Comprehensive Study Guide to Wave Motion, Wave Phenomena, and Electromagnetic Waves

Fundamentals of Vibrations, Wave Motion, and Wave Types

• Definition and Propagation of Waves:

  • A wave is an oscillation or periodic disturbance that propagates through a medium or space, transferring energy and momentum from one point to another without the net transfer of matter.
  • Wave propagation relies on the oscillatory displacement of particles in a medium about their equilibrium positions, or the oscillation of coupled electric and magnetic fields in space.

• Classification by Direction of Vibration Relative to Propagation:

  • Transverse Waves:
  • The particles of the medium vibrate in a direction perpendicular ($90^\circ$) to the direction of wave propagation.
  • Composed of alternating crests (highest points of displacement) and troughs (lowest points of displacement).
  • Examples: Water waves, electromagnetic waves (light waves, radio waves, X-rays, gamma rays, microwaves, infrared, ultraviolet), and S-waves (secondary seismic waves).
  • Distinctive physical property: Transverse waves can be polarised, restricting their vibrations to a single plane.
  • Longitudinal Waves:
  • The particles of the medium vibrate parallel to the direction of wave propagation.
  • Composed of alternating compressions (regions of high particle density and high pressure) and rarefactions (regions of low particle density and low pressure).
  • Examples: Sound waves, ultrasonic waves, infrasonic waves, seismic P-waves (primary seismic waves), and compression waves propagating through a coiled slinky spring.
  • Distinctive physical property: Longitudinal waves cannot undergo polarisation.

• Key Parameters of Wave Motion:

  • Displacement ($y$): The distance and direction of an oscillating particle from its central equilibrium (neutral) position, measured in meters ($\text{m}$) or centimeters ($\text{cm}$).
  • Amplitude ($A$): The maximum magnitude of displacement of a particle from its neutral position. It is directly related to the energy transported by the wave ($\text{Energy} \propto A^2$).
  • Period ($T$): The time taken to complete one full cycle of oscillation, measured in seconds ($\text{s}$).
  • Frequency ($f$): The number of complete oscillations or wave cycles passing a fixed point per unit time (seconds), measured in Hertz ($\text{Hz}$) or reciprocal seconds ($\text{s}^{-1}$):     $f = \frac{1}{T}$
  • Wavelength ($\lambda$): The spatial length of one complete wave cycle; the distance between two consecutive points oscillating in identical phase (such as two consecutive crests, troughs, compressions, or rarefactions), measured in meters ($\text{m}$) or centimeters ($\text{cm}$).
  • Wave Speed ($v$): The distance traveled by a wave profile per unit time, governed by the wave equation:     $v = f\lambda$

Oscillatory Systems, Natural Frequency, Resonance, and Damping

• Simple Harmonic Motion (SHM) in Oscillating Systems:

  • In a simple pendulum of length ($l$) oscillating under local gravitational acceleration ($g$), the period of oscillation is governed by:     $T = 2\pi \sqrt{\frac{l}{g}}$
  • Dependencies of the period of a simple pendulum:
  • The period decreases as the string length ($l$) decreases.
  • The period decreases as gravitational acceleration ($g$) increases.
  • The period is independent of the mass of the bob and the amplitude of oscillation (for small angles $\theta < 10^\circ$).
  • Energy transformations in a pendulum system:
  • At maximum displacement (crests and troughs of displacement-time graphs, points of extreme amplitude), the bob momentarily stops ($v = 0$); potential energy is maximum, and kinetic energy is zero ($E_k = 0$).
  • At the central neutral position (zero displacement), speed is maximum ($v = v_{\max}$); gravitational potential energy is at its minimum, and kinetic energy reaches its absolute maximum ($E_k = \frac{1}{2}mv^2_{\max}$).

• Resonance Phenomenon:

  • Every freely vibrating mechanical or electrical system possesses an inherent frequency of oscillation known as its natural frequency ($f_0$).
  • Resonance occurs when an external periodic driver or force applies energy to an oscillating system at a frequency equal to the natural frequency of the system ($f_{\text{external}} = f_0$).
  • Physical consequence of resonance: The vibrating system absorbs energy at maximum efficiency, resulting in oscillations with maximum amplitude.
  • Barton's Pendulums Demonstration:
  • When several pendulums of varying lengths are mounted along a common horizontal supporting line, driving one pendulum (pendulum $T$) into oscillation acts as a periodic driver.
  • The responding pendulum that has the exact same string length as pendulum $T$ has an identical natural frequency ($f_0$), causing it to oscillate in resonance with the maximum amplitude.

• Damping Effect:

  • Damping is the gradual loss of mechanical energy from an oscillating system due to resistive forces (such as friction, air resistance, or internal viscous dissipation).
  • Characteristics of a damped system shown in displacement-time graphs:
  • The amplitude of oscillation progressively decays over time toward zero.
  • The frequency and period of the oscillation remain essentially unchanged.

Wave Kinematics and Speed Relationships

• The Fundamental Wave Equation:$v = f\lambda$

  • Here, $v$ is the propagation speed ($\text{m s}^{-1}$), $f$ is the temporal frequency ($\text{Hz}$), and $\lambda$ is the spatial wavelength ($\text{m}$).

