**Definition:** Waves are patterns of disturbance that propagate from one point to another without the actual physical transfer of the medium as a whole.
When a pebble drops in still water, circular ripples move outward. Cork pieces on the water surface move up and down but do NOT move away from the disturbance centre. This proves the disturbance (wave) travels, not the water mass itself.
Electromagnetic waves do not require a medium and travel through vacuum at constant speed:
**c = 299,792,458 m/s ≈ 3 × 10⁸ m/s**
**Mechanical Waves**
**Electromagnetic Waves**
**Matter Waves**
Waves in elastic media connect directly to harmonic oscillations. Consider springs connected end-to-end: when one spring is disturbed, the disturbance travels to the next spring while each spring oscillates about equilibrium—the spring coupling system in trains demonstrates this principle.
**EXAM TIP:** Always distinguish between motion of medium particles (oscillatory, about equilibrium) and motion of wave itself (progressive travel).
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**Transverse Wave:** Constituents of medium oscillate perpendicular to direction of wave propagation.
**Longitudinal Wave:** Constituents of medium oscillate parallel to direction of wave propagation.
Water waves are combinations of both transverse and longitudinal components:
**Important:** Transverse and longitudinal waves travel at different speeds in the same medium.
**EXAM TIP:** Identify wave type by asking: Do particles move perpendicular (transverse) or parallel (longitudinal) to wave direction?
**Example 14.1 Application:**
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**y(x,t) = a sin(kx – ωt + φ)** ... (14.2)
where:
**At fixed instant (t = t₀):** Equation becomes y = a sin(kx + constant) → spatial sinusoidal shape
**At fixed location (x = x₀):** Equation becomes y = a sin(constant – ωt) → temporal sinusoidal oscillation
**As time increases:** x must increase to maintain constant phase → wave travels in **positive x-direction**
**Wave travelling in negative direction:** y(x,t) = a sin(kx + ωt + φ) ... (14.4)
**y(x,t) = A sin(kx – ωt) + B cos(kx – ωt)** ... (14.3)
Amplitude relation: **a² = A² + B²**
Phase relation: **tan φ = B/A**
**Amplitude (a):**
**Phase = (kx – ωt + φ):**
**Crest:** Point of maximum positive displacement
**Trough:** Point of maximum negative displacement
**Definition:** Wavelength (λ) is minimum distance between two points having same phase (e.g., consecutive crests or troughs).
For Eq. (14.2) at t = 0:
**y(x,0) = a sin(kx)**
Sine function repeats when angle changes by 2π:
sin(kx) = sin(k(x + λ))
Therefore: **kλ = 2π**
**Wavelength: λ = 2π/k** or **k = 2π/λ** ... (14.6)
**Angular Wave Number (k):**
**Relationship Check:** λk = 2π → Constant for fixed wave
For fixed location (x = 0) in Eq. (14.2):
**y(0,t) = a sin(-ωt) = -a sin(ωt)**
Sine function repeats when argument changes by 2π:
sin(ωt) = sin(ω(t + T))
Therefore: **ωT = 2π**
**Period: T = 2π/ω** or **ω = 2π/T** ... (14.7)
**Angular Frequency (ω):**
**Frequency: ν = 1/T = ω/(2π)** ... (14.8)
**Relationship Check:** νT = 1 → Complete oscillation in one period
For longitudinal waves (displacement parallel to propagation):
**s(x,t) = a sin(kx – ωt + φ)** ... (14.9)
**Given:** y(x,t) = 0.005 sin(80.0x – 3.0t), where all constants in SI units
**Part (a) Amplitude:**
By comparison with standard form y = a sin(kx – ωt):
**a = 0.005 m = 0.5 cm**
**Part (b) Wavelength:**
Angular wave number: k = 80.0 rad m⁻¹
Using λ = 2π/k:
**λ = 2π/80.0 = 0.0785 m = 7.85 cm**
**Part (c) Period and Frequency:**
Angular frequency: ω = 3.0 rad s⁻¹
Period: **T = 2π/ω = 2π/3.0 = 2.09 s**
Frequency: **ν = 1/T = 3.0/(2π) = 0.477 Hz**
**Part (d) Displacement at x = 30.0 cm = 0.30 m, t = 20 s:**
Substitute in wave equation:
y(0.30, 20) = 0.005 sin(80.0 × 0.30 – 3.0 × 20)
y(0.30, 20) = 0.005 sin(24.0 – 60.0)
y(0.30, 20) = 0.005 sin(-36.0)
Converting -36.0 rad to standard form:
-36.0 = -36.0 + 2π(3) = -36.0 + 18.85 ≈ -17.15 rad → sin(-1.019π) ≈ -0.848
**y = 0.005 × (-0.848) = -0.00424 m = -0.424 cm**
**EXAM TIP:** Always check angular units (radians); use sin(-θ) = -sin(θ); simplify arguments modulo 2π for easier calculation.
