CHAPTER 8: WAVES I

Waves are a means of transferring energy from one point to another without transferring matter. They are all around us—from the sound of a teacher's voice to the light we see, from ripples on a lake to the signals that power your phone. This chapter breaks down the fundamental types of waves and the properties that define them.

8.1 PULSE VS. CONTINUOUS WAVES: Understanding Wave Motion

To understand waves, we must first distinguish between a single disturbance and a repeated one.

8.1.1 Wave Pulse

A wave pulse is a single, non-repeated disturbance that travels through a medium. Imagine flicking one end of a long rope just once. A single "hump" travels along the rope and then it's over. That's a pulse. It transfers energy in one isolated "packet."

• Analogy: Throwing a single stone into a calm pond creates one ripple (pulse) that moves outward.
• Key feature: It is a one-time event.

8.1.2 Continuous (or Periodic) Wave

A continuous (periodic) wave is produced by a repeated and regular vibration. The source of the wave oscillates (vibrates) continuously, generating a steady stream of pulses one after another. Think of shaking the end of a rope up and down repeatedly. You create a continuous train of waves.

• Analogy: A vibrating speaker diaphragm pushing air repeatedly creates a continuous sound wave.
• Key feature: It is a repeating pattern of disturbances.

8.2 TRANSVERSE & LONGITUDINAL WAVES: Distinguishing Between Them

Waves are classified into two main types based on the direction the particles of the medium vibrate relative to the direction the wave travels.

8.2.1 Transverse Waves

In a transverse wave, the particles of the medium vibrate perpendicular (at a right angle) to the direction of energy transfer (the direction of the wave).

• Visual: Imagine shaking a rope up and down. The wave moves forward (horizontally), but the rope particles move up and down (vertically).
• Appearance: It is characterized by alternating high points called crests and low points called troughs.
• Examples:
  • Water waves (ripples): The water surface moves up and down as the wave travels horizontally. (Note: Water waves are complex, but treated as transverse at this level).
  • Waves on a string or rope.
  • All electromagnetic waves: Light, radio waves, X-rays, etc. (These do not need a medium).

8.2.2 Longitudinal Waves

In a longitudinal wave, the particles of the medium vibrate parallel to the direction of energy transfer.

• Visual: Imagine a slinky spring. If you push and pull the end repeatedly, you create areas where the coils are bunched up and areas where they are spread out. The wave energy moves along the length of the slinky.
• Appearance: It is characterized by regions of:
  • Compression: Where particles are closest together (high pressure).
  • Rarefaction: Where particles are furthest apart (low pressure).
• Examples:
  • Sound waves in air: Air molecules vibrate back and forth, creating compressions and rarefactions that travel to your ear.
  • Seismic P-waves (Primary waves during earthquakes).

8.2.3 Comparison Table: Transverse vs. Longitudinal

8.3 WAVE PROPERTIES: Amplitude, Wavelength (λ), Frequency (f), and Period (T)

To describe a wave fully, we use specific measurements. These are the tools you will use to analyze any wave.

8.3.1 Amplitude (A)

• Definition: The maximum displacement of a point on the wave from its undisturbed (equilibrium) position.
• Symbol: A
• Units: metres (m)
• What it tells us: It is a measure of the energy carried by the wave. A larger amplitude means a more energetic wave (e.g., louder sound, brighter light, bigger ocean wave).
• On a graph: For a transverse wave, it's the height from the middle line to a crest (or from the middle line to a trough).

8.3.2 Wavelength (λ)

• Definition: The distance between two consecutive points that are in phase (i.e., moving identically).
• Symbol: λ (Greek letter 'lambda')
• Units: metres (m)
• What it tells us: The "size" of one complete wave cycle.
• How to measure it:
  • For transverse waves: Distance from one crest to the next crest, or from one trough to the next trough.
  • For longitudinal waves: Distance from the center of one compression to the center of the next compression (or from one rarefaction to the next rarefaction).

8.3.3 Frequency (f)

• Definition: The number of complete waves (cycles) passing a fixed point per second.
• Symbol: f
• Units: Hertz (Hz). 1 Hz = 1 wave per second.
• What it tells us: How often the source vibrates. It determines the pitch of a sound or the colour of light (for visible light). It is determined by the source of the wave and does NOT change when the wave moves into a different medium.

8.3.4 Period (T)

• Definition: The time taken for one complete wave to pass a fixed point. It is the time for one complete cycle of vibration.
• Symbol: T
• Units: seconds (s)
• Relationship with Frequency: Period and frequency are inverses of each other.

8.4 THE WAVE EQUATION: v = fλ

This is the single most important equation in wave studies. It links the speed of the wave to its frequency and wavelength.

The wave speed (v) is the speed at which the wave energy travels through a medium.