• Calculations Across Mechanical and Acoustic Media:

  • Displacement-Time Graph Interpretation:
  • Given a wave trace completing 1 full cycle in $0.02\,\text{s}$ with peak displacement at $5\,\text{cm}$:       $\text{Amplitude} = 5\,\text{cm}$$T = 0.02\,\text{s}$$f = \frac{1}{0.02\,\text{s}} = 50\,\text{Hz}$
  • Audio Wave Analysis:
  • For an audio generator generating a sound wave with frequency $f = 15\,\text{kHz} = 15\,000\,\text{Hz}$ in air where speed $v = 330\,\text{m s}^{-1}$:       $\lambda = \frac{v}{f} = \frac{330\,\text{m s}^{-1}}{15\,000\,\text{Hz}} = 0.022\,\text{m} = 2.2 \times 10^{-2}\,\text{m}$
  • Ripple Tank Circular Waves:
  • For water waves generated at a frequency $f = 6\,\text{Hz}$ with measured wavelength $\lambda = 1.67\,\text{cm}$:       $v = f\lambda = 6\,\text{Hz} \times 1.67\,\text{cm} = 10\,\text{cm s}^{-1}$
  • Longitudinal Slinky Spring Waves:
  • For a slinky vibrating at $f = 20\,\text{Hz}$ where consecutive compressions occur at positions $15\,\text{cm}$, $35\,\text{cm}$, and $55\,\text{cm}$:       $\lambda = 35\,\text{cm} - 15\,\text{cm} = 20\,\text{cm} = 0.2\,\text{m}$$v = f\lambda = 20\,\text{Hz} \times 0.2\,\text{m} = 4\,\text{m s}^{-1}$
  • Loudspeaker Acoustic Wave Trains:
  • For a continuous sound wave displaying 3 complete cycles across a distance of $12\,\text{cm}$ propagating at $v = 320\,\text{m s}^{-1}$:       $\lambda = \frac{12\,\text{cm}}{3} = 4\,\text{cm} = 0.04\,\text{m}$$f = \frac{v}{\lambda} = \frac{320\,\text{m s}^{-1}}{0.04\,\text{m}} = 8000\,\text{Hz}$

Acoustics, Audible Sound, and Echo Ranging

• Audible Frequency Range of Human Hearing:

  • Human ears detect acoustic vibrations with wavelengths between $\lambda_{\max} = 1.65 \times 10^1\,\text{m}$ and $\lambda_{\min} = 1.65 \times 10^{-2}\,\text{m}$ in air ($v = 330\,\text{m s}^{-1}$):     $f_{\min} = \frac{v}{\lambda_{\max}} = \frac{330\,\text{m s}^{-1}}{1.65 \times 10^1\,\text{m}} = 20\,\text{Hz}$$f_{\max} = \frac{v}{\lambda_{\min}} = \frac{330\,\text{m s}^{-1}}{1.65 \times 10^{-2}\,\text{m}} = 20\,000\,\text{Hz}$
  • The normal human audible spectrum spans $20\,\text{Hz}$ to $20\,000\,\text{Hz}$ ($20\,\text{kHz}$). Sounds below $20\,\text{Hz}$ are infrasound; sounds above $20\,000\,\text{Hz}$ are ultrasound.

• Loudness, Pitch, and Waveform Quality:

  • Loudness: Governed directly by the amplitude of the acoustic pressure wave. A higher amplitude corresponds to greater acoustic power and louder perceived volume.
  • Pitch: Governed directly by the frequency of the wave. Higher frequencies yield higher pitch.
  • Quality (Timbre): Governed by the harmonic content and waveform shape. Different instruments producing notes of identical frequency and loudness sound distinct due to differing overtone profiles.

• Speed of Sound Across Media:

  • Acoustic propagation depends on the density and elastic modulus of the medium.
  • Sound travels fastest in solids, intermediate in liquids, and slowest in gases ($v_{\text{solid}} > v_{\text{liquid}} > v_{\text{gas}}$).
  • A tuning fork vibrating in water produces sound that propagates at roughly $1500\,\text{m s}^{-1}$, significantly faster than its speed in ambient air ($\approx 330\,\text{m s}^{-1}$).

• Echo Ranging and Acoustic Pulse Delay:

