(b) graphical methods to illustrate the principle of superposition

The idea

Graphical methods show superposition by plotting displacement against position or time for each wave separately, then adding the displacements point by point to obtain the resultant graph. For two waves of equal amplitude and wavelength, plotting them on the same axes makes it clear where crests align to give a doubled displacement and where a crest meets a trough to give zero. For waves of different amplitudes, the resultant graph still follows from adding ordinates at each position. The method works for any wave shapes, including pulses, and helps predict the resultant displacement without needing a formula. Care is needed to add displacements with their signs, since displacement above the axis is positive and below is negative.

What earns the marks

• Plots each wave separately as displacement against position or time on the same axes.
• Adds the displacements of the individual waves at each point to obtain the resultant displacement.
• Shows constructive superposition where displacements have the same sign, giving a larger resultant.
• Shows destructive superposition where displacements have opposite signs, giving a smaller or zero resultant.
• Applies the method to waves of unequal amplitude or different shape, not only identical sine waves.

Common mistakes and how to put them right

• Adding amplitudes without regard to sign.

  Put it right: Add displacements algebraically, treating displacements below the axis as negative.

• Plotting the resultant first and trying to infer the components.

  Put it right: Plot each component wave first, then combine them point by point.

• Assuming the resultant must be a sine wave of the same frequency.

  Put it right: The resultant shape depends on the component waves and may be complex.

• Reading the graph at only one point and generalising.

  Put it right: The resultant displacement varies along the graph, so combine at many points.

Exam tips

• Use the same scale and axes for both component waves so ordinates can be added directly.
• Mark a few key points, such as crests and troughs, and add ordinates there before sketching the full resultant.
• Label positive and negative displacement clearly to avoid sign errors.