2l μ The formation of stationary waves by two waves of the same frequency…

The idea

A stationary wave forms when two progressive waves of the same frequency and similar amplitude travel in opposite directions through the same region and superpose. On a stretched string, a wave travelling towards a fixed end reflects and travels back, so the incident and reflected waves meet. Where the waves are in phase they reinforce to give an antinode; where they are in antiphase they cancel to give a node. The pattern does not travel along the string: the nodes and antinodes stay in fixed positions, and energy is stored rather than transferred. The frequency is the same for both waves, so the pattern has that frequency, but the distance between adjacent nodes is half the wavelength. The speed of transverse waves on a string is given by v = √(T/μ), where T is the tension and μ is the mass per unit length, so the frequency of a given mode depends on tension, length and μ.

What earns the marks

• Stationary waves form when two waves of the same frequency travel in opposite directions and superpose.
• The waves must have the same frequency; if the frequencies differ, the pattern of nodes and antinodes moves and is not stationary.
• Reinforcement occurs where the waves arrive in phase, giving an antinode; cancellation occurs where they arrive in antiphase, giving a node.
• The nodes and antinodes remain in fixed positions, so the stationary wave does not transfer energy along the string.
• On a string, reflection at a fixed end produces the oppositely travelling wave, and the fixed end is always a node.
• The wave speed on a string is v = √(T/μ), where T is the tension and μ is the mass per unit length, linking the mode frequency to the physical properties of the string.

Common mistakes and how to put them right

• Saying that a stationary wave is produced by a single wave bouncing back and forth.

  Put it right by describing two waves of the same frequency travelling in opposite directions that superpose.

• Believing that energy travels along a stationary wave from one end to the other.

  Put it right by stating that energy is stored between the nodes and there is no net transfer along the string.

• Using v = fλ with the node-to-node distance as λ.

  Put it right by using the full wavelength, which is twice the node-to-node distance.

Exam tips

• State explicitly that the two waves have the same frequency and travel in opposite directions; both conditions are needed for a stationary wave.
• Link any calculation to the boundary condition that a fixed end is a node, then use L = nλ/2 before applying v = fλ.
• If a question gives tension and mass per unit length, calculate v = √(T/μ) first, then combine it with the allowed wavelengths to find frequencies.