Shape Parameters
Amplitude A and wavelength λ describe the visible shape of a sinusoidal wave on a y-x graph. A is vertical maximum displacement; λ is horizontal repeat distance.
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Wave Parameters and Wave Equation
Class 11 Physics notes covering amplitude, time period, frequency, angular frequency, wavelength, wave number, phase, wave equation, wave velocity, numericals and PYQs.
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Time Parameters
Time period T, frequency f and angular frequency ω describe how fast a particle oscillates at a fixed point. The key links are f = 1/T and ω = 2πf.
Equation Parameters
Wave number k, phase and wave velocity v connect the graph to motion. The key exam equation is y = A sin(ωt − kx).
Parameter
Amplitude A
Amplitude is the maximum displacement of a vibrating particle from its mean position. In y = A sin(ωt − kx), A is the amplitude.
Physical meaning: amplitude measures how far the particle moves from equilibrium, not how far the wave travels. In a rope wave, a larger hand movement produces a larger amplitude.
Real-life example: when a guitar string is plucked strongly, its amplitude is larger and the sound is louder. If it is plucked gently, amplitude is smaller.
Exam perspective: amplitude affects energy and intensity, but it does not change frequency, wavelength or wave speed in a given linear medium.
Common trap: amplitude is not the distance between crest and trough. Crest-to-trough distance is 2A.
Parameter
Time Period T
Time period is the time taken by one particle of the medium to complete one full oscillation. It is also the time between two successive crests passing a fixed point.
Formula: T = 1/f. Unit of T is second.
Real-life example: if a water surface at one point goes up, comes down and returns to the same state in 0.5 s, the time period is 0.5 s.
Exam perspective: in a y-t graph, the horizontal distance between two consecutive crests gives T.
Common trap: T is a time interval, not a distance. Do not confuse it with wavelength.
Parameter
Frequency f
Frequency is the number of complete oscillations performed by a particle in one second. It is also the number of complete waves crossing a fixed point per second.
Formula: f = 1/T. Unit is hertz, written Hz or s−1.
Real-life example: a 256 Hz tuning fork makes each nearby air particle complete 256 oscillations every second.
Exam perspective: frequency is decided by the source. When a wave goes from one medium to another, frequency remains unchanged.
Common trap: students often think frequency changes because speed changes. In refraction or medium change, wavelength changes, frequency remains the same.
Parameter
Angular Frequency ω
Angular frequency is the rate of change of phase with time. It tells how many radians of phase are covered per second.
Formula: ω = 2πf = 2π/T. Unit is rad s−1.
Real-life example: if a rotating reference circle completes f rotations per second, its angular speed is 2πf rad s−1; SHM and wave equations use the same idea.
In y = A sin(ωt − kx), ω is the coefficient of t.
Common trap: angular frequency is not frequency. Frequency counts cycles per second; angular frequency counts radians per second.
Parameter
Wavelength λ
Wavelength is the distance between two nearest particles vibrating in the same phase. For a transverse wave it is crest-to-crest or trough-to-trough distance; for sound it is compression-to-compression distance.
Formula: v = fλ, so λ = v/f. Unit is metre.
Real-life example: in water ripples, the distance between two neighbouring bright crest lines is the wavelength.
Exam perspective: in a y-x graph, the horizontal distance between two consecutive crests gives wavelength.
Common trap: wavelength is measured along the direction of propagation, not vertically.
Parameter
Wave Number k
Wave number is the phase change per unit distance. It tells how rapidly the wave phase changes as we move along x.
Formula: k = 2π/λ. Unit is rad m−1.
In y = A sin(ωt − kx), k is the coefficient of x.
Real-life example: closely spaced ripples have small wavelength and therefore large wave number.
Common trap: ordinary spatial frequency is 1/λ, but angular wave number is 2π/λ.
Parameter
Phase
Phase describes the state of oscillation of a particle at a given position and time. It tells whether the particle is at mean position, crest, trough, moving upward, or moving downward.
For y = A sin(ωt − kx), phase is φ = ωt − kx.
Two particles are in the same phase when their phase difference is 0, 2π, 4π and so on. They are in opposite phase when phase difference is π, 3π and so on.