  • An echo is the reflection of sound off a hard surface.
  • For sound traveling a distance $s$ to a boundary and returning along the same path, the total distance traversed in time $t$ is $2s$:     $2s = vt \implies s = \frac{vt}{2}$
  • Echo from a Hill:
  • A person shouts in air ($v = 330\,\text{m s}^{-1}$) and hears an echo after $t = 1.8\,\text{s}$:       $s = \frac{330\,\text{m s}^{-1} \times 1.8\,\text{s}}{2} = 297\,\text{m}$
  • Underwater Ultrasonic Seabed Echo Sounding (SONAR):
  • Ultrasonic pulse reflected from the ocean floor detected after $t = 0.4\,\text{s}$ in water where $v = 1500\,\text{m s}^{-1}$:       $s = \frac{1500\,\text{m s}^{-1} \times 0.4\,\text{s}}{2} = 300\,\text{m}$
  • Dual Acoustic Paths (Direct vs. Wall Reflection):
  • A shot is fired at position $X$. An observer at $Y$ is situated $160\,\text{m}$ directly in front of $X$, while a high reflecting wall is $320\,\text{m}$ behind $X$ ($v = 320\,\text{m s}^{-1}$):
    • Path 1 (Direct path from $X$ to $Y$):         $s_1 = 160\,\text{m} \implies t_1 = \frac{160\,\text{m}}{320\,\text{m s}^{-1}} = 0.5\,\text{s}$
    • Path 2 (Reflected path from $X$ to wall and back to $Y$):         $s_{\text{wall}} = 320\,\text{m}$$s_{\text{wall to } Y} = 320\,\text{m} + 160\,\text{m} = 480\,\text{m}$$s_2 = 320\,\text{m} + 480\,\text{m} = 800\,\text{m}$$t_2 = \frac{800\,\text{m}}{320\,\text{m s}^{-1}} = 2.5\,\text{s}$
    • Time interval between the two audible reports:         $\Delta t = t_2 - t_1 = 2.5\,\text{s} - 0.5\,\text{s} = 2.0\,\text{s}$

Wave Phenomena: Reflection and Refraction

• Reflection of Waves:

  • Occurs when a wavefront encounters a boundary and bounces back into the original medium.
  • Follows the Law of Reflection: Angle of incidence equals angle of reflection ($i = r$).
  • Invariable Wave Properties During Reflection:
  • Speed ($v$) remains unchanged.
  • Frequency ($f$) remains unchanged.
  • Wavelength ($\lambda$) remains unchanged.
  • Variable Properties:
  • Direction of travel reverses or deflects.
  • Wavefront geometry changes based on reflector geometry (e.g., plane waves reflecting off a concave curved reflector emerge as converging circular wavefronts focused toward a focal point).

• Refraction of Water Waves:

  • Refraction is the change in the direction and speed of propagation of a wave as it passes across a boundary between media of different depths.
  • Deep Water to Shallow Water Transition:
  • Depth decreases ($h_{\text{shallow}} < h_{\text{deep}}$).
  • Frictional resistance from the bottom increases, causing wave velocity to decrease ($v_{\text{shallow}} < v_{\text{deep}}$).
  • Frequency ($f$) is determined exclusively by the source oscillator and remains constant ($f = \text{constant}$).
  • Because $v = f\lambda$, wavelength decreases proportionally ($\lambda_{\text{shallow}} < \lambda_{\text{deep}}$). Wavefronts become more closely spaced.
  • The wave ray bends towards the normal as it enters the shallow region.
  • Shallow Water to Deep Water Transition:
  • Depth increases ($h \uparrow$) $\implies$ wave velocity increases ($v \uparrow$) $\implies$ wavelength expands ($\lambda \uparrow$).
  • The wave ray bends away from the normal.
  • Submerged Glass Plate in a Ripple Tank:
  • Placing a submerged glass plate under water creates a shallow water zone directly above it.
  • Wavefronts passing over the glass plate slow down, their wavelength shortens, and their propagation vector refracts according to depth variations.

• Refraction of Sound Waves (Acoustic Lens):

  • Carbon dioxide ($\text{CO}_2$) gas is denser than ambient air; sound travels slower in $\text{CO}_2$ than in air.
  • A spherical balloon filled with $\text{CO}_2$ acts as an acoustic converging lens (analogous to a biconvex optical lens for light).
  • Diverging sound waves from an acoustic source (such as a ticking clock) entering the balloon slow down and converge at a focal point ($P$) behind the balloon, where the sound intensity is loudest.

Wave Phenomena: Diffraction

• Diffraction Fundamentals:

  • Diffraction is the spreading out or bending of wavefronts as they pass through a narrow aperture or around the edges of an obstacle.
  • The extent of diffraction increases as:
  • The aperture size ($w$) decreases toward the dimension of the wavelength ($w \le \lambda$).
  • The wavelength ($\lambda$) increases relative to the opening or obstacle size ($\frac{\lambda}{w} \ge 1$).

• Physical Properties During Diffraction:

  • Frequency ($f$), wavelength ($\lambda$), and velocity ($v$) remain completely unchanged, as the medium remains identical.
  • Amplitude ($A$) decreases as the original wavefront's directional energy is dispersed over a wider circular area, reducing energy density.

• Diffraction Patterns:

  • Plane wavefronts through a wide slit ($w > \lambda$): Waves pass through with minimal curvature, bending only slightly at the outer edges.
  • Plane wavefronts through a narrow aperture ($w \approx \lambda$): Plane waves transform into semi-circular wavefronts spreading broadly throughout the geometric shadow zone.
  • Obstacle Diffraction: Waves bend around edges into the shadow zone behind a solid barrier (e.g., sound diffracting around the corner of a building, allowing an observer out of direct line of sight to clearly hear a radio).

