### Episode 324: Stationary or standing waves

Most musical instruments depend upon standing waves, as does the operation of a laser. In a sense, a diffraction pattern is a standing wave pattern.

 SummaryDemonstration: Setting up waves on a rope. (15 minutes)Discussion: How superposition results in standing waves. (20 minutes)Demonstration: Melde’s experiment. (20 minutes)Discussion: Stringed instruments. (10 minutes)Demonstration: Standing sound waves. (20 minutes)Student experiments: Measuring λ and c. (30 minutes)Demonstration: Measuring c using a microwave oven. (15 minutes)Student questions: On Melde’s experiment, and on waves in pipes. (30 minutes)Demonstration: Standing waves in 2 and 3 dimensions. (20 minutes)

Demonstration: Setting up waves on a rope
Standing waves do not travel from point to point. They are formed from the superposition of two identical waves travelling in opposite directions. This is most easily achieved by reflecting a travelling wave back upon itself.

Reflection from a denser medium gives a phase change of 180° (= π rad) for the reflected wave. Show this using a rope or stick wave machine held between two people– pull down and let go to make pulse. See that there is a phase change if reflecting from a fixed end, but no effect if reflecting off a less dense medium (e.g. a free end).

The incoming reflected wave superposes with the outgoing wave. The result (if you vibrate the one end of the rope at the correct frequency) is a standing wave. The mid-point of the rope vibrates up and down with a large amplitude – this is an antinode. Other points vibrate with smaller amplitude.

Double the frequency of vibration and you get two antinodes, with a node in between. The amplitude at a node is zero.

Discussion: How superposition results in standing waves
Discuss how two travelling waves superpose to give a standing wave. A node (where there is NO DisplacEment) is a special point, where a positive displacement from one wave is always cancelled by an equal, negative displacement from the other wave.

TAP 324-1: Standing waves

Demonstration: Melde’s experiment

A stretched string or rubber cord can be made to vibrate using a vibrator (of the type illustrated) connected to a signal generator. This is known as Melde’s experiment.

Show that different numbers of ‘loops’ can be formed; identify the pattern of frequencies (e.g. one loop at 40 Hz; two loops at 80 Hz; three at 120 Hz, etc.)

Point out that each point on the string oscillates with simple harmonic motion – if the frequency is high you just see the envelope. Freeze the wave using a stroboscope. To do this, place an electronic strobe between the students and the string, so that it illuminates the string. Adjust the frequency of the strobe until the string appears stationary. (Safety precaution: warn about the flashing light in case you have any students present who may suffer from photo-induced epilepsy.)

The distance between successive nodes (or antinodes) is λ / 2. Deduce the wavelength.

Notice that only certain values of wavelength are possible. If the string has length L, the fundamental has λ / 2 = L, first harmonic λ / L, second harmonic 3 λ / 2 = L and so on. The allowed wavelengths are quantised.

(This experiment can be extended to look at how the pattern changes with length and tension of the string.)

TAP 324-2: Standing waves on a rubber cord

Discussion: Stringed instruments
Relate what you have observed to the way in which stringed instruments work. (Wind instruments are covered later.)

TAP 324-3: Standing waves on a guitar

TAP 324-4: What factors affect the note produced by a string?

Demonstration: Standing sound waves
You can observe the same effect with sound waves. To generate two identical waves travelling in opposite directions, reflect one wave off a hard board.

TAP 324-5: Standing waves in sound

Student experiments: Measuring λ and c
Standing waves make it easy to determine wavelengths (twice the distance between adjacent nodes), and hence wave speed c (since c = f λ ). These experiments allow students to investigate wavelength and speed for sound waves, microwaves and radio waves.

Have a short plenary session for the different groups to report to the whole class.

TAP 324-6: Kundt's experiment

TAP 324-7: Standing waves with microwaves

TAP 324-8: A stationary 1 GHz wave pattern

Demonstration: Measuring c using a microwave oven
Use a microwave oven to measure the speed of light. This makes a memorable demo if you have a microwave oven to hand in the lab. Without using the turntable, place marsh mallows or slices of processed cheese in the oven. Observe that the heating occurs in definite places – the cooking starts first at the anti-nodes, where the standing waves have the maximum amplitude. The purpose of the turntable it to even out this localised heating. Measure the distance between nodes, deduce λ , and multiply by f to find c.

NB It is a widely repeated misconception that microwave heating is a resonance effect, i.e. that the frequency of the microwaves is chosen to be one of the vibration frequencies of the water molecule. It is not. Water molecules resonate at rather higher frequencies than the 2.5 GHz of microwave ovens (22 GHz is one such frequency for free water molecules). The frequency used in ovens is a compromise between too low a value, when no microwaves would be absorbed by the food at all, and a frequency too close to the resonant frequency, when the microwaves would all be absorbed in the outside layer, instead of penetrating a few cm. Microwaves are attenuated exponentially by foodstuffs with a half-depth of about 12 mm.

Student questions: On Melde’s experiment, and on waves in pipes
Some questions based on Melde’s vibrating string experiment.

TAP 324-9: Stationary waves in a string

 Questions on standing waves in air in a pipe; note that you will have to explain that there is a node at the closed end of a pipe, and an antinode at an open end. (The air molecules cannot vibrate at a closed end.)TAP 324-10: Standing waves in pipesTAP 324-11: Standing waves in pipes questionsDemonstrations: Standing waves in 2 and 3 dimensionsYou can demonstrate standing waves in two and three dimensions. Mount a wire loop or a metal plate (a Chladni plate) on top of a vibrator.A wire loop shows a sequence of nodes and antinodes around its length, and is an analogue for electron standing waves in an atom.

TAP 324-12: More complicated standing waves

Chladni plate: Fine sand sprinkled on the plate gathers at the nodes.

Jelly (make up a large cubical shape) on a vibrating plate can look fantastic.

Rubber sheet stretched over a loud speaker: the low frequency standing waves are clearly visible.