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For a progressive wave, wavelength λ, frequency ν, period T, and angular frequency ω relate through wave speed.
**Consider:** In time T (one complete period), wave travels distance λ (one wavelength).
**Wave Speed: v = λ/T = λν** ... (14.10)
Since ν = ω/(2π) and λ = 2π/k:
**v = λν = (2π/k) × (ω/2π) = ω/k** ... (14.11)
**For mechanical waves:** Speed depends on ONLY medium properties (elastic and inertial), independent of wave properties.
v = √(T/μ) where μ = mass per unit length
v = √(Y/ρ)
v = √(B/ρ)
**Wave speed CANNOT be changed by changing amplitude or frequency of source. Speed depends only on medium properties.**
A harmonic wave on string: y = 0.02 sin(10πx – 4πt) m
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**When two or more waves travel through the same medium simultaneously, the net displacement at any point at any instant equals the vector sum of displacements due to individual waves.**
For two waves at same point:
**y_total = y₁ + y₂**
If waves are:
Then: **y(x,t) = a₁ sin(kx – ωt + φ₁) + a₂ sin(kx – ωt + φ₂)**
This resultant can be written as:
**y(x,t) = A sin(kx – ωt + Φ)** ... (14.12)
where resultant amplitude: **A² = a₁² + a₂² + 2a₁a₂ cos(φ₁ – φ₂)**
and phase: **tan Φ = (a₁ sin φ₁ + a₂ sin φ₂)/(a₁ cos φ₁ + a₂ cos φ₂)**
**Constructive Interference (Maximum Amplitude):**
**Destructive Interference (Minimum Amplitude):**
When phase difference changes randomly with time, interference pattern averages over time.
Intensity I ∝ (Amplitude)²
For two interfering waves:
**I_total = I₁ + I₂ + 2√(I₁I₂) cos(Δφ)**
**EXAM TIP:** Recognize interference type by comparing phases or path differences relative to wavelength.
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**Reflection:** When a wave encounters a boundary, part of wave energy bounces back into the original medium.
**Boundary Condition:** Particle at boundary cannot move (constrained).
**Incident wave:** y₁ = a sin(kx + ωt) [moving in –x direction toward boundary]
**Reflected wave:** y₂ = –a sin(kx – ωt) [moving in +x direction from boundary]
**Phase change:** Reflected wave has **180° phase reversal** (π radians)
**Reason:** At boundary, displacement must be zero always.
y_total = y₁ + y₂ = 0 at boundary
Therefore: reflected amplitude = –incident amplitude
**Boundary Condition:** Particle at boundary moves freely; slope of string is zero at boundary (no force perpendicular to boundary).
**Incident wave:** y₁ = a sin(kx + ωt)
**Reflected wave:** y₂ = a sin(kx – ωt) [same phase, no phase change]
**No phase change:** Reflected wave maintains sign
**Reason:** At free end, no constraint on particle motion, slope condition gives reflection without phase reversal.
When incident and reflected waves overlap:
**y_total = y_incident + y_reflected**
For fixed end: y = a sin(kx + ωt) – a sin(kx – ωt)
Using sin A – sin B = 2 cos((A+B)/2) sin((A–B)/2):
**y = 2a cos(ωt) sin(kx)** ... (14.13)
This is a **standing wave** (not a travelling wave):
**Nodes:**
**Antinodes:**
**EXAM TIP:** Standing wave pattern depends on boundary conditions—fixed (phase reversal), free (no phase reversal), or mixed.
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**Beats:** Periodic variation in intensity when two waves of slightly different frequencies interfere.
**Condition:** When two sinusoidal waves of nearly equal frequencies overlap, their phase difference changes periodically with time, causing alternating constructive and destructive interference.