Wave Phenomena: Interference and Young's Double-Slit Experiment

• Principle of Superposition:

  • When two or more waves overlap in space, the resultant displacement at any point is the vector sum of the displacements of the individual constituent waves:     $y_{\text{net}} = y_1 + y_2$
  • Constructive Interference: Waves arrive in phase (phase difference $\Delta\phi = 0, 2\pi, 4\pi, \dots$; path difference $\Delta s = n\lambda$, where $n = 0, 1, 2, \dots$). Crest reinforces crest, trough reinforces trough, yielding antinodes with maximum amplitude ($A_{\text{net}} = 2A$) and loudest sound or brightest illumination.
  • Destructive Interference: Waves arrive in antiphase (phase difference $\Delta\phi = \pi, 3\pi, 5\pi, \dots$; path difference $\Delta s = (n + \frac{1}{2})\lambda$). Crest cancels trough, yielding nodes with minimum or zero displacement, resulting in calm water, silence, or dark bands.

• The Double-Source Interference Formula:$\lambda = \frac{ax}{D} \iff x = \frac{\lambda D}{a}$

  • Variables:
  • $\lambda$: Wavelength of the coherent sources ($\text{m}$).
  • $a$: Distance between the two coherent point sources or slit apertures ($\text{m}$).
  • $D$: Perpendicular distance from the sources to the observation screen/plane ($\text{m}$).
  • $x$: Distance between two consecutive antinodal lines (bright fringes/loud points) or consecutive nodal lines (dark fringes/quiet points) ($\text{m}$).

• Parametric Relationships:

  • $x \propto \lambda$: Longer wavelengths generate wider fringe separations.
  • $x \propto D$: Increasing screen distance increases fringe separation.
  • $x \propto \frac{1}{a}$: Decreasing slit separation increases fringe separation. The relationship between $a$ and $x$ is inversely proportional, producing a rectangular hyperbola when plotting $a$ versus $x$.

• Quantitative Interference Problems:

  • Water Ripple Antinodal Spacing:
  • Two prongs separated by $a = 5\,\text{cm}$ generating $\lambda = 2.5\,\text{cm}$ at screen distance $D = 30\,\text{cm}$:       $x = \frac{\lambda D}{a} = \frac{2.5\,\text{cm} \times 30\,\text{cm}}{5\,\text{cm}} = 15\,\text{cm}$
  • Two Loudspeakers Interference:
  • Loudspeakers spaced $a = 1.2\,\text{m}$ emit sound at $f = 500\,\text{Hz}$ into air ($v = 330\,\text{m s}^{-1}$); observation line is $D = 5\,\text{m}$ away:       $\lambda = \frac{v}{f} = \frac{330\,\text{m s}^{-1}}{500\,\text{Hz}} = 0.66\,\text{m}$$x = \frac{\lambda D}{a} = \frac{0.66\,\text{m} \times 5\,\text{m}}{1.2\,\text{m}} = \frac{3.3}{1.2} = 2.75\,\text{m}$
  • Young's Double-Slit Optical Fringes (Yellow Light):
  • Yellow light with $\lambda = 5.9 \times 10^{-7}\,\text{m}$, slit separation $a = 0.35\,\text{mm} = 0.35 \times 10^{-3}\,\text{m}$, screen distance $D = 2\,\text{m}$:       $x = \frac{5.9 \times 10^{-7}\,\text{m} \times 2\,\text{m}}{0.35 \times 10^{-3}\,\text{m}} = 3.37 \times 10^{-3}\,\text{m} = 3.37\,\text{mm}$
  • Determination of Unknown Optical Wavelength:
  • Double-slit with $a = 0.2\,\text{mm} = 0.2 \times 10^{-3}\,\text{m}$ and $D = 1.5\,\text{m}$. A measurement across 4 fringe intervals spans $20\,\text{mm}$ ($4x = 20\,\text{mm} \implies x = 5\,\text{mm} = 5 \times 10^{-3}\,\text{m}$):       $\lambda = \frac{ax}{D} = \frac{(0.2 \times 10^{-3}\,\text{m}) \times (5 \times 10^{-3}\,\text{m})}{1.5\,\text{m}} = \frac{1.0 \times 10^{-6}}{1.5} \approx 6.7 \times 10^{-7}\,\text{m}$
  • Wavelength Dependency of Color Band Spacing:
  • In the visible spectrum: $\lambda_{\text{red}} > \lambda_{\text{yellow}} > \lambda_{\text{green}}$.
  • In Young's double-slit experiment, fringe separation tracks wavelength ($x_{\text{red}} > x_{\text{yellow}} > x_{\text{green}}$).
  • Narrowest spacing ($P$) corresponds to green light, intermediate ($Q$) to yellow light, and widest ($R$) to red light.

Electromagnetic Waves and Spectrum Characteristics

• Nature and Characteristics of Electromagnetic (EM) Waves:

  • Composed of mutually perpendicular oscillating electric ($\mathbf{E}$) and magnetic ($\mathbf{B}$) fields that oscillate at $90^\circ$ to each other and to the direction of wave travel.
  • All electromagnetic waves are transverse waves.
  • Travel through a vacuum (free space) at an identical, constant invariant speed:     $c = 3.0 \times 10^8\,\text{m s}^{-1}$
  • The speed of an electromagnetic wave in a vacuum is independent of its frequency or wavelength.
  • Do not require a mechanical material medium for propagation.
  • Exhibit all core wave phenomena: reflection, refraction, diffraction, and interference.
  • Can be polarised.
  • Transfer energy and momentum. Photon energy is governed by the Planck-Einstein relation:     $E = hf = \frac{hc}{\lambda}$     where $h = 6.63 \times 10^{-34}\,\text{J s}$.