Real-life example: two points on a rope separated by one wavelength rise and fall together, so they are in the same phase.
Common trap: same displacement does not always mean same phase. One particle may be moving upward while the other is moving downward.
Parameter
Wave Velocity v
Wave velocity is the speed with which a particular phase, crest, compression, pulse or wave pattern travels through the medium.
Formula: v = fλ = ω/k. Unit is m s−1.
Real-life example: sound wave velocity in air is about 340 m s−1, while individual air molecules only vibrate locally.
Exam perspective: wave velocity depends on medium properties, not on amplitude in ordinary linear waves.
Common trap: wave velocity is not particle velocity. Wave velocity is pattern speed; particle velocity is dy/dt.
Core Derivation
Wave Equation
The displacement of a sinusoidal progressive wave travelling along positive x-direction can be written as y = A sin((2π/λ)(vt − x)).
Using ω = 2πf, v = fλ and k = 2π/λ, the same equation becomes y = A sin(ωt − kx).
Symbols: y is displacement of the particle, A is amplitude, ω is angular frequency, t is time, k is wave number, x is position, λ is wavelength and v is wave velocity.
For y = A sin(ωt − kx), the wave travels in positive x-direction. For y = A sin(ωt + kx), the wave travels in negative x-direction. Forms containing vt − x are equivalent to positive x-direction, while forms containing x − vt may differ only by an overall sign depending on sine convention; always reduce the phase and check whether x increases or decreases with time for constant phase.
dy/dt = Aω cos(ωt − kx)Particle velocity is the velocity of the medium particle at a fixed x. It changes with time and can be positive, negative or zero.
vp,max = AωThe maximum value of cos(ωt − kx) is 1, so maximum particle velocity is Aω.
dy/dx = −Ak cos(ωt − kx)dy/dx is the slope of the wave profile at a given instant. It is not particle velocity.
d2y/dt2 = −Aω2 sin(ωt − kx)Since y = A sin(ωt − kx), acceleration becomes a = −ω2y.
v = fλ = ω/kWave velocity is the speed of the wave pattern or phase. It is not dy/dt.
ωt − kx: +x
ωt + kx: −xKeep phase constant and see whether x increases or decreases as t increases.
Difference
Particle Velocity vs Wave Velocity
| Point | Particle Velocity | Wave Velocity |
|---|---|---|
| Meaning | Velocity of an individual oscillating particle of the medium. | Velocity of propagation of the wave pattern, crest, compression or phase. |
| Formula | vp = dy/dt | v = fλ = ω/k |
| Direction | Along the direction of particle vibration. | Along the direction of wave propagation. |
| Value | Changes continuously with time; maximum is Aω. | Constant for a given wave in a uniform medium. |
| Example | A point on a rope moves up and down. | The crest on the rope moves horizontally. |
NCERT Style
Wave Parameter Diagrams
All diagrams use black graph lines and red arrows/labels so they match board-work and NCERT-style wave drawings.
Wave Showing Amplitude and Wavelength
Phase Diagram
Wave Number Meaning
Wave Propagation Direction
Particle Velocity vs Wave Velocity
y-t Graph
y-x Graph
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Exam Patterns
Special Question Types
- Find wave velocity from equation: identify ω and k, then use v = ω/k.
- Find frequency from equation: identify ω, then use f = ω/2π.
- Find wavelength from equation: identify k, then use λ = 2π/k.
- Find wave number from equation: coefficient of x is k.
- Find angular frequency from equation: coefficient of t is ω.
- Find maximum particle velocity: use vp,max = Aω.
- Ratio of maximum particle velocity to wave velocity: vp,max/v = Ak.
- Identify direction: ωt − kx gives positive x, ωt + kx gives negative x.
- For y = A sin((2π/λ)(vt − x)), compare directly with positive x-direction.
- For x − vt and vt − x forms, use constant phase method before deciding direction.
Solved
40 Solved Numericals
These numericals cover wave velocity, frequency, wavelength, wave number, angular frequency, particle velocity, acceleration, phase difference, direction of propagation and equation comparison.
Numerical 1CBSE Easy
A wave has frequency 50 Hz and wavelength 4 m. Find wave velocity.