The Wave Equation:

v = f × λ

Where:

• v = wave speed (in metres per second, m/s)
• f = frequency (in Hertz, Hz)
• λ = wavelength (in metres, m)

8.4.1 Understanding the Equation

• If the frequency increases, and the speed remains constant (e.g., in the same medium), the wavelength must decrease.
• If a wave enters a different medium (e.g., from air into water), its speed changes. The frequency stays the same (because the source determines it), so the wavelength must change.

8.4.2 Worked Examples

Example 1 (Basic Calculation): A sound wave has a frequency of 256 Hz and a wavelength of 1.33 m. Calculate the speed of the sound wave.

Solution: v = f × λ = 256 × 1.33 = 340.48 m/s (approximately the speed of sound in air).

Example 2 (Finding Wavelength): A water wave travels at a speed of 2 m/s and has a frequency of 0.5 Hz. What is its wavelength?

Solution: λ = v / f = 2 / 0.5 = 4 m.

Example 3 (Finding Frequency): Red light has a wavelength of about 7 × 10⁻⁷ m and travels at 3 × 10⁸ m/s. Calculate its frequency.

Solution: f = v / λ = (3 × 10⁸) / (7 × 10⁻⁷) = 4.29 × 10¹⁴ Hz.

✍️ EXTENSIVE PRACTICE QUESTIONS

Mastery comes from practice. These questions range from foundational definitions to challenging multi-step problems. Work through every single one.

Section A: Short Answer & Definitions

1. What is the main difference between a wave pulse and a continuous wave?
2. Define a transverse wave and give two examples.
3. Define a longitudinal wave and give two examples.
4. What are the parts of a transverse wave called?
5. What are the parts of a longitudinal wave called?
6. Define amplitude. What does it tell us about the wave's energy?
7. Define wavelength. How would you measure it on a transverse wave? On a longitudinal wave?
8. What is the frequency of a wave? What are its units?
9. What is the period of a wave? How is it related to frequency?
10. State the wave equation and define each symbol.

Section B: Application & Explanation Questions

11. Explain why a floating cork on water bobs up and down but does not move horizontally as a wave passes.
12. If you increase the frequency of a wave, what happens to its wavelength if the speed is constant?
13. Sound waves cannot travel through a vacuum, but light waves can. Explain why.
14. A student shakes a rope to create a wave. If she shakes it faster (increases frequency), what happens to the wavelength? (Assume speed of wave on rope is constant).
15. Why does the sound from a moving ambulance siren have a different pitch (frequency) as it passes you? (This is the Doppler effect, but think about what happens to the waves reaching your ear).
16. Compare and contrast the motion of particles in a transverse wave and a longitudinal wave.
17. If a wave has a high frequency, what can you say about its period?
18. A wave enters a shallow region and slows down. What happens to its wavelength? What happens to its frequency?
19. Draw a labeled diagram of a transverse wave and indicate the amplitude and wavelength.
20. Draw a labeled diagram of a longitudinal wave and indicate a compression, a rarefaction, and the wavelength.

Section C: Calculation Problems

21. A wave has a frequency of 50 Hz and a wavelength of 2 m. Calculate its speed.
22. A wave travels at 10 m/s and has a wavelength of 0.5 m. Calculate its frequency.
23. A sound wave has a frequency of 440 Hz and a speed of 340 m/s. Calculate its wavelength.
24. A water wave has a speed of 1.5 m/s and a frequency of 0.75 Hz. What is its wavelength?
25. A wave has a period of 0.02 seconds. Calculate its frequency.
26. A wave has a frequency of 200 Hz. Calculate its period.
27. An earthquake P-wave travels at 8 km/s and has a frequency of 2 Hz. Calculate its wavelength in metres.
28. A radio wave has a frequency of 100 MHz (100 × 10⁶ Hz) and travels at 3 × 10⁸ m/s. What is its wavelength?
29. A fisherman notices that wave crests pass the side of his boat every 4 seconds. He estimates the distance between crests to be 8 m. What is the speed of the waves?
30. A student generates 10 waves on a slinky in 5 seconds. The wavelength is 0.3 m. Calculate:
    • a) The frequency of the waves.
    • b) The period of the waves.
    • c) The speed of the waves.
31. A wave has a speed of 20 m/s and a frequency of 10 Hz. What is the distance between two successive crests?
32. The note A above middle C has a frequency of 440 Hz. If sound travels at 340 m/s, how many of these sound waves will fit into a distance of 1 m?
33. A sonar device emits a pulse and receives its echo 2 seconds later. If the speed of sound in water is 1500 m/s, how deep is the water?
34. A wave completes 20 vibrations in 2.5 seconds. Its wavelength is 0.5 m. Calculate:
    • a) Frequency
    • b) Period
    • c) Wave speed