• Structure of the Electromagnetic Spectrum:

  • Ordered by increasing frequency ($f$) and photon energy ($E$) / decreasing wavelength ($\lambda$):
  1. Radio Waves: Lowest frequency, lowest energy, longest wavelength ($\lambda > 10^{-1}\,\text{m}$).
  2. Microwaves: Millimeter-to-centimeter wavelengths ($10^{-3}\,\text{m} \le \lambda \le 10^{-1}\,\text{m}$).
  3. Infrared Radiation (IR): Thermal radiation ($700\,\text{nm} < \lambda < 1\,\text{mm}$).
  4. Visible Light: Spectral band detectable by human eyes ($400\,\text{nm} \le \lambda \le 700\,\text{nm}$, from Red down to Violet).
  5. Ultraviolet Radiation (UV): Ionizing/chemical activity ($10\,\text{nm} \le \lambda \le 400\,\text{nm}$).
  6. X-rays: High-energy ionizing penetrating radiation ($0.01\,\text{nm} \le \lambda \le 10\,\text{nm}$).
  7. Gamma Rays ($\gamma$-rays): Highest frequency, shortest wavelength ($\lambda < 0.01\,\text{nm}$), highest penetrating power and energy.

• Ascending Order of Wavelength for Common Bands:$\text{Gamma rays } (R) < \text{Ultraviolet } (P) < \text{Infrared } (S) < \text{Microwaves } (Q)$$\text{Order: } R \rightarrow P \rightarrow S \rightarrow Q$

• Wavelength vs. Frequency Trade-off:

  • Because $c = f\lambda$ is constant in a given medium, wavelength and frequency are inversely related. As wavelength increases, frequency decreases.

Electronic Oscillatory Circuits and Telecommunications

• Generation of Electromagnetic Waves:

  • Produced whenever charged particles (such as electrons) experience acceleration or periodic oscillation.
  • Plucking strings, striking gongs, or moving mechanical paddles generate mechanical sound or water waves, not electromagnetic radiation.
  • Oscillating electrons in a radio frequency (RF) electrical oscillator circuit generate electromagnetic radio waves.

• The Basic $LC$ Oscillatory Tank Circuit:

  • Composed of a capacitor ($C$) connected in parallel to an inductor coil ($L$).
  • Electrical charge surges periodically back and forth between the electric field of the capacitor and the magnetic field of the inductor.
  • This high-frequency oscillation of charge produces radio waves radiated through an antenna.

• Applications of Electromagnetic Bands:

  • Radio Waves: Terrestrial and satellite telecommunications, audio radio broadcasting, and radar transmission.
  • Microwaves: Point-to-point satellite communication, cellular telephone networks, Wi-Fi, and microwave heating.
  • Infrared Rays: Remote controls, thermal imaging cameras, night vision equipment, and heat therapies.
  • X-rays: Airport baggage security scanners (capable of penetrating luggage fabric and cases to image metal/dense contents) and medical radiography (bone imaging).
  • Gamma Rays: Radiotherapy for cancer sterilization, industrial weld testing, and food irradiation.

Complete Solutions: SPM Physics Paper 1 (Questions 1 to 51)

• Question 1 [Knowledge]:

  • Prompt: Which of the following characteristics differentiate a transverse wave from a longitudinal wave?
  • Options: A (Frequency of the wave compared with the direction in which the wave travels), B (Amplitude of the wave compared with the direction in which the wave travels), C (Direction of the vibrations of medium compared with the direction in which the wave travels), D (The length of the wave compared with the frequency of the wave).
  • Correct Answer: C
  • Explanation: Transverse waves have particle oscillations perpendicular to propagation; longitudinal waves have oscillations parallel to propagation.

• Question 2 [Knowledge]:

  • Prompt: Diagram 1 shows a displacement-time graph for a wave. Find the frequency and amplitude of the wave.
  • Data from Graph: Peak displacement = $5\,\text{cm}$; time for 1 complete cycle = $0.02\,\text{s}$.
  • Solution: Amplitude = $5\,\text{cm}$; Period $T = 0.02\,\text{s} \implies f = \frac{1}{0.02\,\text{s}} = 50\,\text{Hz}$.
  • Correct Answer: B (Amplitude: $5\,\text{cm}$, Frequency: $50\,\text{Hz}$)

• Question 3 [Application]:

  • Prompt: Diagram 2 shows a water wave travelling from $P$ to $Q$. Find the velocity of the wave given $\lambda = 12\,\text{cm}$ and period for 2 cycles is $4\,\text{s}$.
  • Solution: Frequency $f = \frac{1}{T} = \frac{1}{2\,\text{s}} = 0.5\,\text{Hz}$. Wavelength $\lambda = 12\,\text{cm} = 0.12\,\text{m}$.     $v = f\lambda = 0.5\,\text{Hz} \times 0.12\,\text{m} = 0.06\,\text{m s}^{-1}$
  • Correct Answer: B ($0.06\,\text{m s}^{-1}$)

• Question 4 [Understanding]:

  • Prompt: Diagram 3 shows a vibration system with an amplitude that decreases steadily over time.
  • Correct Answer: C (a damping effect)
  • Explanation: The continuous exponential reduction in amplitude over time indicates mechanical energy loss due to damping.

