Waves

This section explores wave characteristics, classifications (e.g., longitudinal, transverse, electromagnetic, and mechanical), and properties like reflection, refraction, and diffraction. Practical applications include determining wave parameters, such as amplitude, frequency, and wavelength, and deriving the wave equation. Sound waves, a key focus, are studied in terms of their production, nature, classification, and applications like echo measurement and resonance, helping you connect theoretical concepts to real-world phenomena.

KEY IDEAS

• A wave is a disturbance which transfers energy between places without the transfer of matter.

• Properties of wave include reflection, refraction, diffraction, interference, polarisation

• Sound is produced by vibrations. Sound is a longitudinal wave. Sound requires a medium to travel, such as air, water, or solids.

Remember the days you went swimming with your friends? When the first person jumped into the pool, what did you observe about the surface of the water? A disturbance was created at the point of impact, and this disturbance spread even to the banks of the pool, even though the water at the origin of the disturbance did not travel to the banks. This indicates that it is possible to transfer energy from place to place through matter without matter accompanying the transfer. This phenomenon describes wave motion.

Definition: A wave is a disturbance which transfers energy between places without the transfer of matter.

Examples of waves include sound, light, water waves, seismic waves, etc.

Figure 7.1: Ripples form during swimming

Activity 7.1 Creating a Concept Map on Waves

Objective: To recall and discuss your prior knowledge of waves.

Materials Needed

1. Large sheets of paper or poster board (for creating the concept map)

2. Markers, coloured pencils, or crayons (for drawing and labelling)

3. Sticky notes (optional, for organising ideas)

4. Reference materials (textbooks or notes on waves, if needed) Keywords to Prompt Discussion

1. Wave

2. Amplitude

3. Wavelength

4. Frequency

5. Speed

6. Reflection

7. Refraction

8. Diffraction

9. Sound Waves

10. Light Waves

11. Electromagnetic Waves

12. Medium

13. Energy Transfer

What to do

1. Organise yourselves into small groups of not more than five.

2. Begin by discussing what you already know about waves. Use the keywords provided above to prompt your discussion.

3. Share any definitions, examples, or concepts related to waves that you can remember.

Organise Your Ideas

a. As you discuss, write down important terms and concepts that come up during your conversation.

b. Consider how these concepts are related to each other. For example, think about how amplitude relates to energy transfer or how frequency relates to wavelength.

On your large sheet of paper or poster board, start creating your concept map

a. Write the main topic “Waves” in the centre of the page.

b. Branch out from the main topic with key concepts you discussed (e.g., Amplitude, Wavelength).

c. Draw lines between related concepts and label those lines with descriptions of their relationships (e.g., “affects,” “is a type of,” “depends on”).

d. Use different colours or shapes to differentiate between types of waves (e.g., sound waves vs. electromagnetic waves).

Figure 7.2: Example of concept map Collaborate and Revise

a. Work together as a group to ensure that everyone’s ideas are included in the concept map.

b. Revise and refine your map as needed, adding any additional connections or concepts that arise during your discussion.

Prepare to Share

a. Once your concept map is complete, prepare to present it to the class.

b. Decide who will explain each part of the map and what key points you want to highlight during your presentation.

Present Your Concept Map

a. Each group will take turns presenting their concept maps to the class.

b. Explain the connections between different concepts and share any insights gained during your discussions.

Properties of Waves

When a wave is travelling through a medium, it interacts with the particles of the medium and exhibits certain characteristics, which we refer to as wave properties.

Waves interact with the medium’s particles, leading to observable phenomena like reflection, refraction, diffraction, and interference. These properties help us understand how waves transfer energy and interact with their surroundings.

Reflection Reflection is the bouncing back of a wave into its original medium after encountering a boundary or obstacle. In this process:

a. The frequency, wavelength, and velocity of the wave remain unchanged, as long as the wave stays in the same medium. Only the direction of the wave changes.

b. The amplitude may change depending on the nature of the boundary (e.g., energy loss at the boundary).

While reflection is often associated with light rays, all types of waves—such as sound waves, water waves, and seismic waves—undergo reflection as well.

For example,

1. Reflection of Light: You can see yourself and the environment behind you in a mirror because light waves reflect off the mirror’s surface. This is one of the most common experiences of wave reflection.

2. Reflection of Sound: Echoes in large, empty halls and reverberations in empty rooms occur because sound waves reflect off hard surfaces like walls, ceilings, and floors. These reflections can enhance or distort the original sound.

Figure 7.3a: Reflection of water waves against a wall

Figure 7.3b: Reflection of light in a lab experiment

Activity 7.2 Demonstrations of reflection Try one or more of the following activities to see reflection in action:

1. Shine a laser pointer along a flat surface directed towards a mirror. Observe the path of the light after it meets the mirror. Experiment with different angles at which the light meets the surface of the mirror.

2. Take a tray or oven dish and fill it with water until it is around 1cm deep.

Put the tray on a flat surface, and then gently lift and drop one end. Observe the wave that is produced and notice what happens when it hits the other end of the tray.

3. Look at yourself in a mirror, and compare the image to that which you see in a photograph of yourself. Are they the same? In what ways are they different?

Refraction Refraction, like reflection, occurs at a boundary. However, in this case, the boundary is permeable to the wave, allowing it to enter the next medium. When a wave passes from one medium to another with a different (optical, in the case of light) density, it changes direction (bends), speed, and wavelength. Importantly, the frequency remains unchanged.

Definition: Refraction is the change in the speed and direction (bending) of a wave when it moves from one medium to another of a different density.

A few daily phenomena resulting from the refraction of waves include:

1. Shallow Appearance of Water: A pool of water appears shallower than it actually is because light rays bend (refract) when they travel from water to air, creating a misleading perception of depth.

2. Fish in a Pond: Fish appear closer to the surface of a pond than they really are due to the refraction of light as it passes from water to air, altering the perceived position of the fish.

3. Sound Intensity Variation: The intensity of sound varies during the day because sound waves refract when passing through layers of air at different temperatures, causing changes in their direction and concentration.

Figure 7.4 a: Light refracting through a rectangular prism Figure 7.4 b: Real and apparent positions of a fish in water

Activity 7.3 Demonstrations of refraction Try one or more of the following activities to see refraction in action:

1. Shine a laser pointer along a piece of paper, directed towards a glass block lying flat on the paper. Observe the path of the light as it enters the glass.

Experiment by changing the angle at which the light hits the surface of the glass.