Consider two waves with same amplitude a but slightly different frequencies ν₁ and ν₂ (where ν₁ > ν₂):
**y₁ = a sin(2πν₁t)**
**y₂ = a sin(2πν₂t)**
**Net displacement:**
**y = y₁ + y₂ = a[sin(2πν₁t) + sin(2πν₂t)]**
Using sum-to-product formula: sin A + sin B = 2 sin((A+B)/2) cos((A–B)/2):
**y = 2a cos(π(ν₁ – ν₂)t) × sin(2π(ν₁ + ν₂)t/2)** ... (14.14)
This can be rewritten as:
**y = [2a cos(π(ν₁ – ν₂)t)] × sin(2πν_avg × t)**
where **ν_avg = (ν₁ + ν₂)/2** (average frequency)
**Physical Meaning:**
The amplitude varies between maximum (2a) and minimum (0) periodically.
**Maximum amplitude:** cos term = ±1 → A = 2a (constructive interference)
**Minimum amplitude:** cos term = 0 → A = 0 (destructive interference)
The amplitude returns to maximum after cos argument changes by 2π:
π(ν₁ – ν₂)t = 2π
**t = 2/(ν₁ – ν₂)** → Time for one complete beat cycle
**Beat frequency (frequency of amplitude variation):**
**ν_beat = ν₁ – ν₂** ... (14.15)
**Beat period:** **T_beat = 1/(ν₁ – ν₂)**
Intensity I ∝ A²(t) = 4a² cos²(π(ν₁ – ν₂)t)
**Maximum intensity:** I_max ∝ 4a² (when cos = ±1)
**Minimum intensity:** I_min = 0 (when cos = 0)
Listener hears **ν_beat beats per second** (intensity maxima per second).
**Tuning Musical Instruments:**
**Example:** Two tuning forks at 256 Hz and 260 Hz
**EXAM TIP:** Remember beat frequency = difference of component frequencies, NOT sum. Beats represent amplitude modulation, not frequency modulation.
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| Parameter | Formula | Units |
|-----------|---------|-------|
| Wave equation | y = a sin(kx – ωt + φ) | m |
| Wavelength | λ = 2π/k | m |
| Angular frequency | ω = 2π/T = 2πν | rad s⁻¹ |
| Frequency | ν = 1/T = ω/(2π) | Hz |
| Wave speed | v = λν = ω/k | m s⁻¹ |
| Period | T = 2π/ω | s |
| Angular wave number | k = 2π/λ | rad m⁻¹ |
| Resultant amplitude | A² = a₁² + a₂² + 2a₁a₂cos(Δφ) | m |
| Standing wave envelope | A(x) = 2a sin(kx) or 2a cos(kx) | m |
| Beat frequency | ν_beat = \|ν₁ – ν₂\| | Hz |
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1. **Wave ≠ Medium Motion:** Wave travels; medium particles oscillate about fixed equilibrium positions only.
2. **Wave Speed is Constant:** Determined by medium alone. Changing source frequency changes wavelength, not speed. v = √(property of medium/inertial property).
3. **Transverse Only in Solids:** Fluids cannot sustain shear; thus only longitudinal waves in fluids.
4. **Superposition Applicability:** Linear media (small amplitude); breaks down for nonlinear media or large amplitudes.
5. **Phase Difference Matters:** Two identical amplitude waves can cancel (180° phase) or reinforce (0° phase) completely.
6. **Standing Waves = Confined Waves:** Result from reflection within boundaries; nodes at fixed ends, variable at free ends.
7. **Beats Require Similar Frequencies:** Beat method practical only if frequency difference << average frequency.
8. **Reflection Phase Reversal:** Fixed boundary → phase reversal; free boundary → no phase reversal.
Q1. A cork piece floats on water. When a wave passes through the water, the cork:
Answer: B — Waves transport energy and pattern, not matter; the cork oscillates with local particle motion but the water mass does not flow outward, so the cork stays at nearly the same position.
Q2. Which of the following is NOT a property that distinguishes transverse from longitudinal waves?
Answer: B — Both transverse and longitudinal waves can travel through the same medium (e.g., seismic P and S waves in Earth's crust); the key difference is particle motion direction, not medium type.
Q3. A wave travels along a stretched string with wavelength 0.5 m and frequency 4 Hz. The wave speed is:
Answer: B — Using v = λf = 0.5 m × 4 Hz = 2 m/s.
Q4. Why do mechanical waves require a material medium for propagation?
Answer: B — Mechanical waves arise from particle oscillations coupled by elastic forces; these properties are only present in material media, not in vacuum.
Q5. Sound is a longitudinal wave in air. Which statement correctly describes the particle motion?
Answer: B — In longitudinal waves like sound, particles oscillate along the direction of wave travel, causing local density and pressure variations (compressions and rarefactions).