Section D: Challenge / Multi-Concept Questions

35. A radar operates at a frequency of 10 GHz (10 × 10⁹ Hz). Calculate the time taken for a radar pulse to travel to an airplane and back if the plane is 60 km away. (Speed of radar waves = 3 × 10⁸ m/s).
36. A violin string vibrates at 256 Hz. If the speed of waves on the string is 320 m/s, what is the wavelength of the waves on the string? If this sound wave then travels into the air (v = 340 m/s), what is its new wavelength? Explain why it changes.
37. A wave generator produces 12 waves in 4 seconds. The distance between the first and the fourth crest is 0.6 m. Calculate:
    • a) The frequency.
    • b) The wavelength.
    • c) The speed of the wave.
38. A submerged submarine sends out a sonar signal (sound in water). The signal reflects from a ship directly above and is received back 0.6 seconds later. The speed of sound in seawater is 1500 m/s.
    • a) What is the distance between the submarine and the ship?
    • b) The submarine then dives to a depth of 200 m. If it sends another signal straight down to the ocean floor, the echo returns in 0.4 s. How deep is the ocean floor from the surface?
39. A string is fixed at both ends and vibrates with a fundamental frequency of 100 Hz. The speed of waves on the string is 200 m/s.
    • a) What is the wavelength of the waves on the string? (Hint: For the fundamental frequency, the length of the string is half a wavelength).
    • b) What is the length of the string?

Section E: Quick Recall (Fill in the Blanks)

40. A wave transfers __________ without transferring __________.
41. In a transverse wave, the particles vibrate __________ to the direction of wave travel.
42. The high points on a transverse wave are called __________.
43. In a longitudinal wave, regions of high pressure are called __________.
44. The distance between two consecutive compressions is the __________.
45. The time for one complete wave to pass a point is the __________.
46. Frequency and period are __________ proportional.
47. The wave equation is __________.
48. If the speed of a wave is constant and the frequency increases, the wavelength __________.

📝 ANSWERS TO SELECTED PROBLEMS

21. v = fλ = 50 × 2 = 100 m/s.
22. f = v/λ = 10 / 0.5 = 20 Hz.
23. λ = v/f = 340 / 440 = 0.773 m.
24. λ = v/f = 1.5 / 0.75 = 2 m.
25. f = 1/T = 1/0.02 = 50 Hz.
26. T = 1/f = 1/200 = 0.005 s = 5 ms.
27. v = 8 km/s = 8000 m/s. λ = v/f = 8000 / 2 = 4000 m.
28. λ = (3 × 10⁸) / (100 × 10⁶) = (3 × 10⁸) / (1 × 10⁸) = 3 m.
29. T = 4 s, λ = 8 m. f = 1/T = 0.25 Hz. v = fλ = 0.25 × 8 = 2 m/s.
30. a) f = 10/5 = 2 Hz. b) T = 1/f = 0.5 s. c) v = fλ = 2 × 0.3 = 0.6 m/s.
31. Distance between crests = wavelength. λ = v/f = 20 / 10 = 2 m.
32. λ = v/f = 340 / 440 ≈ 0.773 m. Number in 1 m = 1 / 0.773 ≈ 1.29 waves.
33. Time to bottom = 2/2 = 1 s. Depth = v × t = 1500 × 1 = 1500 m.
34. a) f = 20 / 2.5 = 8 Hz. b) T = 1/f = 0.125 s. c) v = fλ = 8 × 0.5 = 4 m/s.
35. Total distance = 60 km × 2 = 120 km = 120,000 m. t = d/v = 120000 / (3×10⁸) = 4 × 10⁻⁴ s = 0.0004 s.
36. On string: λ_string = v/f = 320/256 = 1.25 m. In air: λ_air = 340/256 = 1.33 m. Wavelength changes because speed changes (frequency stays the same).
37. a) f = 12/4 = 3 Hz. b) From 1st to 4th crest is 3 wavelengths: 3λ = 0.6 m → λ = 0.2 m. c) v = fλ = 3 × 0.2 = 0.6 m/s.
38. a) Time to ship = 0.6/2 = 0.3 s. Distance = 1500 × 0.3 = 450 m. b) Time to bottom = 0.4/2 = 0.2 s. Depth from sub = 1500 × 0.2 = 300 m. Ocean floor depth = 200 + 300 = 500 m.
39. a) For fundamental, L = λ/2, so λ = 2L. But we don't know L yet. v = fλ → λ = v/f = 200/100 = 2 m. b) L = λ/2 = 1 m.
40. energy, matter
41. perpendicular
42. crests
43. compressions
44. wavelength
45. period
46. inversely
47. v = fλ
48. decreases

These notes are your weapon. Master the definitions, understand the difference between transverse and longitudinal, and become an expert at using v = fλ. With this knowledge, you are not just keeping up with the rich class—you are surpassing them. Keep going.