2. Place a penny into a transparent container, like a drinking glass, and observe the penny from the top. How does its apparent depth compare with true depth?

Diffraction Diffraction is the spreading out of waves when they pass through small openings (apertures) or the bending of waves around obstacles.

It is a common phenomenon observed with all types of waves, such as light, sound, and water waves.

Key points about diffraction

1. The degree of diffraction depends on the wavelength of the wave and the size of the aperture or obstacle.

2. Waves with longer wavelengths (e.g., low-frequency sounds) diffract more effectively.

3. If the wavelength is significantly smaller than the gap or obstacle, little to no diffraction occurs, and the wave passes straight through as if in free space.

Figure 7.5a: Diffraction of water waves through an opening in a wall

Figure 7.5b: Diffraction of light through a slightly ajar door Examples of Diffraction in Daily Life

1. Sound Diffraction: Low-frequency sounds (bass) diffract around buildings and through small gaps, allowing them to be heard in areas not directly exposed to the sound source.

2. Light Diffraction: Light rays diffract through small gaps between leaves in a forest canopy, scattering to light up the forest floor.

3. Water Waves: When ocean waves pass through a narrow harbour entrance or around a breakwater, they spread out and form circular wave patterns beyond the gap.

4. Radio Waves: Radio signals, especially those in the AM frequency range (longer wavelengths), can bend around hills, buildings, and other obstacles, allowing reception in areas not in direct line-of-sight of the transmitter.

5. Shadow Edges: The edges of shadows are not perfectly sharp because light diffracts slightly around the edges of objects, causing a gradual transition from light to shadow.

Activity 7.4 Demonstrations of diffraction Try one or more of the following activities to see diffraction in action:

1. Shine a laser pointer at a diffraction grating so that the pattern emerging can be seen on the wall opposite. Describe the pattern that you see.

Experiment by changing the distance to the wall.

2. As in activity 7.2, create a water wave in a tray. This time, place an object in the centre of the tray. How does the wave behave as it passes the object?

3. As above, but this time create a barrier in the middle of the try with a small gap in the centre. How does the wave behave as it passes through the gap?

Interference of Waves

Interference occurs when two or more waves meet, overlap, and combine as they travel through the same medium. This phenomenon can happen with any type of wave—such as sound, light, water, or even electromagnetic waves.

The resulting wave pattern depends on how the individual waves interact. This interaction is governed by the principle of superposition, which states: the resultant displacement at any point is the algebraic sum of the displacements of the individual waves at that point.

Types of Interference

Constructive Interference: Occurs when waves meet in phase (their crests and troughs align). The amplitudes of the waves add up, creating a wave with a greater amplitude.

Example: Louder sound at certain points in a room when sound waves from multiple sources reinforce each other.

Destructive Interference: Occurs when waves meet out of phase (a crest aligns with a trough). The amplitudes cancel each other out, reducing or completely nullifying the resultant wave.

Example: Noise-cancelling headphones use destructive interference to reduce unwanted sound.

Figure 7.6a: Interference of two water waves

Figure 7.6b: Interference pattern observed in a double slit experiment on light.

Applications and Examples

1. Light Interference: Thin-film interference creates colourful patterns on soap bubbles or oil spills due to the overlapping of light waves reflected from different layers of the film.

2. Sound Interference: When sound waves overlap, they can create regions of louder (constructive interference) and quieter (destructive interference) sound, like beats in musical tuning.

3. Water Wave Interference: When two stones are dropped into a pond, the ripples overlap, creating constructive and destructive interference patterns visible as areas of higher or lower wave heights.

Interference explains how waves interact to create complex wave patterns, whether amplifying or cancelling each other out. This principle is fundamental to understanding many natural and technological phenomena.

Polarisation In the next discussion, you will learn about a type of wave called a transverse wave.

In transverse waves, the direction of the medium’s displacement is perpendicular to the direction in which the wave travels. A common example of transverse waves is electromagnetic waves.

Electromagnetic waves consist of oscillating electric and magnetic fields that are perpendicular to each other and the direction of wave propagation. These fields typically oscillate in multiple planes. Through a process called polarisation, it is possible to filter or restrict these oscillations to a single plane, allowing only waves in that plane to propagate while blocking others.

This filtering process is essential in various applications, such as reducing glare in sunglasses or enhancing signal clarity in communication systems.

Figure 7.7: magnetic (blue and horizontal) and electric (red and vertical) fields propagating in the same direction but perpendicular in orientation.

Activity 7.5 Demonstrations of polarisation Try the following activity to see polarisation in action:

Go out into the street on a sunny day and look at the windscreen of a car.

Now look again, but this time with polarising sunglasses on. How are the two images different?

Classification of waves

1. Based on the Medium

a. Mechanical Waves: Require a medium (solid, liquid, or gas) to propagate and cannot travel through a vacuum. Examples include sound waves and water waves.

b. Electromagnetic Waves: Do not require a medium and can propagate through a vacuum. However, they interact with matter when travelling through a medium, such as air or water. Examples include light, X-rays, and microwaves.

2. Based on the Direction of Propagation Relative to Medium Particles

a. Longitudinal Waves: The wave propagates parallel to the displacement of particles in the medium (e.g., sound waves).

b. Transverse Waves: The wave propagates perpendicular to the displacement of particles in the medium (e.g., electromagnetic waves).

Note: Water waves are surface waves with both transverse and longitudinal components.

Figure 7.8: using a slinky to demonstrate transverse and longitudinal waves

3. Based on Energy Transfer

a. Progressive Waves: Transfer energy through a medium without permanently displacing the medium’s particles. Examples include electromagnetic waves and surface water waves.

b. Standing Waves: Confine energy between fixed points where incident and reflected waves superimpose, creating nodes (no motion) and antinodes (maximum motion). These waves do not transport energy over long distances, as seen in vibrating guitar strings.

Activity 7.6 Modelling Transverse and Longitudinal Waves

In groups of 10-20 students, follow the instructions below in order to model the behaviour of transverse and longitudinal waves.

1. Half of the group (group A) should stand in a line, shoulder to shoulder.

The rest of the group (group B) should observe.

2. The person on the far left of group A should act as the source of a longitudinal wave (e.g., sound). To do this, they should begin by vibrating side-to-side.

3. They should gently collide with their neighbour, causing them to also vibrate from side to side.

4. This process should continue until the final person, on the far right hand side, is vibrating. The wave has been transferred longitudinally, with the oscillations parallel to the transfer of energy.