• Question 5 [Application]:

  • Prompt: An audio generator produces a sound wave with a frequency of $15\,\text{kHz}$. Find the wavelength if the velocity is $330\,\text{m s}^{-1}$.
  • Solution: $f = 15\,000\,\text{Hz}$, $v = 330\,\text{m s}^{-1}$.     $\lambda = \frac{v}{f} = \frac{330}{15\,000} = 0.022\,\text{m} = 2.2 \times 10^{-2}\,\text{m}$
  • Correct Answer: D ($2.2 \times 10^{-2}\,\text{m}$)

• Question 6 [Understanding]:

  • Prompt: Which quantity becomes maximum when a vibrating system experiences resonance?
  • Correct Answer: A (Amplitude)

• Question 7 [Knowledge]:

  • Prompt: A vibrating system has a natural frequency of $f$. For resonance, external frequency must be:
  • Correct Answer: D ($f$)

• Question 8 [Knowledge]:

  • Prompt: The period of oscillation of a simple pendulum will decrease with:
  • Analysis: From $T = 2\pi \sqrt{\frac{l}{g}}$, $T$ decreases as the string length ($l$) decreases.
  • Correct Answer: D (the length of the string to the bob)

• Question 9 [Understanding]:

  • Prompt: Barton's pendulum setup with pendulums $P, Q, R, S$ of different lengths connected to driver pendulum $T$. Which will swing with maximum amplitude?
  • Correct Answer: Pendulum having the identical length to pendulum $T$ (resonance condition).

• Question 10 [Application]:

  • Prompt: Diagram 5 shows a circular wave pattern produced at $6\,\text{Hz}$ in a ripple tank with measured wavelength $\lambda = 1.67\,\text{cm}$.
  • Solution: $v = f\lambda = 6\,\text{Hz} \times 1.67\,\text{cm} = 10\,\text{cm s}^{-1}$.
  • Correct Answer: A ($10\,\text{cm s}^{-1}$)

• Question 11 [Application]:

  • Prompt: Diagram 6 shows a simple pendulum bob swinging and its displacement graph. Which position ($A, B, C,$ or $D$) has maximum kinetic energy?
  • Correct Answer: B
  • Explanation: Kinetic energy is maximum at the neutral equilibrium position ($y = 0$), where potential energy is minimum and speed is greatest.

• Question 12 [Understanding]:

  • Prompt: Which of the following is not a transverse wave?
  • Correct Answer: A (Sound waves)
  • Explanation: Sound waves are longitudinal mechanical waves.

• Question 13 [Understanding]:

  • Prompt: What happens to the wavelength and magnitude of velocity of water waves when reflected?
  • Correct Answer: D (Wavelength: Unchanged; Magnitude of velocity: Unchanged)

• Question 14 [Understanding]:

  • Prompt: Diagram 7 shows a submerged glass plate in water. What direction does the wave travel after passing through the plate?
  • Correct Answer: B
  • Explanation: The shallower water over the glass plate slows the wave down, shortening the wavelength and refracting the propagation direction toward the normal.

• Question 15 [Understanding]:

  • Prompt: Which water wave graph represents the wave with the greatest energy?
  • Correct Answer: C
  • Explanation: Energy carried by a periodic wave is proportional to the square of its amplitude and frequency ($E \propto f^2 A^2$).

• Question 16 [Application]:

  • Prompt: Ali shouts in front of a hill and hears an echo $1.8\,\text{s}$ later ($v = 330\,\text{m s}^{-1}$). Distance to the hill?
  • Solution: $s = \frac{vt}{2} = \frac{330 \times 1.8}{2} = 297\,\text{m}$.
  • Correct Answer: C ($297\,\text{m}$)

• Question 17 [Knowledge/Application]:

  • Prompt: Coherent sources $S_1$ and $S_2$ produce an interference pattern in Diagram 8. The water is calm at:
  • Correct Answer: Positions on nodal lines (destructive cancellation where path difference is an odd half-integer multiple of $\lambda$).

• Question 18 [Application]:

  • Prompt: Two prongs $5\,\text{cm}$ apart ($a = 5\,\text{cm}$) generate $\lambda = 2.5\,\text{cm}$ waves. Find distance between antinode lines at distance $D = 30\,\text{cm}$.
  • Solution: $x = \frac{\lambda D}{a} = \frac{2.5 \times 30}{5} = 15\,\text{cm}$.
  • Correct Answer: D ($15\,\text{cm}$)

• Question 19 [Knowledge]:

  • Prompt: Plane wavefronts pass through a small aperture (diffraction). Which quantity changes?
  • Correct Answer: C (Amplitude)
  • Explanation: Wavelength, speed, and frequency are unchanged, but energy spreads over a circular front, lowering amplitude.