Show Solution
Given: f = 50 Hz, λ = 4 m
Formula: v = fλ
Solution: v = 50 × 4 = 200 m s−1
Final Answer: 200 m s−1
Numerical 2CBSE Easy
A wave has time period 0.02 s. Find frequency.
Show Solution
Given: T = 0.02 s
Formula: f = 1/T
Solution: f = 1/0.02 = 50 Hz
Final Answer: 50 Hz
Numerical 3NEET Easy
A wave has wavelength 2 m and speed 340 m s−1. Find frequency.
Show Solution
Given: v = 340 m s−1, λ = 2 m
Formula: f = v/λ
Solution: f = 340/2 = 170 Hz
Final Answer: 170 Hz
Numerical 4JEE Main Easy
Find angular frequency if frequency is 10 Hz.
Show Solution
Given: f = 10 Hz
Formula: ω = 2πf
Solution: ω = 20π rad s−1
Final Answer: 20π rad s−1
Numerical 5CBSE Easy
Find wave number if wavelength is 0.5 m.
Show Solution
Given: λ = 0.5 m
Formula: k = 2π/λ
Solution: k = 2π/0.5 = 4π rad m−1
Final Answer: 4π rad m−1
Numerical 6NEET Easy
For y = 0.04 sin(100t − 5x), identify amplitude.
Show Solution
Given: Equation: y = 0.04 sin(100t − 5x)
Formula: A is coefficient before sine
Solution: A = 0.04 m
Final Answer: 0.04 m
Numerical 7JEE Main Easy
For y = 0.02 sin(80t − 4x), find wave velocity.
Show Solution
Given: ω = 80 rad s−1, k = 4 rad m−1
Formula: v = ω/k
Solution: v = 80/4 = 20 m s−1
Final Answer: 20 m s−1
Numerical 8CBSE Medium
For y = 5 sin(20t − 2x), find angular frequency.
Show Solution
Given: Equation has coefficient of t = 20
Formula: ω is coefficient of t
Solution: ω = 20 rad s−1
Final Answer: 20 rad s−1
Numerical 9NEET Medium
For y = 0.01 sin(40t − 8x), find wave number.
Show Solution
Given: Equation has coefficient of x = 8
Formula: k is coefficient of x
Solution: k = 8 rad m−1
Final Answer: 8 rad m−1
Numerical 10JEE Main Medium
For y = 0.03 sin(60t − 3x), find frequency.
Show Solution
Given: ω = 60 rad s−1
Formula: f = ω/2π
Solution: f = 60/2π = 30/π Hz
Final Answer: 30/π Hz
Numerical 11JEE Main Medium
For y = 0.03 sin(60t − 3x), find wavelength.
Show Solution
Given: k = 3 rad m−1
Formula: λ = 2π/k
Solution: λ = 2π/3 m
Final Answer: 2π/3 m
Numerical 12NEET Medium
A wave has A = 0.02 m and ω = 100 rad s−1. Find maximum particle velocity.
Show Solution
Given: A = 0.02 m, ω = 100 rad s−1
Formula: vp,max = Aω
Solution: vp,max = 0.02 × 100 = 2 m s−1
Final Answer: 2 m s−1
Numerical 13JEE Main Medium
For y = 0.05 sin(200t − 10x), find maximum particle velocity.
Show Solution
Given: A = 0.05 m, ω = 200 rad s−1
Formula: vp,max = Aω
Solution: vp,max = 0.05 × 200 = 10 m s−1
Final Answer: 10 m s−1
Numerical 14JEE Main Medium
For y = A sin(ωt − kx), state direction of propagation.
Show Solution
Given: The phase is ωt − kx
Formula: Constant phase gives x = (ω/k)t − constant/k
Solution: x increases with t, so wave travels in positive x-direction.
Final Answer: Positive x-direction
Numerical 15JEE Main Medium
For y = A sin(ωt + kx), state direction of propagation.
Show Solution
Given: The phase is ωt + kx
Formula: Constant phase gives x = constant/k − (ω/k)t
Solution: x decreases with t, so wave travels in negative x-direction.