5. Group B should now take their turn and stand side by side, observed by group A.

6. The person on the far left hand side of group A should perform a ‘Mexican Wave’, moving up and down with exaggerated movements.

7. After a second, the person next to them should also being moving in the same way.

8. This process should continue until the final person, on the far right hand side, is moving. The wave has been transferred transversely, with the oscillation perpendicular to the transfer of energy.

Activity 7.7 Exploring Wave Properties Through Demonstrations

Objective: Explore different types of waves by conducting hands-on demonstrations using a rope, a slinky, and a vibrating guitar string.

Materials Needed

1. Long rope or string (approximately 3-5 metres)

2. Slinky spring

3. Guitar (or any instrument with a vibrating string)

4. Flat surface (for the slinky demonstration)

5. Paper and pen/pencil (for taking notes and creating classification charts) What to do Part A: Rope Wave Demonstration

1. Organise yourselves into small groups.

2. Stretch the long rope or string between two fixed points (e.g., two chairs or trees) so that it is taut.

3. One member of your group should quickly move one end of the rope up and down to create waves. Observe how the wave travels along the rope.

4. After the demonstration, discuss as a group:

a. What type of wave was created?

b. How did the wave travel through the rope?

Part B: Slinky Wave Demonstration

4. Stretch the slinky across a flat surface.

5. One member of your group should hold one end of the slinky stationary while another member quickly pushes and pulls the other end to create waves.

6. Next, while one end of the slinky is held stationary, have another member move the other end up and down or side-to-side to create a different type of wave.

7. After each type of wave demonstration with the slinky, discuss as a group:

a. What characteristics did you observe for each type of wave?

b. How did each wave behave differently?

c. What do you think distinguishes these types of waves?

Part C: Guitar String Wave Demonstration

8. Use a guitar or similar instrument to demonstrate waves.

9. Pluck a string to create vibrations and observe how these vibrations produce sound waves.

10. Discuss as a group:

a. What characteristics did you notice about the sound waves produced by the vibrating guitar string?

b. How does this wave differ from those created with the rope and slinky?

c. What properties did you observe?

Part D: Classification Discussion

9. On a piece of paper, create a classification chart with columns for:

a. Longitudinal Waves

b. Transverse Waves

c. Electromagnetic Waves

d. Mechanical Waves

e. Progressive Waves

f. Stationary Waves

10. Based on your observations from each demonstration, categorise each type of wave into the appropriate columns in your classification chart.

11. Discuss as a group why you classified each wave in that way.

Activity 7.8 Classifying Waves Based on Definitions

Objective: work in pairs or small groups to classify different types of waves based on their definitions. You will discuss the characteristics of each wave type and categorise them accordingly.

Materials Needed

1. Lists of definitions (provided below)

2. Lists of different types of waves (provided below)

3. Paper and pen/pencil

4. Whiteboard or large paper (optional, for organising your classifications) Definitions List

1. Transverse Wave: A wave in which the particle’s displacement is perpendicular to the direction of wave propagation.

2. Longitudinal Wave: A wave in which the particle’s displacement is parallel to the direction of wave propagation.

3. Progressive Wave: A wave that travels through a medium, transferring energy from one point to another.

4. Stationary Wave: A wave that remains in a constant position, formed by the interference of two waves traveling in opposite directions.

5. Mechanical Wave: A wave that requires a medium (solid, liquid, or gas) to travel through.

6. Electromagnetic Wave: A wave that does not require a medium and can travel through a vacuum; it consists of oscillating electric and magnetic fields.

Types of Waves List

1. Sound Wave

2. Light Wave

3. Water Wave

4. Seismic Wave

5. Radio Wave

6. Microwaves

7. String Wave (e.g., on a guitar string)

8. Ultrasound Wave

What to do

1. Organise yourselves into groups of no more than five.

2. Review Definitions

a. Each group should read through the definitions list carefully.

b. Discuss the meaning of each term and ensure that everyone understands the differences between transverse waves, longitudinal waves, progressive waves, stationary waves, mechanical waves, and electromagnetic waves.

3. Review Types of Waves

a. Next, review the list of different types of waves provided.

b. Discuss each type and consider its characteristics.

4. Classify the Waves

a. As a group, classify each type of wave from the “Types of Waves” list into appropriate categories based on the definitions:

i. Identify which waves are transverse and which are longitudinal.

ii. Determine which are mechanical and which are electromagnetic.

iii. Discuss whether they are progressive or stationary.

b. You can create a table or chart to organise your classifications.

5. Write down your classifications on paper or use a whiteboard to visually represent your findings.

6. Present Your Classifications

a. Each group will take turns presenting their classifications to the class.

b. Explain how you categorised each type of wave based on its definition and characteristics.

Activity 7.9 Exploring Wave Types Using PhET Simulations

Objective: Use the PhET wave simulation tools to explore different types of waves.

Materials Needed

1. Computer or tablet with internet access

2. Access to PhET simulations, specifically:

a. Wave on a String

b. Sound

c. Fourier: Making Waves

d. Wave Interference

3. Paper and pen/pencil (for taking notes and classifying waves) What to do

1. Open your web browser and navigate to the PhET website. Choose one of the wave simulations from the list above.

2. Explore Different Types of Waves

a. Sound Simulation

i. Click on “Play” to generate sound waves.

ii. Observe how the waves propagate through the medium.

iii. Take notes on the characteristics you observe, such as patterns, movement, and behaviours.

b. Wave on a String Simulation

i. Use the controls to create waves by moving one end of a string up and down.

ii. Observe how the wave travels along the string.

iii. Note any characteristics such as shape, amplitude, and speed.

c. Fourier: Making Waves Simulation

i. Experiment with creating different waveforms using sine waves.

ii. Observe how these combinations can produce various wave shapes.

iii. Take note of how different parameters affect wave behaviour.

d. Wave Interference Simulation

i. Set up two sources of waves (e.g., point sources) and observe how they interact.

ii. Note how the waves combine and create patterns.

iii. Discuss any observations about how these interactions occur.

3. After exploring each simulation, create a table in your notebook to organize your observations:

Wave Type Characteristics Observed Classification

Example Wave 1 Describe shape, direction, amplitude, etc.

Example Wave 2 Describe shape, direction, amplitude, etc.

Example Wave 3 Describe shape, direction, amplitude, etc.

4. Discuss Findings with Peers

a. After completing your exploration and classification, discuss your findings with classmates in small groups.

b. Share insights about how you categorised each type of wave based on your observations.

