CIE AS/A Level Physics 8.1 Stationary waves Study Notes- 2025-2027 Syllabus
CIE AS/A Level Physics 8.1 Stationary waves Study Notes – New Syllabus
CIE AS/A Level Physics 8.1 Stationary waves Study Notes at IITian Academy focus on specific topic and type of questions asked in actual exam. Study Notes focus on AS/A Level Physics latest syllabus with Candidates should be able to:
- explain and use the principle of superposition
- show an understanding of experiments that demonstrate stationary waves using microwaves, stretched strings and air columns (it will be assumed that end corrections are negligible; knowledge of the concept of end corrections is not required)
- explain the formation of a stationary wave using a graphical method, and identify nodes and antinodes
- understand how wavelength may be determined from the positions of nodes or antinodes of a stationary wave
Principle of Superposition
The principle of superposition states that when two or more waves of the same type meet at a point, the resultant displacement at that point is the vector sum of the individual displacements produced by each wave.
\( \mathrm{y = y_1 + y_2 + y_3 + \dots} \)
Explanation:
- Superposition occurs for waves that overlap in the same medium, such as sound, light, or water waves.
- Each wave continues to travel as if the others were not present — they do not permanently alter each other.
- The resulting displacement depends on the amplitude and phase relationship of the interacting waves.
Types of Interference from Superposition:
Constructive Interference: Occurs when waves meet in phase (their crests and troughs coincide). The resultant amplitude is the sum of individual amplitudes.
\( \mathrm{A_{resultant} = A_1 + A_2} \)
Destructive Interference: Occurs when waves meet in antiphase (crest of one coincides with trough of another). The resultant amplitude is the difference between the individual amplitudes.
\( \mathrm{A_{resultant} = |A_1 – A_2|} \)
Conditions for Observable Interference:
- Waves must be of the same type.
- Waves must have the same frequency (or nearly so).
- Waves must have a constant phase relationship — they must be coherent.
Example
Two waves of equal amplitude \( \mathrm{2 \ cm} \) meet in phase at a point. What is the resultant displacement at that point? What would it be if they were in antiphase?
▶️ Answer / Explanation
In phase: \( \mathrm{y = 2 + 2 = 4 \ cm} \) (constructive interference)
In antiphase: \( \mathrm{y = 2 – 2 = 0 \ cm} \) (destructive interference)
Hence, the resultant displacement depends on whether the waves meet in phase or in antiphase.
Stationary (Standing) Waves — Experimental Demonstrations
A stationary wave (or standing wave) is formed when two progressive waves of the same frequency and amplitude travel in opposite directions and superpose. The result is a wave pattern with points of zero displacement (nodes) and points of maximum displacement (antinodes).
Key Properties of Stationary Waves:
- Energy is not transferred along the wave — it is stored in oscillations.
- Nodes (N): points of zero amplitude, where waves always cancel destructively.
- Antinodes (A): points of maximum amplitude, where waves interfere constructively.
- Adjacent nodes (or antinodes) are separated by \( \mathrm{\dfrac{\lambda}{2}} \).
\( \mathrm{Distance \ between \ two \ successive \ nodes = \dfrac{\lambda}{2}} \)
Experimental Demonstrations:
| Apparatus | Wave Type | Setup Description | Observation |
|---|---|---|---|
| Microwaves | Electromagnetic | A microwave transmitter and receiver are placed facing each other. Standing waves form due to reflection from a metal plate. The receiver detects alternating maxima and minima in signal intensity as it moves along the path. | Nodes correspond to minima in intensity; spacing between adjacent minima gives \( \mathrm{\dfrac{\lambda}{2}} \). |
| Stretched string (using a signal generator and vibration generator) | Mechanical (transverse) | A string under tension is connected to a vibration generator. When the driving frequency matches one of the string’s natural frequencies, stationary waves form. | Nodes occur at fixed ends; antinodes occur in between. Lengths of the string correspond to \( \mathrm{n \dfrac{\lambda}{2}} \). |
| Air column (e.g., resonance tube or closed pipe) | Sound (longitudinal) | A tuning fork is held above an air column in a tube. By adjusting the air column length, resonance occurs when standing waves form inside the column. | Nodes occur at the closed end, antinodes at the open end. The resonant length gives the wavelength relation \( \mathrm{L = \dfrac{(2n-1)\lambda}{4}} \) for a closed pipe. |
Stationary waves are formed by superposition of two waves of the same frequency and amplitude moving in opposite directions. They exhibit a fixed pattern of nodes and antinodes, demonstrating energy storage rather than transfer.
