show an understanding of experiments that demonstrate stationary waves using microwaves, stretched strings and air columns (it will be assumed that end corrections are negligible; knowledge of the concept of end corrections is not required)

Stationary (Standing) Waves

A stationary wave is formed when two waves of the same frequency travel in opposite directions and interfere

constructively and destructively at fixed points. The result is a pattern of nodes (points of zero displacement)

and antinodes (points of maximum displacement) that does not travel along the medium.

The general condition for a stationary wave of wavelength λ in a medium of length L is

\$\$

L = n\frac{\lambda}{2}\qquad (n = 1,2,3,\dots)

\$\$

where n is the harmonic number. The wave speed v is related to the frequency f and wavelength by

\$\$

v = f\lambda .

\$\$

Key Features

• Nodes: points of zero amplitude.
• Antinodes: points of maximum amplitude.
• Distance between successive nodes (or antinodes) = λ/2.
• Only certain discrete frequencies (harmonics) satisfy the boundary conditions.

Experiments Demonstrating Stationary Waves

1. Microwaves in a Waveguide

Microwaves (electromagnetic waves) can be reflected from a metal plate placed inside a rectangular waveguide.

The incident and reflected waves interfere to produce a standing‑wave pattern that can be detected with a

microwave detector.

1. Set up a microwave transmitter at one end of a straight waveguide.
2. Place a movable metal plate at the opposite end; the plate acts as a reflector.
3. Insert a small detector probe that can be moved along the length of the waveguide.
4. Record the detector reading as a function of position.

The detector shows alternating maxima and minima corresponding to antinodes and nodes. The distance

between successive minima (or maxima) is λ/2, allowing the wavelength and thus the frequency to be

determined using the known speed of light c.

2. Stretched String Fixed at Both Ends

A string under tension T and linear mass density μ supports transverse standing waves when

driven at one of its resonant frequencies.

For a string of length L fixed at both ends, the allowed wavelengths are

\$\$

\lambda_n = \frac{2L}{n}\qquad (n = 1,2,3,\dots)

\$\$

and the corresponding frequencies are

\$\$

f_n = \frac{n}{2L}\sqrt{\frac{T}{\mu}} .

\$\$

Experimental procedure:

1. Clamp a string of known length L and measure its mass per unit length μ.
2. Apply a known tension T using a hanging mass.
3. Drive the string with a mechanical oscillator (e.g., a violin bow or a speaker) and gradually

   increase the driving frequency.

4. Observe the formation of stationary patterns (nodes and antinodes) using a strobe light or

   by feeling the vibration.

5. Record the frequencies at which clear standing‑wave patterns appear; these correspond to the

   harmonics n.

Because the ends are fixed, both are nodes. The distance between adjacent nodes is λ/2.

3. Air Columns (Acoustic Resonance)

Air columns in tubes can support longitudinal standing waves. Two common configurations are:

• Both ends open (open–open tube).
• One end closed, one end open (closed–open tube).

When end corrections are negligible, the allowed wavelengths are:

Here v is the speed of sound in air (≈ 340 m s⁻¹ at room temperature).

Experimental setup (e.g., using a Kundt’s tube or a resonance tube):

1. Place a speaker at one end of the tube to generate a continuous tone.
2. Vary the frequency of the tone using a signal generator.
3. Detect resonance by observing a pronounced increase in sound intensity (or by using a

   microphone connected to an oscilloscope).

4. Measure the length L of the air column and note the frequencies at which resonance occurs.
5. Compare the measured frequencies with the theoretical values from the table above.

Comparative Summary of the Three Experiments

Common Points and Conceptual Understanding

• All three systems exhibit nodes and antinodes that are fixed in space.
• The distance between adjacent nodes (or antinodes) is always half a wavelength.
• Only discrete frequencies satisfy the boundary conditions; these are the resonant frequencies.
• Neglecting end corrections simplifies the analysis: the physical length of the medium is taken as the

  exact distance between the reflecting boundaries.

Typical Examination Questions

1. Derive the expression for the fundamental frequency of a string fixed at both ends, given its tension

   T, length L, and linear mass density μ.

2. A microwave source of frequency 10 GHz is used in a rectangular waveguide. The distance between two

   successive minima measured with a detector is 1.5 cm. Calculate the speed of the microwaves in the

   guide and comment on the result.

3. In a closed–open tube of length 0.25 m, the second resonance is observed at 1020 Hz. Assuming the speed

   of sound is 340 m s⁻¹, determine whether the observed resonance corresponds to the expected harmonic

   and calculate the end correction that would be required if the measured value differed.

Conclusion

Stationary waves provide a clear illustration of wave interference and the quantisation of allowed

frequencies in bounded media. By performing the three classic experiments—microwave standing waves,

vibrating strings, and acoustic resonance in air columns—students can directly observe nodes and antinodes,

measure wavelengths, and verify the theoretical relationships that underpin the concept of standing

waves in the Cambridge A‑Level Physics syllabus.