• Understand the principle of superposition in the context of wave interference.
• Explain the formation of a stationary wave as the superposition of two progressive waves of the same frequency, amplitude, and speed, travelling in opposite directions.
• Use a graphical method to illustrate the formation of a stationary wave.
• Identify nodes and antinodes on a stationary wave.
• Describe the properties of stationary waves (e.g., energy transfer, phase relationships).
• Recall and use the relationship between the distance between adjacent nodes (or antinodes) and the wavelength (distance = λ/2).

• Define and use key vocabulary related to stationary waves (e.g., superposition, node, antinode, interference, wavelength, frequency).
• Describe the process of stationary wave formation using appropriate scientific language.
• Explain the difference between progressive waves and stationary waves.
• Discuss the characteristics of nodes and antinodes.

Here are some important terms for this lesson. Pay attention to their meanings and translations.

Review the key terms using flashcards. You can create your own set on Quizlet or use a pre-existing one.

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Stationary Wave (Standing Wave): A wave pattern that results when two waves of the same frequency, wavelength, and amplitude travel in opposite directions and interfere. It does not transfer energy.

Node: A point in a stationary wave where the amplitude of oscillation is always zero.

Antinode: A point in a stationary wave where the amplitude of oscillation is maximum.

Superposition Principle: When two or more waves meet at a point, the resultant displacement at that point is the vector sum of the displacements due to each individual wave.

Interference: The phenomenon that occurs when two or more waves meet and combine to form a resultant wave of greater, lower, or the same amplitude.

A stationary waveтұрақты толқын, also known as a standing wave, is formed by the superpositionсуперпозиция of two progressive waves of the same frequencyжиілік, wavelengthтолқын ұзындығы, and amplitudeамплитуда, travelling in opposite directions. This typically happens when a wave is reflectedшағылған back along its original path from a boundary.

Consider an incident waveтүскен толқын travelling to the right and a reflected waveшағылған толқын travelling to the left. As these two waves pass through each other, they interfere.

• At some points, the waves meet in phaseфаза (crest meets crest, trough meets trough), resulting in constructiveконструктивті interferenceинтерференция. These points are called antinodes, and they oscillate with maximum amplitude (2A, where A is the amplitude of individual waves).
• At other points, the waves meet out of phase by π radians (180°) (crest meets trough), resulting in destructiveдеструктивті interference. These points are called nodes, and they have zero amplitude; they remain stationary.

Graphical Method:
Imagine taking snapshots of the two waves and their resultant at different moments in time:

1. Time t = 0: The incident wave and reflected wave might align such that constructive interference occurs at certain points (antinodes) and destructive interference at others (nodes).
2. Time t = T/4 (where T is the period): Both waves advance by a quarter of a wavelength. The pattern of interference shifts. Points that were at maximum displacement might now be at zero displacement, and vice-versa, but the positions of nodes and antinodes remain fixed.
3. Time t = T/2: Both waves advance by half a wavelength. The resultant wave is an inverted version of the wave at t=0 (for particles between nodes).
4. Time t = 3T/4: Similar to T/4, but with further phase shifts.
5. Time t = T: The pattern returns to the state at t=0.

(Ideally, this section would be accompanied by diagrams showing the superposition at these time intervals.)
The key is that nodes are always points of zero displacement, and antinodes are always points of maximum displacement. All particles between two adjacent nodes oscillate in phase, but their amplitudes vary from zero at the nodes to maximum at the antinode. Particles in an adjacent segment (between the next pair of nodes) oscillate in antiphase with the first segment.

Properties of Stationary Waves:

• No net energy transfer: Energy is stored in the oscillating segments between nodes.
• Fixed nodes and antinodes: Their positions do not change.
• Amplitude variation: Amplitude varies from zero at nodes to maximum at antinodes.
• Phase: All particles between two adjacent nodes oscillate in phase. Particles in adjacent segments are in antiphase (π radians out of phase).
• The distance between two adjacent nodes (or two adjacent antinodes) is λ/2.
• The distance between a node and an adjacent antinode is λ/4.