Final Answer: Negative x-direction
Numerical 16CBSE Medium
If T = 0.01 s and λ = 3 m, find wave velocity.
Show Solution
Given: T = 0.01 s, λ = 3 m
Formula: v = λ/T
Solution: v = 3/0.01 = 300 m s−1
Final Answer: 300 m s−1
Numerical 17NEET Medium
A source makes 120 oscillations in 4 s. Find frequency and time period.
Show Solution
Given: N = 120, t = 4 s
Formula: f = N/t, T = 1/f
Solution: f = 120/4 = 30 Hz; T = 1/30 s
Final Answer: 30 Hz, 1/30 s
Numerical 18JEE Main Medium
Find phase difference between two points separated by λ/4.
Show Solution
Given: Separation = λ/4
Formula: Δφ = 2πΔx/λ
Solution: Δφ = 2π(λ/4)/λ = π/2
Final Answer: π/2 rad
Numerical 19JEE Main Medium
Find phase difference between two points separated by 0.25 m if λ = 1 m.
Show Solution
Given: Δx = 0.25 m, λ = 1 m
Formula: Δφ = 2πΔx/λ
Solution: Δφ = 2π × 0.25 = π/2
Final Answer: π/2 rad
Numerical 20NEET Medium
If k = 6 rad m−1 and ω = 300 rad s−1, find wave velocity.
Show Solution
Given: k = 6, ω = 300
Formula: v = ω/k
Solution: v = 300/6 = 50 m s−1
Final Answer: 50 m s−1
Numerical 21JEE Main Medium
For y = 0.02 sin(2π(50t − x/4)), find frequency.
Show Solution
Given: Compare with y = A sin 2π(ft − x/λ)
Formula: f is coefficient of t inside 2π
Solution: f = 50 Hz
Final Answer: 50 Hz
Numerical 22JEE Main Medium
For y = 0.02 sin(2π(50t − x/4)), find wavelength.
Show Solution
Given: Compare with y = A sin 2π(ft − x/λ)
Formula: x/λ = x/4
Solution: λ = 4 m
Final Answer: 4 m
Numerical 23JEE Main Medium
For y = 0.02 sin(2π(50t − x/4)), find velocity.
Show Solution
Given: f = 50 Hz, λ = 4 m
Formula: v = fλ
Solution: v = 50 × 4 = 200 m s−1
Final Answer: 200 m s−1
Numerical 24JEE Advanced
For y = 0.01 sin(100t − 2x), find ratio of maximum particle velocity to wave velocity.
Show Solution
Given: A = 0.01, ω = 100, k = 2
Formula: vp,max/v = Aω/(ω/k) = Ak
Solution: Ratio = 0.01 × 2 = 0.02
Final Answer: 0.02
Numerical 25JEE Advanced
For y = 0.04 sin(50t − 5x), find maximum particle acceleration.
Show Solution
Given: A = 0.04 m, ω = 50 rad s−1
Formula: amax = Aω2
Solution: amax = 0.04 × 502 = 100 m s−2
Final Answer: 100 m s−2
Numerical 26CBSE Medium
A y-t graph has consecutive crests separated by 0.2 s. Find frequency.
Show Solution
Given: T = 0.2 s
Formula: f = 1/T
Solution: f = 1/0.2 = 5 Hz
Final Answer: 5 Hz
Numerical 27CBSE Medium
A y-x graph has consecutive crests separated by 0.8 m. Find wave number.
Show Solution
Given: λ = 0.8 m
Formula: k = 2π/λ
Solution: k = 2π/0.8 = 2.5π rad m−1
Final Answer: 2.5π rad m−1
Numerical 28NEET Difficult
If f is doubled in the same medium, what happens to wavelength?
Show Solution
Given: Same medium means v constant; f becomes 2f
Formula: v = fλ
Solution: λ becomes λ/2
Final Answer: Wavelength becomes half
Numerical 29JEE Main Difficult
For y = 3 sin π(0.5x − 100t), find speed and direction.
Show Solution
Given: Phase can be written 0.5πx − 100πt = −(100πt − 0.5πx)
Formula: Speed = ω/k = 100π/0.5π
Solution: v = 200 m s−1; form kx − ωt also travels positive x.