• Question 20 [Understanding]:

  • Prompt: Wavefront travels from shallow region $P$ to deep region $Q$ in Diagram 9. Which physical quantity will increase?
  • Correct Answer: B (Velocity of the wave) and D (Wavelength)

• Question 21 [Knowledge]:

  • Prompt: Plane wavefronts travel towards a concave reflector in uniform depth (Diagram 10). What changes after reflection?
  • Correct Answer: A (The wavefront pattern)
  • Explanation: Plane wavefronts become converging curved circular wavefronts.

• Question 22 [Knowledge/Understanding]:

  • Prompt: Table 1 shows wave spreading around obstacles and apertures. Which patterns are caused by diffraction?
  • Correct Answer: C ($P$ and $Q$)

• Question 23 [Application]:

  • Prompt: Slinky vibrating at $20\,\text{Hz}$ has compressions at $15\,\text{cm}$, $35\,\text{cm}$, and $55\,\text{cm}$. Find speed.
  • Solution: $\lambda = 35 - 15 = 20\,\text{cm} = 0.2\,\text{m}$. $v = f\lambda = 20 \times 0.2 = 4\,\text{m s}^{-1}$.
  • Correct Answer: B ($4\,\text{m s}^{-1}$)

• Question 24 [Application]:

  • Prompt: Hearing range wavelengths are $1.65 \times 10^1\,\text{m}$ to $1.65 \times 10^{-2}\,\text{m}$ in air ($v = 330\,\text{m s}^{-1}$). Which sound can be heard?
  • Solution: Range is $20\,\text{Hz}$ to $20\,000\,\text{Hz}$. $500\,\text{Hz}$ falls directly in this interval.
  • Correct Answer: C ($500\,\text{Hz}$)

• Question 25 [Knowledge]:

  • Prompt: Which produces longitudinal waves?
  • Correct Answer: B (Hitting of a gong - produces sound waves)

• Question 26 [Application]:

  • Prompt: Two speakers $1.2\,\text{m}$ apart emit $500\,\text{Hz}$ tone ($v = 330\,\text{m s}^{-1}$). Find distance between loudest points at $D = 5\,\text{m}$.
  • Solution: $\lambda = \frac{330}{500} = 0.66\,\text{m}$. $x = \frac{0.66 \times 5}{1.2} = 2.75\,\text{m}$.
  • Correct Answer: D ($2.75\,\text{m}$)

• Question 27 [Knowledge]:

  • Prompt: Sound from a ticking clock focuses at $P$ behind a balloon filled with $\text{CO}_2$ (Diagram 12). Why?
  • Correct Answer: A (refracted)

• Question 28 [Application]:

  • Prompt: Shot fired at $X$. Man at $Y$ is $160\,\text{m}$ away; wall is $320\,\text{m}$ behind $X$ ($v = 320\,\text{m s}^{-1}$). Time interval between reports?
  • Solution: Direct time $t_1 = \frac{160}{320} = 0.5\,\text{s}$. Reflected distance $= 320 + (320 + 160) = 800\,\text{m} \implies t_2 = \frac{800}{320} = 2.5\,\text{s}$. Interval $\Delta t = 2.5 - 0.5 = 2.0\,\text{s}$.
  • Correct Answer: D ($2.0\,\text{s}$)

• Question 29 [Knowledge/Understanding]:

  • Prompt: Student around a corner hears music from a radio (Diagram 14). Why?
  • Correct Answer: C (diffracted)

• Question 30 [Knowledge]:

  • Prompt: Which quantity increases when sound becomes louder?
  • Correct Answer: B (Amplitude)

• Question 31 [Application]:

  • Prompt: Young's double slit with yellow light ($\lambda = 5.9 \times 10^{-7}\,\text{m}$), $a = 0.35\,\text{mm}$, $D = 2\,\text{m}$. Find fringe spacing $x$.
  • Solution: $x = \frac{5.9 \times 10^{-7} \times 2}{0.35 \times 10^{-3}} = 3.37 \times 10^{-3}\,\text{m} = 3.37\,\text{mm}$.
  • Correct Answer: C ($3.37\,\text{mm}$)

• Question 32 [Application]:

  • Prompt: Double-slit experiment with $a = 0.2\,\text{mm}$, $D = 1.5\,\text{m}$, and 4 fringe intervals spanning $20\,\text{mm}$. Find $\lambda$.
  • Solution: $x = \frac{20\,\text{mm}}{4} = 5\,\text{mm} = 5 \times 10^{-3}\,\text{m}$.     $\lambda = \frac{(0.2 \times 10^{-3}) \times (5 \times 10^{-3})}{1.5} = 6.67 \times 10^{-7}\,\text{m} \approx 6.7 \times 10^{-7}\,\text{m}$
  • Correct Answer: A ($6.7 \times 10^{-7}\,\text{m}$)

• Question 33 [Understanding]:

  • Prompt: Color bands $P, Q, R$ show increasing fringe widths ($x_P < x_Q < x_R$). Identify the color filters.
  • Solution: Because $x \propto \lambda$ and $\lambda_{\text{green}} < \lambda_{\text{yellow}} < \lambda_{\text{red}}$, $P = \text{green}$, $Q = \text{yellow}$, $R = \text{red}$.
  • Correct Answer: A (green, yellow, red)

• Question 34 [Understanding]:

  • Prompt: A tuning fork vibrates in air, then vibrates in water. What happens to sound speed?
  • Correct Answer: A (Increases)
  • Explanation: Elastic modulus to density ratio is much higher in liquid water, increasing speed from $\approx 330\,\text{m s}^{-1}$ to $\approx 1500\,\text{m s}^{-1}$.