Wave Terminology and Representation

Representation of Waves

In Figure 7.9, a longitudinal wave and a transverse wave are compared.

All waves, whether transverse or longitudinal, can be represented graphically in the form of a transverse wave when displacement is plotted against time or position. This graphical representation produces a sinusoidal (trigonometric) curve, regardless of the wave type.

Figure 7.9: relating longitudinal and transverse waves.

Key Terms in Wave Properties

1. Crest: The highest point of a transverse wave above the equilibrium position.

It alternates with troughs as the wave propagates.

2. Trough: The lowest point of a transverse wave below the equilibrium position. Troughs alternate with crests throughout the wave’s motion.

3. Compression: A region in a longitudinal wave where particles of the medium are closely packed, resulting in high pressure. It is analogous to the crest of a transverse wave.

4. Rarefaction: A region in a longitudinal wave where particles of the medium are spread out, resulting in low pressure. It alternates with compressions and is analogous to the trough of a transverse wave.

5. Displacement (y): The distance moved by a particle of the medium from its equilibrium position at any time due to wave propagation. Displacement can occur in any direction, depending on the wave type.

6. Amplitude (a): The maximum displacement of a particle from its equilibrium position. It indicates the wave’s energy and is measured in metres (m).

7. Wavelength (λ): The distance between two successive points in phase on a wave, such as two crests, two troughs, or two compressions. It is measured in metres (m).

8. Period (T): The time taken for one complete oscillation or cycle of a wave.

It is measured in seconds (s).

Activity 7.10 Identifying the feature from a graph From the images below, match up the correct feature with the letter from the image. Note that one letter will not be needed.

Figure 7.10: Features of a wave Feature Letter Wavelength Time period Amplitude

Activity 7.11 Drawing Wave Profiles on Graph Paper

Objective: Create wave profiles on graph paper by specifying amplitudes and wavelengths.

Materials Needed

1. Graph paper (sheets or a notebook)

2. Ruler

3. Pencil or pen

4. Eraser (optional)

5. Coloured pencils or markers (optional, for added detail) What to do

1. Before you start drawing, familiarise yourself with the key parameters of a wave:

a. Amplitude (A): The maximum height of the wave from the rest position (baseline).

b. Wavelength (λ): The distance between two consecutive points in phase on the wave (e.g., crest to crest or trough to trough).

2. Choose Your Parameters. Decide on the amplitude and wavelength for your wave profile. For example:

c. Amplitude = 3 units

d. Wavelength = 6 units

3. Set Up Your Graph Paper:

a. Place your graph paper on a flat surface.

b. Choose a baseline (horizontal line) to represent the rest position of the wave. This will be your starting point.

4. Draw the Wave Profile:

a. Start at the left edge of the graph paper and mark your baseline.

b. Using your ruler, measure and mark points for one complete wavelength:

i. From the starting point, move right by the wavelength distance (e.g., 6 units).

c. Now, draw the wave:

i. From the baseline, move up to represent the crest of the wave at an amplitude of 3 units.

ii. Move back down to the baseline, then down to represent the trough at -3 units (if using a vertical scale).

iii. Return to the baseline and complete one full cycle by moving back up to create another crest.

d. Repeat this process to draw multiple wavelengths across your graph paper.

3. Clearly label your wave profile with:

i. Amplitude (e.g., “Amplitude = 3 units”)

ii. Wavelength (e.g., “Wavelength = 6 units”)

iii. Any other relevant information (e.g., “Wave Profile”).

4. Use coloured pencils or markers to differentiate between crests and troughs, or to highlight specific features of your wave profile.

The key terms describing waves that we have met so far can all be directly measured. However, there are also some properties of waves which can be calculated, such as their velocity or frequency.

Activity 7.12 Deriving the Wave Equation from the Definition of Speed Objective: mathematically derive the wave equation using the definition of speed as distance divided by time.

Definitions

1. Speed (v): The distance travelled per unit of time.

2. Wavelength (λ): The distance between two consecutive points in phase on a wave (e.g., crest to crest or trough to trough).

3. Frequency (f): The number of complete wave cycles that pass a point in one second (measured in Hertz, Hz).

4. Period (T): The time taken for one complete cycle of the wave to pass a point.

What to do

1. Recall the formula for speed (v) which is given by:

v = distance_______ time

2. Consider a Wave

a. For a wave, the distance travelled in one complete cycle is equal to the wavelength (λ).

b. Therefore, when one complete wave cycle travels past a point, it covers a distance of λ.

3. Relate Time to Frequency

a. The frequency (f) is defined as the number of cycles per second.

If T is the period (the time for one complete cycle), then:

f = 1/T

b. This means that if you know the frequency, you can determine the period:

T = 1/f

4. Substituting into the Speed Formula

a. When one complete cycle passes a point, it takes a time equal to the period (T).

b. Therefore, substituting λ for distance and T for time in the speed formula gives:

v = λ__ T

5. Expressing Period in Terms of Frequency

a. Now substitute T = 1/f into the equation:

v = λ__ 1/f

6. Rearranging gives you the wave equation:

v = f λ More Key Terms in Wave Properties

1. Frequency (f): The number of complete oscillations made by a particle per second or the number of wavelengths passing through a point per second.

Frequency is measured in hertz (Hz) or s⁻¹.

f = 1/T

2. Velocity (v): The distance travelled by a wave in one second. Velocity is measured in metres per second (m/s).

v = λf Finding the displacement of a particle at any given time The motion of a wave follows the form of a simple harmonic motion. As such, the general wave equation is given as y = asinθ , Where θ is the phase (angle) of the cycle through which the particle has oscillated.

θ could also be seen as the angular displacement, if the wave/particle were in a circular motion. As such, ω = θ __and θ = ωt .

Hence y = asinωt, where ω is the angular frequency; the angle per second through which the particle oscillates, and t is the time that has passed.

For example, a particle at the beginning of an oscillation, in the rest position has a phase angle of 0⁰. A particle a quarter of the way through a cycle has a phase angle of 90⁰or π__ 2 radians).

Progressive wave equation

Figure 7.11: Phase lag Wave Moving Left to Right (Positive x-Direction)

1. Displacement at Origin (O): The displacement of the particle at O is given by:

y = a sin ω t Where:

• a is the amplitude,

• ω is the angular frequency,

• t is the time.

2. Displacement at a Point P: A particle P at a distance x from O lags behind O by a phase angle ϕ. Its displacement is given by: y = a sin (ω t – φ), The negative sign means that P lags behind O.