Questions on Theory:

1. (Easy) What is a node in a stationary wave?
   
   Answer

   A node is a point in a stationary wave where the amplitude of oscillation is always zero.
2. (Medium) Describe the conditions necessary for the formation of a stationary wave.
   
   Answer

   Two progressive waves of the same frequency, wavelength (and therefore speed), and similar amplitude must travel in opposite directions and superpose (interfere).
3. (Medium) How does the amplitude of particles vary along a stationary wave?
   
   Answer

   The amplitude of particles varies from zero at the nodes to a maximum value (twice the amplitude of the individual progressive waves) at the antinodes. Between a node and an antinode, the amplitude gradually increases.
4. (Hard — Critical Thinking) A string is fixed at both ends. If the tension in the string is increased, while its length and mass per unit length remain constant, how does this affect the frequencies of the stationary waves that can be formed on the string? Explain your reasoning.
   
   Answer

   The speed of a wave on a string is given by v = √(T/μ), where T is the tension and μ is the mass per unit length. If tension (T) increases, the wave speed (v) increases.
   For a string fixed at both ends, the possible wavelengths of stationary waves are given by L = n(λ/2), so λ = 2L/n, where L is the length of the string and n is an integer (1, 2, 3,…).
   The frequency is f = v/λ. Substituting the expressions for v and λ:
   f = (√(T/μ)) / (2L/n) = (n / 2L) * √(T/μ).
   Since L, μ, and n are constant for a given harmonic, if T increases, v increases, and therefore the frequency (f) of each possible stationary wave (harmonic) will increase.
   

Exercise 1: Match the Term with its Definition

1. Node
2. Antinode
3. Superposition
4. Stationary Wave

A. A point of maximum amplitude in a stationary wave.
B. A wave pattern that does not transfer energy.
C. A point of zero amplitude in a stationary wave.
D. The principle that states that the resultant displacement is the sum of individual displacements.

Exercise 2: Fill in the Blanks

1. A stationary wave is formed by the __________ of two progressive waves travelling in opposite directions.
2. Points of zero displacement in a stationary wave are called __________.
3. The distance between two adjacent antinodes is equal to __________ the wavelength.
4. In a stationary wave, there is no net transfer of __________.

Watch this video to better understand the formation and characteristics of stationary waves:

Alternative good video:

Problem 1: A string of length 0.8 m is fixed at both ends. It vibrates in a stationary wave pattern with 4 antinodes. What is the wavelength of the waves forming this pattern?

Explore stationary waves using the PhET Interactive Simulation «Waves on a String».

Simulation Link: Wave on a String

(Instructor: Ensure students have access to this simulation. You might need to embed it directly if your WordPress setup allows, or provide the link as above.)

Instructions & Questions:

1. Open the simulation. Select «Fixed End» for the right end of the string. Set Damping to «None» or very low. Choose «Oscillate» mode.
2. Start with a low frequency (e.g., Amplitude around 0.50 cm, Frequency around 0.5 Hz). Slowly increase the frequency. Observe what happens.
3. Question 1: Adjust the frequency until you see a clear stationary wave pattern with the minimum number of antinodes (this is the fundamental frequency or first harmonic). How many nodes (including the ends) and antinodes do you observe? Describe the pattern.
   
   Answer Q1

   For the fundamental frequency (first harmonic) on a string fixed at both ends, you should observe:
   — 2 nodes (one at each fixed end).
   — 1 antinode (in the middle of the string).
   The string vibrates in a single loop.
   
4. Question 2: Now, carefully increase the frequency until you find the next clear stationary wave pattern (second harmonic). How many nodes and antinodes do you observe now? How does this frequency compare to the fundamental frequency?
   
   Answer Q2

   For the second harmonic, you should observe:
   — 3 nodes (two at the ends and one in the middle).
   — 2 antinodes.
   The string vibrates in two loops. This frequency should be approximately double the fundamental frequency.
   
5. Question 3: Try to create a stationary wave with 3 antinodes (third harmonic). What is the relationship between the length of the string (L) visible in the simulation and the wavelength (λ) of the wave for this pattern? (Hint: How many half-wavelengths fit into L?)
   
