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Superposition overview

Topic 3 of 9

Measure string and microwave patterns

A stable standing-wave pattern turns node or minimum positions into a wavelength measurement. Identify what the apparatus detects and count the intervals between matching positions.

In the ideal pattern, neighbouring nodes are λ/2 apart. Repeated detector minima can locate the corresponding fixed pattern for microwaves. A span covering several intervals usually makes a position-reading uncertainty a smaller fraction of the total distance.

A driven stretched string

Stretch a string between effectively fixed supports. Drive it weakly near an end and vary the driving frequency until stable loops appear. The wave and its reflection overlap; stationary low-amplitude positions are displacement nodes.

The supports and driver have different roles. An attachment that moves appreciably is not an exact fixed node. In the ideal fixed-end model, the support separation L contains an integer n of half-wavelength loops:

L = nλ/2

Keep the support separation, tension and string unchanged when comparing the patterns at different frequencies. After selecting a steady pattern, locate nodes against a fixed ruler rather than reading a moving antinode as a fixed endpoint.

A reflected microwave pattern

Point a microwave transmitter towards a reflecting metal sheet. The incident and reflected waves overlap. Move a detector along the propagation direction, keeping its orientation fixed so that it samples the same electric-field component.

The detector shows repeated signal maxima and minima at fixed positions. It measures signal strength, not signed electric-field displacement. Its minima are not sound-pressure nodes. For the stationary pattern, successive minima or successive maxima are λ/2 apart.

Locate several corresponding nodes or minima

A weakly driven stretched string

The measured two-loop string has fixed end nodes and a near-end driverA string is held by two fixed supports at ruler positions 13.0 and 93.0 centimetres. Its two-loop standing profile has a middle displacement node at 53.0 centimetres. The ruler uses the same horizontal scale as the string, three drawing units per centimetre. Vertical displacement is exaggerated. A vibrator weakly drives a point just inside the left support, rather than claiming that an exactly fixed endpoint moves. A labelled cable connects the vibrator to a signal generator. Dashed node references meet the three ruler readings. The 80-centimetre span contains two adjacent-node intervals.13.053.093.0Ruler position / cmVibratorSignalgeneratorTwo loops, three nodesFixedsupportFixedsupport

A moving attachment is not an exact fixed node. The weak near-end drive approximates the fixed-boundary model. Subtract ruler readings before counting the two node intervals.

Microwaves overlap with their reflection

An aligned movable detector locates fixed microwave minimaA transmitter at left faces a metal reflector at right. Separate arrows indicate incident waves to the right and reflected waves to the left. A vertical receiving probe detects the same vertical electric-field component while it is translated along the propagation direction. Its cable leads to a signal meter. The ruler marks four supplied successive minima at 8.0, 9.5, 11.0 and 12.5 centimetres. Their 4.5-centimetre span contains three half-wavelength intervals. The drawn detector is at the 11.0-centimetre position. The meter is a signal-strength instrument, not a signed field-displacement display.IncidentReflectedScan; keep probe orientation fixedTransmitterReflectorProbe8.09.511.012.5Minimum positions / cmSignal meter

Keep the source, reflector and detector orientation fixed while finding positions. Real minima can remain above zero; a raw detector reading is not a signed electric-field displacement.

The string view identifies the fixed supports, weak driver and supplied node positions. The microwave view separates the transmitter, reflector and scanning detector. The supplied position readings determine separations; the detector's orientation stays fixed during the scan.

Estimate before calculating

String nodes separated by roughly 0.4 m suggest a wavelength of about 0.8 m. Microwave minima separated by about 1.5 cm suggest a wavelength of a few centimetres. The factor of two comes from the standing pattern.

Worked string readings

Three successive displacement nodes have supplied ruler positions 13.0, 53.0 and 93.0 cm. Their span contains two node intervals:

Span = 93.0 - 13.0 = 80.0 cm = 0.800 m
Node interval = 0.800/2 = 0.400 m
λ = 2 × 0.400 = 0.800 m

The ruler origin is not a node. Subtract the position readings before assigning a separation. The 0.80 m support separation contains two loops, so L = 2λ/2 = λ; it does not contain two complete wavelengths.

Worked microwave readings

Four successive detector minima are at 8.0, 9.5, 11.0 and 12.5 cm. There are three intervals:

Span = 12.5 - 8.0 = 4.5 cm = 0.045 m
λ = 2 × 0.045/3 = 0.030 m

The readings are supplied positions in a model experiment. An absolute field or intensity calibration is not needed to locate minima, but the position scale and the identity of successive minima must be reliable.

Keep the measurement conditions controlled

For the microwave scan, keep source frequency, reflector position and detector alignment fixed. Record position units and the ruler or position readout's resolution. Repeat the scan to judge broad minima; changing detector orientation can change its response without representing a new interference minimum.

Real minima may remain above zero because of background signals or unequal incident and reflected amplitudes. Repeated measurements can reveal variation and improve minimum location; they do not correct a wrong scale or an uncontrolled setup.

For either experiment, label the first and last matching positions and count the intervals between them. Use the full measured span in the wavelength calculation, rather than assuming the apparatus starts at the ruler's zero.