Phase Angle and Wavelength Relationship

For a full wavelength λ, the total phase angle θ = 2π, At the position x along the wave at , θ = ϕ Therefore, λ_ x = 2π___ ϕ Making ϕ the subject, ϕ = 2πx____ λ Substituting ϕ = 2πx____ λ into y = a sin (ω t – φ) y = a sin (ωt − 2πx____ λ ) Wave number and frequency y = a sin (ω t – kx) · The wave number is defined as k = 2π___ λ , · The velocity (v) is related to the wavelength (λ) and period (T) by:

v = λ__ T, where T = λ_ v and ω = 2π___ T , ω = 2πv___ λ Substituting ϕ and ω into y = a sin (ω t – φ) y = a sin (2πvt____ λ ± 2πx___ λ ) OR y = a sin (2πft ± 2πx___ λ ) The progressive wave equation

1. For a left-to-right motion (positive x-direction): of the wave, y = a sin (2πvt____ λ − 2πx____ λ ) = a sin (2πft− 2πx___ λ )

2. For a right-to-left motion (negative x-direction):

y = a sin (2πvt____ λ + 2πx___ λ ) = a sin (2πft + 2πx___ λ ) Parameters · y: Displacement of the particle.

· a: Amplitude of the wave.

· v: Wave velocity.

· f: Frequency of vibration (f = 1/T).

· x: Distance of the particle from the origin.

· λ: Wavelength of the wave.

· k: Wave number (k = 2π___ λ ).

· ω: Angular frequency (ω = 2πf ).

Activity 7.13 Exploring Wave Properties Using PhET Simulation

Objective: use the PhET wave simulation to obtain the frequency, amplitude, and wavelength of sinusoidal waves.

Materials Needed

1. Computer or tablet with internet access

2. Access to the PhET wave simulation tool (e.g., PhET Waves Simulation)

3. Paper and pen/pencil (for recording measurements and calculations)

4. Calculator (optional, for calculations) What to do

1. Access the Simulation

a. Open your web browser and navigate to the PhET wave simulation tool.

b. Familiarise yourself with the interface and controls of the simulation.

2. Generate Sinusoidal Waves

a. Use the simulation to create sinusoidal waves by adjusting parameters such as amplitude, frequency, and wavelength.

b. Start with a default setting and observe how changing these parameters affects the wave.

3. Measure Wave Properties

a. Amplitude (A): Use the simulation’s tools to measure the maximum height of the wave from its rest position. Record this value.

b. Frequency (f): Determine how many complete cycles of the wave pass a point in one second. Record this value in Hertz (Hz).

c. Wavelength (λ): Measure the distance between two consecutive crests (or troughs) of the wave. Record this value in meters (m).

d. Period (T): Calculate the period of the wave using the formula:

T = 1/f where f is the frequency you measured.

4. Create a table in your notes to organize your measurements:

Wave Setting Amplitude (A) Frequency (f) Wavelength (λ) Period (T) 1 2 3

5. Use the recorded frequency and wavelength to calculate wave velocity

(vv) using the formula:

v = f λ Record your calculated values in your table.

6. Based on your measurements, write down the general progressive wave equation by substituting the measurements in the progressive wave equation below:

y = a sin (2πft ∓ 2πx___ λ )

Activity 7.14 Calculating amplitude, velocity, frequency, period, wavelength of a wave Study the worked examples below carefully before attempting the example questions that follow.

Worked Example 1

A wave travelling along a string is described by the equation:

y(x, t) = 0.04sin (8πt − 4πx), where y is in meters, t is in seconds, and x is in meters.

a. Determine the following:

b. Amplitude (a)

c. Angular frequency ω

d. Frequency f

e. Wavelength λ

f. Wave speed v

g. Wave number k

h. The direction of wave propagation

i. The displacement (y) of a particle at x = 0.5 m and t = 0.25 s.

Solution:

Wave Setting Amplitude (A) Frequency (f) Wavelength

(λ) Period (T) 1 2 3

Step 1: Write down the general wave equation and the given wave equation y(x, t) = asin(ωt − kx) y(x, t) = 0.04sin(8πt − 4πx),

Step 2: compare the equations, extract the necessary data and use it to calculate the parameters as follows:

a. Amplitude a = 0.04 m

b. Angular Frequency ω = 8π = 25.13 rad / s

c. Frequency 2πf = 8π, f = 8π___ 2π = 4.0 Hz

d. Wavelength k = 2π___ λ = 4π, λ = 2π___ 4π = 0.5 m

e. Wave Speed v = λf = 0.5 × 4 = 2.0 m / s

f. Wave Number k = 2π___ λ = 2π___ 0.5 = 12.57 rad / m

g. Displacement at x= 0.5 m and t = 0.25 s y(x, t) = 0.04sin(8πt − 4πx), y(x, t) = 0.04sin(8π(0.25) − 4π(0.5)), y(x, t) = 0.0 m

Worked Example 2

A sound wave propagating in air is described by the equation:

y(x, t) = 0.02sin(300t + 0.5x) Where y is in meters, t is in seconds, and x is in meters. Determine the following:

a. Amplitude (a)

b. Angular frequency (ω)

c. Frequency (f)

d. Wave number (k)

e. Wavelength (λ)

f. Wave speed (v)

g. The direction of wave propagation

h. The displacement (y) of a particle at x= 2 m and t = 0.01 s.

Solution

Step 1: Write down the general wave equation and the given wave equation y(x, t) = asin(ωt − kx) y(x, t) = 0.02sin(300t + 0.5x)

Step 2: compare the equations, extract the necessary data and use it to calculate the parameters as follows:

a. Amplitude a = 0.02 m

b. Angular Frequency ω = 300 rad / s

c. Frequency f = ω__ 2π= 300/2π = 47.75 Hz

d. Wave Number k = 0.5 rad / m

e. Wavelength λ = 2π___ k = 2π___ 0.5 = 12.57 m

f. Wave Speed v = λf = 12.57 × 47.75 = 600.0 m / s

g. Since we have a + sign in the brackets, the wave is in the negative x-direction

h. Displacement y at x = 2 m, t = 0.01 s y(2,0.01) = 0.02sin(300(0.01) + 0.5(2)) = − 0.0151 m Practice problems Now, using the worked example as a guide, solve the following problems individually or in groups.