Final Answer: 200 m s−1, positive x-direction
Numerical 30JEE Main Difficult
For y = 0.02 sin(5x + 100t), find speed and direction.
Show Solution
Given: k = 5, ω = 100, plus sign
Formula: v = ω/k; plus sign gives negative x-direction
Solution: v = 100/5 = 20 m s−1
Final Answer: 20 m s−1, negative x-direction
Numerical 31JEE Advanced
For y = A sin(ωt − kx), find particle velocity expression.
Show Solution
Given: y = A sin(ωt − kx)
Formula: vp = dy/dt
Solution: vp = Aω cos(ωt − kx)
Final Answer: Aω cos(ωt − kx)
Numerical 32JEE Advanced
For y = A sin(ωt − kx), find acceleration expression.
Show Solution
Given: y = A sin(ωt − kx)
Formula: a = d2y/dt2
Solution: a = −Aω2 sin(ωt − kx) = −ω2y
Final Answer: −ω2y
Numerical 33JEE Advanced
For y = A sin(ωt − kx), find slope expression.
Show Solution
Given: y = A sin(ωt − kx)
Formula: dy/dx = derivative with respect to x
Solution: dy/dx = −Ak cos(ωt − kx)
Final Answer: −Ak cos(ωt − kx)
Numerical 34NEET Difficult
A wave has A = 2 cm and f = 5 Hz. Find maximum particle velocity.
Show Solution
Given: A = 2 cm = 0.02 m, f = 5 Hz
Formula: vp,max = Aω = A(2πf)
Solution: vp,max = 0.02 × 10π = 0.2π m s−1
Final Answer: 0.2π m s−1
Numerical 35JEE Main Difficult
A wave has λ = 0.4 m and T = 0.02 s. Find ω, k and v.
Show Solution
Given: λ = 0.4 m, T = 0.02 s
Formula: ω = 2π/T, k = 2π/λ, v = λ/T
Solution: ω = 100π rad s−1; k = 5π rad m−1; v = 20 m s−1
Final Answer: ω = 100π, k = 5π, v = 20 m s−1
Numerical 36CBSE Difficult
A wave covers 600 m in 2 s and frequency is 100 Hz. Find wavelength.
Show Solution
Given: distance = 600 m, time = 2 s, f = 100 Hz
Formula: v = s/t, λ = v/f
Solution: v = 300 m s−1; λ = 300/100 = 3 m
Final Answer: 3 m
Numerical 37NEET Difficult
Distance between nearest same phase particles is 1.5 m. If f = 20 Hz, find speed.
Show Solution
Given: λ = 1.5 m, f = 20 Hz
Formula: v = fλ
Solution: v = 20 × 1.5 = 30 m s−1
Final Answer: 30 m s−1
Numerical 38JEE Advanced
Two points separated by 0.3 m have phase difference 3π/2. Find wavelength.
Show Solution
Given: Δx = 0.3 m, Δφ = 3π/2
Formula: Δφ = 2πΔx/λ
Solution: λ = 2π × 0.3 /(3π/2) = 0.4 m
Final Answer: 0.4 m
Numerical 39JEE Advanced
For y = 0.02 sin(314t − 3.14x), estimate f, λ and v.
Show Solution
Given: ω = 314, k = 3.14
Formula: f = ω/2π, λ = 2π/k, v = ω/k
Solution: f = 50 Hz; λ = 2 m; v = 100 m s−1
Final Answer: 50 Hz, 2 m, 100 m s−1
Numerical 40Reasoning
If wave velocity is 100 m s−1 and maximum particle velocity is 5 m s−1, find A/λ relation.
Show Solution
Given: vp,max/v = Aω/(ω/k) = Ak = 2πA/λ
Formula: 5/100 = 2πA/λ
Solution: A/λ = 1/(40&pi)
Final Answer: A/λ = 1/(40π)
Numerical 41JEE Advanced
For y = 0.01 sin 2π(100t − x/0.5), find vp,max.