   Answer Q3

   For a stationary wave with 3 antinodes, the string vibrates in three loops.
   This means that 3 half-wavelengths fit into the length of the string L.
   So, L = 3 * (λ/2).
   The wavelength λ = (2/3)L.
   
6. Question 4 (Exploration): Change the «Tension» slider. How does increasing or decreasing tension affect the frequencies at which stationary waves are formed?
   
   Answer Q4

   Increasing the tension increases the speed of the wave on the string (v = √(T/μ)). Since f = v/λ, and the wavelengths for stationary patterns are fixed by the length of the string (λ = 2L/n), an increase in wave speed (v) will lead to an increase in the frequencies (f) at which stationary waves are formed. Decreasing tension will have the opposite effect, lowering these frequencies.
   

Work with a partner or in a small group to complete an activity on stationary waves.

(Instructor: Insert an embedded activity from Quizizz, LearningApps.org, Formative, or GoConqr here. For example, a collaborative quiz or a matching game.)

Example Activity Idea (if embedding is not possible, describe it):
«With your group, draw diagrams representing the first three harmonics of a stationary wave on a string fixed at both ends. Label the nodes and antinodes for each. Discuss how the wavelength changes for each harmonic.»

Placeholder for embedded activity. Please replace this with your chosen interactive tool or describe the activity.

Answer the following questions. Show your working where necessary.

1. A microwave oven uses microwaves of frequency 2.45 GHz (2.45 x 10⁹ Hz) to heat food. These microwaves form stationary waves inside the oven. The speed of microwaves is 3.00 x 10⁸ m/s.
   
   (a) Calculate the wavelength of these microwaves.
   
   (b) The distance between antinodes in the stationary wave pattern determines where «hot spots» occur. What is the distance between adjacent antinodes?
   
   Answer 1

   (a) Given: f = 2.45 x 10⁹ Hz, v = 3.00 x 10⁸ m/s.
   
   Formula: v = fλ => λ = v/f
   
   λ = (3.00 x 10⁸ m/s) / (2.45 x 10⁹ Hz)
   
   λ &approx; 0.1224 m or 12.24 cm.

   (b) The distance between adjacent antinodes is λ/2.
   
   Distance = 0.1224 m / 2 &approx; 0.0612 m or 6.12 cm.
   

2. A guitar string of length 75 cm is fixed at both ends. When plucked, it can vibrate with a fundamental frequency of 220 Hz.
   
   (a) What is the wavelength of the fundamental mode of vibration?
   
   (b) What is the speed of the transverse waves on this string?
   
   (c) What is the frequency of the third harmonic for this string?
   
   Answer 2

   (a) For the fundamental mode (n=1) on a string fixed at both ends, L = λ/2.
   
   Given L = 75 cm = 0.75 m.
   
   λ₁ = 2L = 2 * 0.75 m = 1.50 m.

   (b) Speed v = f₁λ₁.
   
   v = 220 Hz * 1.50 m = 330 m/s.

   (c) The third harmonic (n=3) has a frequency f₃ = 3 * f₁.
   
   f₃ = 3 * 220 Hz = 660 Hz.
   
   (Alternatively, for the third harmonic, L = 3(λ₃/2) => λ₃ = 2L/3 = 2*0.75/3 = 0.50 m. Then f₃ = v/λ₃ = 330 m/s / 0.50 m = 660 Hz.)
   

3. Explain, using the principle of superposition, how nodes and antinodes are formed in a stationary wave. You may include a simple diagram sketch in your explanation if it helps (describe it if you can’t draw here).
   
   Answer 3

   A stationary wave is formed by the superposition of two identical progressive waves (same frequency, wavelength, amplitude) travelling in opposite directions.
   
   Nodes: At certain points, the two waves always meet completely out of phase (phase difference of π radians or 180°). This means a crest from one wave meets a trough from the other. Due to destructive interference, the resultant displacement at these points is always zero. These are the nodes.
   