1. The displacement of a particle in a progressive wave is described by:

y = 0.05sin(10πt − 2πx) Where y and x are in metres, t is in seconds.

a. What is the displacement of the particle at x = 2 m, t = 0.5 s ?

b. What is the amplitude of the wave?

2. A wave travelling along a string is represented by the equation: y = 0.1sin (50t − 4x) Where y and x are in metres, t is in seconds,

a. Find the angular frequency ω and wave number k from the equation.

b. Determine the frequency f of the wave.

c. Calculate the velocity v of the wave.

3. A sound wave has a wavelength of 2 m and a speed of 340 m/s.

a. Calculate the frequency of the wave.

b. Determine the angular frequency.

4. The displacement-time graph of a particle vibrating in a wave is sinusoidal, with a maximum displacement of 0.08 m and a period of 0.02 s.

a. What is the amplitude of the wave?

b. Calculate the angular frequency (ω) and frequency (f).

5. A snapshot of a wave travelling in the positive x-direction shows the following characteristics: The crest is located at x = 2 m with a maximum displacement of 0.05 m, the wavelength is 4 m, and the wave moves at a speed of 20 m/s.

a. Write the wave equation.

b. Determine the displacement of a particle at x = 1 m and t = 0.1 s.

6. A progressive wave is described by: y = 0.03sin(5t + 3x), y and x are in metres, t is in seconds

a. What is the wave’s amplitude?

b. What is the direction of propagation of the wave?

c. Find the angular frequency (ω) and wave number (k).

d. Calculate the frequency and velocity of the wave.

Production, Nature, and Transmission of Sound

Production of Sound

1. Sound is produced by vibrations.

2. When an object vibrates, it creates a disturbance in the surrounding medium (usually air).

3. This disturbance travels outwards in the form of waves, which we perceive as sound.

Examples:

1. Musical Instruments: In a guitar, plucking a string causes it to vibrate. The vibrations transfer to the air, producing sound waves that we hear as music.

2. Voice Production: In humans, the vocal cords vibrate as air passes through them, producing sound waves that form speech and singing.

3. Speakers: In a loudspeaker, an electrical signal causes a diaphragm to vibrate, creating sound waves that propagate through the air.

4. Vibrating tuning fork producing sound waves Nature of Sound

1. Sound is a longitudinal wave.

2. In a longitudinal wave, the particles of the medium vibrate parallel to the direction of wave propagation.

Figure 7.12a: Vibrating tuning fork producing sound waves

3. Sound waves consist of compressions (regions of high pressure) and rarefactions (regions of low pressure).

Figure 7.12b: Longitudinal wave showing compressions and rarefactions Transmission of Sound

1. Sound requires a medium to travel, such as air, water, or solids.

2. It cannot travel through a vacuum.

3. The speed of sound varies in different media. It is generally faster in solids than in liquids and gases.

Activity 7.15 Exploring the Speed of Sound in Different States of Matter Objective: engage in a group discussion to compare the speed of sound in solids, liquids, and gases.

Materials Needed

1. Access to textbooks or online resources about sound and its properties (if available)

2. Paper and pens/pencils (for taking notes)

3. Whiteboard or large paper (optional, for recording key points during discussions) What to do

1. Divide yourselves into small groups of 4-5 students. Each group will choose one state of matter from the states of matter below:

a. Solids

b. Liquids

c. Gases

2. Each group should spend 10-15 minutes researching and discussing the following points about their assigned state:

d. What is the approximate speed of sound in your assigned state of matter? (For reference, consider general values: approximately 5000 m/s for solids, 1500 m/s for liquids, and 343 m/s for gases.)

e. Why does sound travel faster in your assigned state compared to the others?

f. Discuss how molecular arrangement, density, and elasticity affect the speed of sound.

g. Think of real-world examples where the speed of sound is important in your assigned state (e.g., underwater communication for liquids, musical instruments for solids).

3. As a group, summarise your findings in a few key points. Write these down on paper.

4. After your discussion time is up, each group should share their findings with the class. One member from each group should present the key points you prepared.

5. After all groups have presented, engage in a class-wide discussion:

h. Compare and contrast the findings from each group.

i. Discuss how molecular structure and material properties influence the speed of sound.

j. Explore any questions or interesting insights that arose during your discussions.

6. Summarise the main points discussed regarding how sound behaves differently in solids, liquids, and gases. Highlight why understanding these differences is important in various fields such as engineering, acoustics, and environmental science.

Activity 7.16 Exploring Sound Transmission with a Tuning Fork Objective: Investigate how sound travels through different materials by using a tuning fork.

Materials Needed

1. Tuning fork

2. Rubber mallet (for striking the tuning fork)

3. Container filled with water (enough to submerge the tuning fork)

4. Metal object (e.g., a metal rod or a piece of metal sheet)

5. Paper and pen/pencil (for taking notes) What to do

1. Strike the Tuning Fork

a. Hold the tuning fork by its handle.

b. Use the rubber mallet to strike the tuning fork gently. Observe how the tines of the fork vibrate and produce sound.

2. Listen to Sound in Air

a. Stand a few feet away from the tuning fork after striking it and listen to the sound produced in air. Pay attention to how loud or clear the sound is.

3. Observe Vibrations

a. While holding the struck tuning fork, place your fingers lightly on the tines of the fork. Feel the vibrations as they resonate through your fingers.

4. Test Sound Transmission in Water

a. Carefully submerge the tines of the tuning fork into the container of water while keeping the handle above water.

b. Strike the tuning fork again while it is submerged and listen closely to how the sound changes compared to when it was in air.

c. Note how the vibrations feel through your fingers when they are submerged in water.

5. Test Sound Transmission in Metal

a. Next, strike the tuning fork again and then gently place one tine against a solid metal object (e.g., a metal rod or sheet).

b. Listen carefully for any differences in sound transmission through the metal compared to air and water.

c. You may also try placing your ear against the metal object while holding the tuning fork against it to hear how sound travels through solid materials.

6. Record Your Observations:

a. Create a table in your notes to summarise your observations about sound transmission through each medium:

Medium Sound Quality/Characteristics Observations on Vibrations

Air Water

Metal

7. After completing your observations, discuss with your classmates what you noticed about how sound travelled through different materials.

Consider questions such as:

a. How did the sound differ between air, water, and metal?

b. What do you think causes these differences in sound transmission?

Classification of Sound

Sound waves can be classified based on their frequency:

1. Infrasonic (Infrasound) Sound: Sound waves with frequencies below 20 Hz. Humans cannot hear these sounds. Infrasound. They can be produced by natural phenomena such as earthquakes, volcanic eruptions, and ocean waves, as well as by some animals for communication.