Show Solution
Given: A = 0.01 m, f = 100 Hz
Formula: ω = 2πf, vp,max = Aω
Solution: vp,max = 0.01 × 200π = 2π m s−1
Final Answer: 2π m s−1
Question Bank
PYQs and Exam Questions
This bank includes CBSE, NEET, JEE Main, JEE Advanced, IB, ICSE, IGCSE, A-Level, assertion-reason, true-false, case-study, reasoning and difficult conceptual questions.
Question 1CBSE
Define amplitude of a wave.
Show Answer
Amplitude is maximum displacement of a particle from its mean position.
Question 2CBSE
Define time period and frequency.
Show Answer
Time period is time for one oscillation; frequency is number of oscillations per second.
Question 3CBSE
Write relation between frequency and time period.
Show Answer
f = 1/T.
Question 4CBSE
Write relation between wave velocity, frequency and wavelength.
Show Answer
v = fλ.
Question 5CBSE
What is wavelength in a transverse wave?
Show Answer
Distance between two consecutive crests or troughs.
Question 6CBSE
What is wavelength in a longitudinal wave?
Show Answer
Distance between two consecutive compressions or rarefactions.
Question 7NEET
What is angular frequency?
Show Answer
Angular frequency is rate of change of phase with time, ω = 2πf.
Question 8NEET
What is wave number?
Show Answer
Wave number is phase change per unit length, k = 2π/λ.
Question 9NEET
For y = A sin(ωt − kx), identify amplitude.
Show Answer
A is amplitude.
Question 10NEET
For y = A sin(ωt − kx), identify angular frequency.
Show Answer
ω is angular frequency.
Question 11NEET
For y = A sin(ωt − kx), identify wave number.
Show Answer
k is wave number.
Question 12NEET
For y = A sin(ωt − kx), find wave velocity.
Show Answer
v = ω/k.
Question 13JEE Main
Which equation travels in positive x-direction: y = A sin(ωt − kx) or y = A sin(ωt + kx)?
Show Answer
y = A sin(ωt − kx) travels in positive x-direction.
Question 14JEE Main
Which equation travels in negative x-direction?
Show Answer
y = A sin(ωt + kx) travels in negative x-direction.
Question 15JEE Main
What is particle velocity?
Show Answer
Particle velocity is dy/dt, the velocity of an oscillating medium particle.
Question 16JEE Main
What is wave velocity?
Show Answer
Wave velocity is speed of propagation of phase or wave pattern, v = fλ = ω/k.
Question 17JEE Main
What is maximum particle velocity?
Show Answer
vp,max = Aω.
Question 18JEE Main
What is slope of a wave?
Show Answer
Slope is dy/dx at a point on the wave profile.
Question 19JEE Advanced
Derive acceleration for y = A sin(ωt − kx).
Show Answer
a = d2y/dt2 = −ω2y.
Question 20JEE Advanced
For y = A sin(ωt − kx), write dy/dx.
Show Answer
dy/dx = −Ak cos(ωt − kx).
Question 21JEE Advanced
For y = A sin(ωt − kx), write dy/dt.
Show Answer
dy/dt = Aω cos(ωt − kx).
Question 22JEE Advanced
What is the ratio vp,max/v for y = A sin(ωt − kx)?
Show Answer
vp,max/v = Aω/(ω/k) = Ak.
Question 23IB
A y-t graph is drawn at a fixed position. Which parameter is read from crest-to-crest spacing?
Show Answer
Time period T.
Question 24IB
A y-x graph is drawn at a fixed time. Which parameter is read from crest-to-crest spacing?
Show Answer
Wavelength λ.
Question 25IB
What does phase describe?
Show Answer
Phase describes the state of oscillation of a particle at a given position and time.
Question 26IB
When are two particles in phase?
Show Answer
When their phase difference is 0 or an integral multiple of 2π.
Question 27ICSE
State the SI unit of frequency.
Show Answer
Hertz or s−1.
Question 28ICSE
State the SI unit of wavelength.
Show Answer
Metre.
Question 29ICSE
State the SI unit of wave velocity.
Show Answer
m s−1.
Question 30ICSE
State the unit of angular frequency.
Show Answer
rad s−1.
Question 31IGCSE
If frequency increases and speed is constant, what happens to wavelength?
