pattern.

A string is stretched horizontally over a pulley, with masses hanging on the end to provide tension. A vibration generator (or a tuning fork) vibrates the other end.

Adjust the length, the tension or the driving frequency until the string settles into a clear pattern of loops. Each loop is one half-wavelength.

What you see: nodes at the two fixed ends and between the loops; an antinode in the middle of every loop. The string in each loop swings while the nodes stay still.

What you measure: the length $L$ between the fixed ends and the number of loops $n$.

$L = n\frac{\lambda}{2} \quad\Rightarrow\quad \lambda = \frac{2L}{n}$

The wave speed follows from $v = f\lambda$, with $f$ the driving frequency.

A microwave transmitter sends a beam towards a metal plate, which reflects it straight back. The incident and reflected waves superpose in the space between, forming a stationary wave.

A small probe (a diode detector) is moved along the line between transmitter and plate. Its reading rises and falls.

• Maximum reading: the probe is at an antinode.
• Minimum reading: the probe is at a node.

Consecutive maxima are separated by $\dfrac{\lambda}{2}$, so

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

The metal plate is a node (a fixed boundary), and the pattern is fixed in space — which is exactly why the probe reading depends on position.

A tube is partly filled with water, so the water surface seals the bottom end. A loudspeaker or tuning fork of known frequency sounds above the open top.

Raise or lower the water level to change the length $L$ of the air column. At certain lengths the note suddenly becomes much louder: the air column is in resonance.

Where the nodes are: the water surface is a fixed boundary, so there is a node there. The open top is free to move, so there is an antinode there.

That means the shortest resonating column is a quarter of a wavelength:

$L_1 = \frac{\lambda}{4}$

The next resonance adds another half-wavelength:

$L_2 = \frac{3\lambda}{4}$

So $L_2 - L_1 = \dfrac{\lambda}{2}$, giving $\lambda = 2(L_2 - L_1)$.

In this course, end corrections are negligible: treat the antinode as exactly at the open end.