2. Audio sonic Sound: Sound waves with frequencies between 20 Hz and 20,000 Hz. Humans can hear these sounds.

3. Ultrasonic Sound: Sound waves with frequencies above 20,000 Hz. Humans cannot hear these sounds. Ultrasound is used in various applications, including medical imaging (ultrasound scans), cleaning, and non-destructive testing. Some animals, like bats and dolphins, use ultrasound for navigation and communication.

Activity 7.17 Frequency and detection of sound using a frequency generator or video Follow the link below to a video. In this video the frequency of a sound is slowly increased. When the video starts you should put your hand in the air and keep it raised until you can no longer hear the sound. Click here Discuss with your neighbour the factors which might affect peoples’ hearing and their ability to hear high pitched sounds.

Echo and Depth Determination

1. An echo is a reflected sound wave.

2. When a sound wave hits a hard surface, it bounces back and reaches our ears as an echo.

3. The time taken for the sound to travel to the surface and back can be used to determine the distance to the surface.

For an echo to be perceived as distinct from the original sound, there needs to be a minimum time delay, typically around 0.1 seconds. This corresponds to a minimum distance of about 17 meters (56 feet) between the source and the reflecting surface. If the distance is shorter, the reflected sound may blend with the original sound, making the echo indistinguishable.

Formula for calculating distance:

Distance = speed of sound × time delay/2

Activity 7.18 Creating and Measuring Echoes

Objective: create an echo by clapping your hands or using a whistle near a wall or in a large open space. You will listen for the reflected sound and use the speed of sound in air to calculate the distance to the reflecting surface.

Materials Needed

1. Open space with a wall or large flat surface (e.g., gymnasium, playground)

2. Stopwatch (or a timer on your phone)

3. Whistle (optional, if you prefer not to clap)

4. Paper and pen/pencil (for recording measurements and calculations) What to do

1. Locate an open space where you can safely make sounds without disturbing others. Ensure there is a wall or large flat surface nearby to create an echo.

2. Stand at a distance of about 10-15 meters from the wall. Clap your hands loudly or blow the whistle towards the wall. Make sure to face the wall while doing this.

3. After making the sound, listen carefully for the echo. The echo is the sound that reflects off the wall and returns to you.

4. Use a stopwatch or timer to measure the time it takes from when you make the sound until you hear the echo. Start the timer when you clap or whistle and stop it as soon as you hear the echo. Record this time in seconds.

5. Use the formula for calculating distance based on the speed of sound:

Distance = speed × time · The average speed of sound in air is approximately 343 meters per second (m/s). However, since the sound travels to the wall and back, you need to divide your recorded time by 2 before using it in your calculation:

Distance to wall = speed of sound × time delay/2

6. Substitute your values into the formula. If your recorded time is t seconds:

Distance to wall = 342 m / s × t/2

7. Create a table in your notes to organise your measurements:

Trial Time Recorded (s) Distance to Wall (m)

1 2 3

8. Repeat steps 2-6 for increasing distances to get different time recordings.

9. After completing your calculations, discuss with classmates what you observed about echoes. Consider questions such as:

a. How did changing your distance from the wall affect the echo?

b. Were there any differences in clarity or timing based on how loudly you clapped or whistled?

Activity 7.19 Exploring Echolocation and Its Applications

Objective: watch a video of a blind person using echolocation to navigate their environment and engage in a group discussion about how echolocation works, its applications in nature, and its relation to ultrasound, audio sound, and infrasound.

Materials Needed

1. Access to a computer or device with internet access

2. Access to the video: Human Echolocation - Daniel Kish (3:41 minutes)

3. Paper and pens/pencils (for taking notes)

4. Reference materials (textbooks or online resources) for definitions of ultrasound, audio sound, and infrasound (optional) What to do

1. Gather your group and watch the video titled “Human Echolocation - Daniel Kish.” Pay close attention to how Daniel uses echolocation to navigate his surroundings.

2. After watching the video, discuss the following questions with your group:

a. How does echolocation work? What sounds does Daniel make, and how do they help him “see” his environment?

b. What are some other applications of echolocation that you know of?

Can you think of any animals that use echolocation?

c. Discuss whether humans can hear the sounds made by animals that use echolocation.

3. Research and share examples of animals that use echolocation, such as bats, dolphins, and certain species of birds.

4. Discuss how these animals utilise echolocation for navigation and hunting.

5. As a group, research the following terms:

a. Ultrasound: Sound waves with frequencies above the audible range for humans (above 20 kHz).

b. Audio Sound: Sound waves within the audible range for humans (approximately 20 Hz to 20 kHz).

c. Infrasound: Sound waves with frequencies below the audible range for humans (below 20 Hz).

6. Discuss how these types of sound relate to echolocation and how they are used in various technologies (e.g., medical imaging with ultrasound).

7. Take notes during your discussion, recording key points about how echolocation works, its applications, and the definitions of ultrasound, audio sound, and infrasound.

8. Summarise your findings as a group. Reflect on what you learned about echolocation and its significance in both nature and technology.

Resonance and Speed of Sound

Resonance occurs when an object vibrates at its natural frequency due to an external force. This phenomenon can be used to determine the speed of sound in air.

Activity 7.20 Exploring Sound Vibrations Through Humming

Objective: To explore how sound vibrations are produced in the body by humming or making simple sounds. You will feel the vibrations in your throat and discuss how changes in pitch and volume affect these vibrations.

What to do

1. Sit or stand comfortably, ensuring you have enough space around you to focus on this activity.

2. Gently place your hands on your neck, specifically over your throat (oesophagus area). Make sure your fingers are lightly touching the skin.

3. Take a deep breath and hum or produce a simple sound like “ah” or “oo.”

Focus on the sensation of the vibrations as you make the sound.

4. Experiment by changing the pitch of your hum (higher or lower) while keeping your hands on your throat. Notice how the vibrations feel different.

5. Next, change the volume of your sound (louder or softer) and observe any changes in the intensity of the vibrations you feel.

6. After you have experimented with different sounds, take a moment to reflect on what you felt during the activity.

7. Gather with your classmates to discuss your observations. Consider the following questions:

a. What did you notice about the vibrations when you changed the sound?

i. Discuss how different pitches and volumes affected the intensity and quality of the vibrations you felt.

b. How do you think this relates to how we produce different sounds when speaking or singing?

i. Explore how our vocal cords and throat contribute to producing various sounds and how changes in pitch and volume occur.