Show Answer
Wavelength decreases.
Question 32IGCSE
If wavelength doubles and frequency is constant, what happens to wave speed?
Show Answer
Wave speed doubles.
Question 33IGCSE
What does a larger amplitude usually mean for sound?
Show Answer
Louder sound, because energy/intensity is greater.
Question 34IGCSE
Does amplitude decide wave speed in a linear medium?
Show Answer
No, wave speed is decided by medium properties.
Question 35A-Level
Explain why same displacement does not necessarily mean same phase.
Show Answer
Two particles may have same displacement but opposite velocities, so their phases can differ.
Question 36A-Level
For constant phase in ωt − kx = constant, show direction.
Show Answer
x = (ω/k)t − constant/k, so x increases with t; direction is positive x.
Question 37A-Level
For constant phase in ωt + kx = constant, show direction.
Show Answer
x = constant/k − (ω/k)t, so x decreases with t; direction is negative x.
Question 38A-Level
What is the physical meaning of k?
Show Answer
It is spatial angular frequency: phase change per metre.
Question 39Assertion-Reason
Assertion: Wave velocity and particle velocity are different. Reason: Wave velocity is pattern speed while particle velocity is dy/dt.
Show Answer
Both are true and the reason correctly explains the assertion.
Question 40Assertion-Reason
Assertion: v = ω/k. Reason: ω = 2πf and k = 2π/λ.
Show Answer
Both are true and the reason correctly explains the assertion.
Question 41Assertion-Reason
Assertion: Frequency changes when a wave enters another medium. Reason: Speed changes in another medium.
Show Answer
Assertion is false; reason can be true.
Question 42Assertion-Reason
Assertion: Maximum particle velocity is Aω. Reason: Particle velocity is dy/dt.
Show Answer
Both are true and the reason correctly explains the assertion.
Question 43True/False
Amplitude is half the crest-to-trough distance.
Show Answer
True.
Question 44True/False
Wave number k = λ/2π.
Show Answer
False. k = 2π/λ.
Question 45True/False
For y = A sin(ωt − kx), the wave travels in positive x-direction.
Show Answer
True.
Question 46True/False
Particle velocity is equal to wave velocity at all times.
Show Answer
False.
Question 47Case Study
A student gets a y-t graph at one point and measures 0.04 s between crests. What is frequency?
Show Answer
T = 0.04 s, so f = 25 Hz.
Question 48Case Study
A student gets a y-x graph and measures 0.5 m between crests. What is k?
Show Answer
k = 2π/0.5 = 4π rad m−1.
Question 49Case Study
A rope wave is y = 0.03 sin(120t − 6x). Find v.
Show Answer
v = ω/k = 120/6 = 20 m s−1.
Question 50Case Study
A sound wave has f = 500 Hz and v = 340 m s−1. Find λ.
Show Answer
λ = v/f = 340/500 = 0.68 m.
Question 51Reasoning
Why is the coefficient of x not wavelength in y = A sin(ωt − kx)?
Show Answer
The coefficient of x is k, and wavelength is 2π/k.
Question 52Reasoning
Why is dy/dt not wave speed?
Show Answer
dy/dt measures vertical or local particle motion, while wave speed measures propagation of phase along x.
Question 53Difficult Conceptual
Can maximum particle velocity be greater than wave velocity?
Show Answer
Yes, mathematically vp,max/v = Ak; depending on amplitude and wave number it can be less or greater, though small-amplitude waves usually have Ak much less than 1.
Question 54Difficult Conceptual
What happens to phase at a fixed x as time increases?
Show Answer
Phase increases at rate ω for y = A sin(ωt − kx).
Revision
Quick Revision Notes
- A is maximum displacement from mean position.
- T is time for one complete oscillation.
- f is oscillations per second and f = 1/T.
- ω = 2πf = 2π/T.
- λ is distance between nearest same-phase points.
- k = 2π/λ is phase change per metre.
- Phase for positive x wave is ωt − kx.
- Wave velocity is v = fλ = ω/k.
- Particle velocity is dy/dt.
- Maximum particle velocity is Aω.
- Slope of wave is dy/dx.
- Particle acceleration is a = −ω2y.
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