Example
A stationary wave is formed on a string of length \( \mathrm{1.2 \ m} \) with two fixed ends. If the string vibrates in its third harmonic, calculate the wavelength of the wave.
▶️ Answer / Explanation
For a string with both ends fixed: \( \mathrm{L = n \dfrac{\lambda}{2}} \)
For the third harmonic (\( \mathrm{n = 3} \)): \( \mathrm{1.2 = 3 \dfrac{\lambda}{2}} \Rightarrow \lambda = \dfrac{2 \times 1.2}{3} = 0.8 \ m} \)
Therefore, the wavelength of the stationary wave is \( \mathrm{0.8 \ m.} \)
Formation of Stationary (Standing) Waves — Graphical Explanation
A stationary wave is formed by the superposition of two progressive waves of the same frequency, amplitude, and wavelength traveling in opposite directions. These waves interfere continuously, producing a pattern of nodes and antinodes.
Mathematical Representation:
For two waves: \( \mathrm{y_1 = A \sin(kx – \omega t)} \) and \( \mathrm{y_2 = A \sin(kx + \omega t)} \)
When they superpose:
\( \mathrm{y = y_1 + y_2 = 2A \sin(kx) \cos(\omega t)} \)
Interpretation:
- \( \mathrm{2A \sin(kx)} \) represents the amplitude of the stationary wave at position \( \mathrm{x} \).
- \( \mathrm{\cos(\omega t)} \) represents the time-dependent oscillation at each point.
- At certain positions, amplitude is always zero (nodes), while at others it is maximum (antinodes).
Identification of Nodes and Antinodes (Graphically):
- Nodes (N): Points of zero displacement — occur where \( \mathrm{\sin(kx) = 0} \).
- Antinodes (A): Points of maximum displacement — occur where \( \mathrm{\sin(kx) = \pm 1} \).
Nodes: \( \mathrm{kx = n\pi} \Rightarrow \mathrm{x = n \dfrac{\lambda}{2}} \)
Antinodes: \( \mathrm{kx = (n + \tfrac{1}{2})\pi} \Rightarrow \mathrm{x = (n + \tfrac{1}{2})\dfrac{\lambda}{2}} \)
Graphical Representation:
- The stationary wave can be visualized as two opposite-moving waves continuously interfering.
- At certain times, the displacement patterns show crests and troughs coinciding — at others, they cancel.
- Over time, the envelope of maximum amplitude (antinodes) and zero amplitude (nodes) remains fixed in space.
Diagram (Conceptual):
Spacing between successive nodes or antinodes = \( \mathrm{\dfrac{\lambda}{2}} \)
Example
Two waves of equal amplitude and frequency travel in opposite directions along a string. Describe how their superposition produces a stationary wave pattern.
▶️ Answer / Explanation
When the waves meet, they interfere. At points where their displacements are always equal and opposite, they cancel out, forming nodes. At other points where they reinforce each other, antinodes appear. The resulting pattern does not move — the nodes and antinodes remain stationary, forming a standing wave.
Determining Wavelength from the Positions of Nodes or Antinodes
The wavelength \( \mathrm{\lambda} \) of a stationary wave can be determined by measuring the distance between adjacent nodes or antinodes, since the spatial separation between them is known to be half a wavelength.
\( \mathrm{Distance \ between \ successive \ nodes \ or \ antinodes = \dfrac{\lambda}{2}} \)
Hence,
\( \mathrm{\lambda = 2 \times (distance \ between \ successive \ nodes)} \)
Practical Measurement:
- In microwave experiments, move the receiver along the line of propagation and measure the distance between successive minima (nodes) of signal intensity.
- In a vibrating string, measure the distance between fixed points (nodes) using a scale or ruler.
- In an air column, measure the distance between successive resonances (antinodes) of sound intensity.
By determining the distance between consecutive nodes or antinodes and multiplying by two, we can find the wavelength of the stationary wave accurately.
Example
In a stationary wave experiment on a string, the distance between five consecutive nodes is measured as \( \mathrm{0.60 \ m} \). Calculate the wavelength of the wave.
▶️ Answer / Explanation
Distance between 5 consecutive nodes = 4 intervals = \( \mathrm{4 \times \dfrac{\lambda}{2}} = 2\lambda \)
\( \mathrm{2\lambda = 0.60 \Rightarrow \lambda = 0.30 \ m} \)
Therefore, the wavelength of the stationary wave is \( \mathrm{0.30 \ m.} \)