8. As a group, summarise what you learned about sound vibrations and their relationship to speech and singing.

Distance = speed × time Distance to wall = speed of sound × time delay/2 Distance to wall = 342 m / s × t/2

Activity 7.21 Measuring the Speed of Sound in Air Using a Resonance Tube or Video Objective: Determine the speed of sound in air using a resonance tube experiment.

Materials Needed

If using a Resonance Tube

1. Resonance tube

2. Water (to fill the resonance tube)

3. Tuning forks of known frequencies (e.g., 256 Hz, 512 Hz)

4. Meter stick or measuring tape

5. Rubber pad (optional, for stability) If using a Video

6. Access to a computer or device with internet access

7. Video demonstrating the speed of sound using a resonance tube (e.g., Speed of Sound Experiment) What to do Option 1: Using a Resonance Tube

1. Set up the resonance tube. Ensure the resonance tube is vertical and stable.

Fill it partially with water, leaving enough air space at the top.

Figure 7.13: Experimental set up

2. Strike the Tuning Fork

a. Take one of the tuning forks and strike it against a rubber pad or hard surface to produce sound.

b. Hold the vibrating tuning fork just above the open end of the resonance tube.

3. Find Resonance

a. Slowly adjust the water level in the tube by adding or removing water until you hear the loudest sound (resonance). This indicates that the length of air column corresponds to a resonant frequency.

b. Measure and record the length of the air column at resonance (let’s call this l₁).

4. Repeat steps 2 and 3 for different tuning forks to find additional resonance lengths (l₂, l₃, etc.).

5. For each tuning fork, calculate the wavelength (λ) using:

v = 4 lₙ where lₙ is the length of the air column at resonance.

6. Calculate Speed of Sound

a. Use the formula:

v = f λ where f is the frequency of the tuning fork and v is the speed of sound.

b. Record your calculations in a table:

Tuning Fork

Frequency (Hz) Length of Air

Column (m) Wavelength

(m) Speed of Sound

(m/s) Option 2: Watching a Video

1. Watch the Video

a. Find and watch a video demonstrating how to measure the speed of sound using a resonance tube.

b. Pay attention to how they set up the experiment, find resonance, and measure distances.

2. While watching, take notes on key steps shown in the video, including how they calculate wavelength and speed.

3. Calculate Speed of Sound Based on Observations

a. If specific frequencies and lengths are mentioned in the video, use those values to calculate speed using:

v = f λ

Activity 7.22 Exploring Sound Production with Musical Instruments

Objective: Explore how sound is produced using various local musical instruments.

Materials Needed

1. Access to local musical instruments (e.g., drums, percussion instruments, flutes, horns, stringed instruments) available within your community or at home

2. Paper and pens/pencils (for taking notes)

3. Space to play the instruments comfortably What to do

1. Organise yourselves into groups of no more than five. Discuss among your group members and select a variety of local musical instruments available.

Each group should aim to include at least one of each type: percussion (e.g., drums), wind (e.g., flute, horns), and stringed instruments.

2. Each group should take turns exploring how each one produces sound.

For each instrument, consider the following:

a. Drums/Percussion

i. How is sound produced? (e.g., by striking the surface)

ii. What happens to the drumhead when it is hit?

b. Flute/Horns

i. How is sound produced? (e.g., by blowing air through the instrument)

ii. How does changing your breath or finger placement affect the pitch?

c. Stringed Instruments

i. How is sound produced? (e.g., by plucking or bowing the strings)

ii. How does tension or length of the string affect the sound?

3. Each group member should take turns demonstrating how their chosen instrument produces sound. Pay attention to how the instrument works and what changes occur when you modify your playing technique.

4. After each demonstration, discuss as a group:

a. What mechanisms are involved in producing sound for each type of instrument?

b. How do different materials and shapes of the instruments influence the sound quality?

5. As a group, discuss how sound travels as a longitudinal wave:

a. In longitudinal waves, particles of the medium move parallel to the direction of wave propagation.

b. Discuss how this relates to sound traveling through air (or other media) when instruments are played.

c. Explore examples of how sound waves compress and rarefy as they travel.

6. Take notes on your findings regarding how different instruments produce sound and how this relates to the nature of sound waves.

1. Halliday, D., Resnick, R., & Walker, J. (2018). Fundamentals of Physics (11th ed.). Wiley. Comprehensive resource on wave motion and sound.

2. Serway, R. A., & Jewett, J. W. (2019). Physics for Scientists and Engineers with Modern Physics (10th ed.). Cengage Learning. Detailed explanation of wave properties and equations.

3. Tipler, P. A., & Mosca, G. (2022). Physics for Scientists and Engineers (7th ed.). Macmillan. Insights on sound waves and resonance phenomena.

4. Hecht, E. (2016). Optics (5th ed.). Pearson. Explores wave behavior like diffraction, reflection, and interference.

5. Giancoli, D. C. (2015). Physics: Principles with Applications (7th ed.).

Pearson. Simplified explanations suitable for introductory wave concepts.

6. Young, H. D., & Freedman, R. A. (2020). University Physics with Modern Physics (15th ed.). Pearson. Advanced wave mechanics and sound wave applications.

7. Walker, J. S. (2019). Physics (6th ed.). Pearson. Discussion on wave propagation and sound properties.

Review questions 7.1

1. Define the following terms:

a. Amplitude

b. Frequency

c. Period

d. Wavelength

2. A wave is described by the equation: y(x, t) = 0.05sin(400t − 2x). Using this equation, determine the following parameters of the wave:

a. The amplitude of the wave.

b. Angular frequency (ω).

c. Wave number (k).

d. Wavelength (λ).

e. Frequency (f).

f. Period (T).

g. Wave velocity (v).

h. The direction of wave propagation.

3. The amplitude of a transverse wave on a string is 0.02 m, and its frequency is 50 Hz. The speed of the wave is 10 m/s.

a. Find the wavelength of the wave.

b. Write the wave equation.

c. Calculate the displacement at x = 3 m and t = 0.2 s.

d. Sketch the waveform on a

i. displacement-time graph

ii. displacement-position graph

Review questions 7.2

1. State the range of frequencies for infrasound, audible sound, and ultrasound.

a. Infrasound

b. Audible sound

c. Ultrasound

2. Define resonance in the context of sound waves.

3. Why can sound not travel through a vacuum?

4. In what way does the resonance method provide an accurate determination of the speed of sound in air?