Q6. A spring is stretched and then released, creating a disturbance that travels along a chain of connected springs. This happens because:
Answer: B — Elasticity provides restoring forces and inertia allows particles to overshoot equilibrium; this back-and-forth coupling transfers the disturbance along the chain without bulk matter flow.
Q7. If the frequency of a wave is doubled while the wave speed remains constant, the wavelength:
Answer: B — From v = λf, if v is constant and f doubles, then λ must halve: λ_new = v/(2f) = λ_old/2.
Q8. Which pair correctly matches wave type with an example?
Answer: B — Seismic S-waves cause particle motion perpendicular to wave direction (transverse); P-waves cause parallel motion (longitudinal), creating compressions and rarefactions in Earth's layers.
Q9. A student observes waves on a pond created by dropping stones. She notes that when the distance between consecutive wave crests increases, the frequency of waves (as measured by a fixed observer) stays the same. Which conclusion is correct? (Assume waves are not being absorbed.)
Answer: B — Using v = λf: if λ increases and f remains constant, then v must increase proportionally; the wave speed depends on the medium properties, which can change (e.g., depth of pond affecting water wave speed).
Q10. A sinusoidal wave is described by y = 0.02 sin(50x − 400t), where y and x are in metres and t is in seconds. The wavelength and frequency of this wave are approximately:
Answer: A — From y = A sin(kx − ωt), k = 50 rad/m and ω = 400 rad/s. λ = 2π/k = 2π/50 ≈ 0.126 m; f = ω/(2π) = 400/(2π) ≈ 63.7 Hz.
What is a mechanical wave?
A disturbance that propagates through a medium by oscillation of particles, requiring elasticity and inertia of the medium.
How do you distinguish transverse from longitudinal waves?
In transverse waves particles oscillate perpendicular to wave propagation; in longitudinal waves they oscillate parallel to it.
Why does a cork piece bob up-down but not move away when a wave passes?
Waves transport energy and pattern, not matter; the cork oscillates with the medium locally but the medium itself does not flow outward.
Give one example each of transverse and longitudinal waves.
Transverse: waves on a string or water surface; Longitudinal: sound waves in air or compression waves in a spring.
What is the relationship between wave speed v, wavelength λ, and frequency f?
v = λf, where v is speed, λ is wavelength, and f is frequency.
Why do mechanical waves require a medium?
They depend on oscillations of particles and elastic restoring forces, which only exist in a material medium.
What is the speed of electromagnetic waves in vacuum?
c = 299,792,458 m/s ≈ 3 × 10⁸ m/s, the same for all EM waves.
How does the disturbance propagate in connected springs?
One spring compresses or stretches, pulling the next spring out of equilibrium, which then pulls the next, creating wave propagation along the chain.
Describe how sound propagates in air using compression and rarefaction.
A compressed air region pushes molecules outward creating compression in the next region and rarefaction in its own; this alternating pattern moves as sound.
What role do elastic forces play in wave propagation?
Elastic forces provide restoring forces that accelerate particles back toward equilibrium after disturbance, enabling energy transfer through the medium.
Define a mechanical wave and explain why it requires a medium for propagation. Give one example. [2 marks]
State that a wave is a disturbance carrying energy without bulk matter flow; explain that elasticity and inertia of particles (only in medium) enable propagation; example: sound in air, water ripples, or string waves.
Distinguish between transverse and longitudinal waves with one example each. Why are water surface waves considered transverse while sound waves in air are longitudinal? [5 marks]
Transverse: particle motion ⊥ to wave direction (e.g., string); Longitudinal: particle motion ∥ to wave direction (e.g., spring compression). For water and sound: explain particle motion relative to disturbance propagation in each case; water surface particles move up-down as waves move horizontally; air molecules oscillate back-forth along sound direction creating compression-rarefaction.
A wave with frequency 10 Hz travels along a string with speed 20 m/s. (a) Calculate the wavelength. (b) If the frequency is doubled to 20 Hz while the speed remains 20 m/s, how does the wavelength change? (c) Explain this relationship using the equation v = λf and discuss the physical meaning of this result. [6 marks]
(a) Use v = λf directly to find λ = 2 m. (b) New λ = v/(2f) = 1 m; wavelength halves. (c) Show that v depends on medium properties (tension, mass density) which do not change; when f increases, λ must decrease proportionally to keep v constant; this reflects that a given medium cannot transmit all frequencies at the same speed in all scenarios—connect to dispersive media if applicable.
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