Waves and superposition

If the relative phase changes unpredictably during observation, maxima and minima shift and the measured pattern can average out. Splitting light from one source provides a reliable way to preserve a phase relationship, whereas independent ordinary lamps do not normally form a stable visible pattern. For initially in-phase sources, use path difference to identify constructive and destructive locations. If the sources already have a nonzero phase difference, include that as well; a whole-wavelength path difference alone no longer guarantees a maximum. Amplitude cancellation refers to displacement or field, while intensity depends on its square.

Between adjacent nodes of a standing wave, points oscillate in phase but with different amplitudes. Crossing a node changes the phase by π. In a progressive sinusoidal wave, phase changes continuously with position, so a separation of half a wavelength gives antiphase. A standing wave can be understood as two equal waves travelling in opposite directions, which explains why energy transport cancels ideally. Real driven systems still require energy input to replace losses, so saying that no energy ever enters a standing-wave apparatus would be too strong.

The principal-maximum directions still satisfy d sin θ = nλ for normal incidence. Here d is the separation between neighbouring slits, not the width of an individual slit or the total grating width. Convert a line density into spacing by taking its reciprocal in consistent units. The finite width of each slit also produces a diffraction envelope that affects the brightness of different orders. A sharper peak helps wavelength measurement, but it does not eliminate angular reading uncertainty, alignment errors or an incorrectly stated line density.

Draw the complete loop before assigning a wavelength. A single loop between two adjacent nodes spans only half a wavelength, even though it may look like one complete hump. For a string fixed at both ends, the fundamental has one loop and wavelength twice the string length. The second harmonic has two loops and wavelength equal to the length. In an air column, identify whether the diagram represents displacement or pressure because their nodes and antinodes exchange roles. The boundary conditions determine which modes can fit; the same formula does not automatically apply to every pipe.

Phase is periodic, so phase differences differing by 2π describe the same relative stage of oscillation. For interference, first decide whether the sources begin in phase; any initial phase difference must be combined with the travel difference. In a time trace, use 2πΔt/T instead of a spatial path difference. The result may describe a phase lead or lag, which depends on propagation direction and which point is used as reference. Do not infer standing-wave phase from position using the progressive-wave rule without accounting for the opposing wave.

The distinction is useful when predicting how a pattern changes. Narrowing one slit generally broadens its diffraction pattern, while reducing the separation of two slits increases their interference-fringe spacing. These are different geometric quantities, even though a double-slit apparatus displays both effects at once. The broad single-slit envelope affects the brightness of the finer double-slit fringes. Strong spreading is particularly apparent when the opening size is comparable with wavelength. Avoid explaining diffraction merely as waves bending around corners without relating the amount of spreading to the relevant size ratio.