• Question 35 [Application]:

  • Prompt: Loudspeaker produces wave where 3 wavelengths span $12\,\text{cm}$ ($v = 320\,\text{m s}^{-1}$). Find $\lambda$ and $f$.
  • Solution: $\lambda = \frac{12\,\text{cm}}{3} = 4\,\text{cm} = 0.04\,\text{m}$. $f = \frac{320}{0.04} = 8000\,\text{Hz}$.
  • Correct Answer: C (Wavelength: $4\,\text{cm}$, Frequency: $8000\,\text{Hz}$)

• Question 36 [Knowledge]:

  • Prompt: How do light waves differ from sound waves?
  • Correct Answer: C (can be polarised)
  • Explanation: Only transverse waves (like light) can be polarised; longitudinal waves (like sound) cannot.

• Question 37 [Understanding]:

  • Prompt: Diagram 19 shows waveforms of notes $P$ and $Q$ with identical period, but $Q$ has smaller amplitude.
  • Correct Answer: C (Note $Q$ is less loud than note $P$)

• Question 38 [Knowledge]:

  • Prompt: Two speakers connected to one audio generator produce alternate loud and soft sounds. This exhibits:
  • Correct Answer: C (interference)

• Question 39 [Knowledge]:

  • Prompt: In Young's double-slit experiment, what is the graphical relationship between slit distance $a$ and fringe spacing $x$?
  • Correct Answer: D
  • Explanation: From $x = \frac{\lambda D}{a}$, $a = \frac{\lambda D}{x}$, demonstrating an inverse curve ($a \propto \frac{1}{x}$).

• Question 40 [Understanding]:

  • Prompt: Which diagram shows the propagation and refraction of waves from deep to shallow water correctly?
  • Correct Answer: B
  • Explanation: Wavefronts in the shallow region must be closer together (shorter wavelength) and refract toward the boundary normal.

• Question 41 [Application]:

  • Prompt: Ship detects ultrasonic echo from seabed after $0.4\,\text{s}$ ($v = 1500\,\text{m s}^{-1}$). Find depth.
  • Solution: $s = \frac{vt}{2} = \frac{1500 \times 0.4}{2} = 300\,\text{m}$.
  • Correct Answer: B ($300\,\text{m}$)

• Question 42 [Knowledge]:

  • Prompt: Which of the following produces electromagnetic waves?
  • Correct Answer: D (An electron oscillating up and down in an electrical oscillator circuit)

• Question 43 [Knowledge]:

  • Prompt: Common characteristic shared by X-rays, radio waves, and ultraviolet rays:
  • Correct Answer: A (velocity - all travel at $c = 3.0 \times 10^8\,\text{m s}^{-1}$ in a vacuum)

• Question 44 [Knowledge]:

  • Prompt: Which electromagnetic wave possesses the highest energy level?
  • Correct Answer: B (Gamma rays)
  • Explanation: Gamma rays have the highest frequency and therefore highest photon energy ($E = hf$).

• Question 45 [Knowledge]:

  • Prompt: Arrange in ascending order of wavelength: $P$: Ultraviolet, $Q$: Microwaves, $R$: Gamma rays, $S$: Infrared.
  • Correct Answer: D ($R \rightarrow P \rightarrow S \rightarrow Q$)

• Question 46 [Knowledge]:

  • Prompt: Correct relationship between wavelength and frequency of an electromagnetic wave:
  • Correct Answer: B (Wavelength increases, Frequency decreases)

• Question 47 [Knowledge]:

  • Prompt: Which is not a characteristic of electromagnetic waves?
  • Correct Answer: D (All have same frequency - frequencies span from $< 10^3\,\text{Hz}$ to $> 10^{20}\,\text{Hz}$ across the spectrum)

• Question 48 [Understanding]:

  • Prompt: Which electromagnetic wave is used in airport security baggage scanners?
  • Correct Answer: A (X-ray)

• Question 49 [Knowledge]:

  • Prompt: Diagram 21 shows an oscillatory $LC$ tank circuit. Name the waves it produces.
  • Correct Answer: D (Radio waves)

• Question 50 [Knowledge]:

  • Prompt: Correct classification of Longitudinal and Transverse waves:
  • Correct Answer: A (Longitudinal: Sound waves; Transverse: Light / X-rays / Radio waves)

• Question 51 [Knowledge]:

  • Prompt: Which statement is not true regarding electromagnetic waves?
  • Correct Answer: C (The velocity of the waves is affected by the frequency)
  • Explanation: The velocity of electromagnetic waves in a vacuum is invariant ($c = 3 \times 10^8\,\text{m s}^{-1}$) regardless of frequency.