   Antinodes: At other points, midway between nodes, the two waves always meet in phase (phase difference of 0 radians). A crest from one wave meets a crest from the other, and a trough meets a trough. Due to constructive interference, the resultant displacement at these points oscillates with maximum amplitude (twice the amplitude of one individual wave). These are the antinodes.
   
   (Diagram sketch description: Two sine waves travelling in opposite directions. Show them at an instant where one is y1 = A sin(kx — ωt) and the other is y2 = A sin(kx + ωt). Show their sum y = y1 + y2 = 2A sin(kx) cos(ωt). Points where sin(kx)=0 are nodes, points where sin(kx)=±1 are antinodes.)
   
4. Distinguish clearly between progressive waves and stationary waves in terms of (a) energy transfer, (b) amplitude of particles, and (c) phase difference between particles.
   
   Answer 4

   Feature	Progressive Wave	Stationary Wave
   (a) Energy Transfer	Energy is transferred in the direction of wave propagation.	No net transfer of energy. Energy is stored in segments.
   (b) Amplitude	All particles oscillate with the same amplitude.	Amplitude varies from zero at nodes to maximum at antinodes.
   (c) Phase Difference	Phase changes continuously from one point to the next. Phase difference between two points can be any value from 0 to 2π depending on their separation.	All particles between two adjacent nodes oscillate in phase. Particles in adjacent segments (separated by a node) oscillate in antiphase (π radians out of phase).

5. A pipe is open at one end and closed at the other. It can produce stationary sound waves. The length of the pipe is 0.68 m. The speed of sound in air is 340 m/s.
   
   (a) For the fundamental mode (lowest frequency), sketch the displacement wave pattern. Mark the node (N) and antinode (A).
   
   (b) Calculate the wavelength of this fundamental mode.
   
   (c) Calculate the fundamental frequency.
   
   (d) What is the wavelength and frequency of the next higher harmonic (first overtone)?
   
   Answer 5

   (a) For a pipe open at one end and closed at the other, the closed end must be a displacement node (N) and the open end must be a displacement antinode (A).
   
   Sketch description for fundamental: The pipe is shown. At the closed end, the wave displacement is zero (N). At the open end, the wave displacement is maximum (A). The wave shape is one-quarter of a full sine wave fitting into the pipe.

   (b) For the fundamental mode in such a pipe, L = λ₁/4.
   
   Given L = 0.68 m.
   
   λ₁ = 4L = 4 * 0.68 m = 2.72 m.

   (c) Fundamental frequency f₁ = v/λ₁.
   
   f₁ = 340 m/s / 2.72 m &approx; 125 Hz.

   (d) For a pipe closed at one end and open at the other, only odd harmonics are present (1st, 3rd, 5th, etc.). The next higher harmonic is the 3rd harmonic (first overtone).
   
   For the 3rd harmonic, L = 3λ₃/4.
   
   λ₃ = 4L/3 = 4 * 0.68 m / 3 &approx; 0.907 m.
   
   Frequency f₃ = 3f₁ = 3 * 125 Hz = 375 Hz.
   
   (Alternatively, f₃ = v/λ₃ = 340 m/s / 0.907 m &approx; 375 Hz).
   

• Save My Exams — Stationary Waves: (Search on Save My Exams for «A-Level Physics Stationary Waves» for notes and questions) Save My Exams
• PhysicsAndMathsTutor — Waves: (Browse the Waves section for relevant notes and past papers) PhysicsAndMathsTutor (Example for OCR A, find your board)
• YouTube — ALevelPhysicsOnline — Standing Waves on a String: Example Video (Link might be broken, search for relevant content) (Note: The provided link was broken, please search for a suitable video like «Standing waves A level physics»)
• HyperPhysics — Standing Waves: HyperPhysics Concepts

Take a few moments to reflect on your learning:

• What are the three most important things you learned about stationary waves today?
• What part of this lesson did you find most challenging? Why?
• How can you use the concept of stationary waves to explain a real-world phenomenon (e.g., musical instruments)?
• What questions do you still have about stationary waves?
• What steps will you take to clarify any remaining doubts?