Waves — Class 11 Physics

1. What Is a Wave?

A wave is a disturbance that travels through space or a medium, transferring energy and information from one region to another without requiring bulk transport of the medium as a whole. Mechanical waves require a material medium, while electromagnetic waves can travel through vacuum.

Important mechanical waves include waves on strings, sound waves and water waves. The particles of a medium generally oscillate about their equilibrium positions while the disturbance propagates.

2. Transverse and Longitudinal Waves

In a transverse wave, particles of the medium oscillate perpendicular to the direction of wave propagation. Waves on a stretched string are an ideal example.

In a longitudinal wave, particles oscillate parallel to the direction of propagation. Sound in air is a longitudinal mechanical wave and consists of compressions and rarefactions.

3. Important Wave Terms

Amplitude A: maximum displacement of a particle from its equilibrium position.
Wavelength λ: distance between two successive points in the same phase, such as two consecutive crests or compressions.
Time period T: time for one complete oscillation of a particle.
Frequency f: number of oscillations per second, f=1/T.
Wave speed v: speed with which a particular phase of the disturbance travels.

The fundamental relation is v=fλ.

4. Progressive Wave

A sinusoidal progressive wave travelling in the positive x-direction can be written as:

y(x,t)=A sin(kx−ωt+φ).

Here k=2π/λ is the wave number, ω=2πf is angular frequency, and φ is the initial phase.

The combination kx−ωt represents a wave travelling in the +x direction. A form containing kx+ωt represents propagation in the −x direction.

5. Phase Difference

Two points separated by a distance Δx on a progressive sinusoidal wave have phase difference:

Δφ=2πΔx/λ.

Points separated by one wavelength are in phase. Points separated by λ/2 are in opposite phase.

6. Particle Velocity and Wave Velocity

Particle velocity is the rate at which a particular particle of the medium oscillates, whereas wave velocity is the speed at which the disturbance propagates through the medium. These are different physical quantities.

For y=A sin(kx−ωt+φ), particle velocity is ∂y/∂t=−Aω cos(kx−ωt+φ).

7. Wave Speed on a Stretched String

For a transverse wave on a stretched string, the wave speed is:

v=√(T/μ),

where T is the tension in the string and μ is its mass per unit length.

Increasing tension increases wave speed. Increasing linear density decreases wave speed.

8. Superposition Principle

When two or more waves overlap in a linear medium, the resultant displacement is the algebraic sum of the individual displacements. This is the principle of superposition.

Superposition leads to interference, standing waves and beats.

9. Interference

When coherent waves overlap, their amplitudes combine to produce regions of reinforcement and cancellation.

For two waves of equal amplitude, constructive interference occurs when the path difference is nλ, while destructive interference occurs when it is (n+½)λ, where n is an integer.

10. Standing Waves

A standing wave is produced by the superposition of two waves having the same frequency, amplitude and speed travelling in opposite directions.

It contains fixed points of zero displacement called nodes and points of maximum displacement called antinodes. There is no net propagation of the wave pattern along the medium.

Distance between two consecutive nodes or two consecutive antinodes is λ/2. Distance between a node and its nearest antinode is λ/4.

11. Standing Waves on a String Fixed at Both Ends

For a string of length L fixed at both ends, the allowed wavelengths satisfy:

L=nλ/2, where n=1,2,3,…

Therefore the frequencies are:

f_n=nv/(2L).

The fundamental frequency is f₁=v/(2L). The higher modes are harmonics of the fundamental for this ideal string.

12. Organ Pipes

Air columns in pipes can support standing sound waves. For an ideal open pipe of length L, both ends behave approximately as displacement antinodes and:

f_n=nv/(2L), n=1,2,3,…

For an ideal closed pipe with one end closed, the closed end is a displacement node and the open end is an antinode. Allowed frequencies are:

f_n=(2n−1)v/(4L), n=1,2,3,…

Thus an ideal closed pipe supports odd harmonics only.

13. Beats

When two sound waves of slightly different frequencies interfere, the listener hears periodic variations in loudness called beats.

The beat frequency is f_beat=|f₁−f₂|.

Beats can be used to compare frequencies and to tune musical instruments.

14. Sound Waves

Sound is a mechanical wave. In gases it propagates mainly as a longitudinal pressure disturbance. Its speed depends on the properties of the medium and, in gases, is approximately proportional to the square root of absolute temperature under suitable conditions.

15. Speed of Sound in a Gas

For a gas under conditions where the adiabatic approximation is appropriate:

v=√(γP/ρ),

where γ=C_P/C_V, P is pressure and ρ is density. Using the ideal-gas relation, the speed can also be related to temperature and molar mass.

16. Reflection of Waves

When a wave reaches a boundary, part or all of it may be reflected. For a string, reflection at a fixed end introduces a phase reversal, while reflection at a free end does not introduce the same inversion. Boundary conditions determine the resulting pattern.

Worked Numerical 1 — Wave Speed

Question: A wave has frequency 50 Hz and wavelength 2 m. Find its speed.

v=fλ=50×2=100 m/s.

Worked Numerical 2 — Wavelength

Question: Sound travels at 340 m/s and has frequency 680 Hz. Find its wavelength.

λ=v/f=340/680=0.50 m.

Worked Numerical 3 — Wave Number

Question: Find the wave number for a wave with wavelength 0.40 m.

k=2π/λ=2π/0.40=5π rad/m, approximately 15.7 rad/m.

Worked Numerical 4 — String Wave Speed

Question: A string has tension 100 N and linear mass density 0.01 kg/m. Find the wave speed.

v=√(T/μ)=√(100/0.01)=100 m/s.

Worked Numerical 5 — Fundamental Frequency of a String

Question: A 2 m string is fixed at both ends. Its wave speed is 120 m/s. Find the fundamental frequency.

f₁=v/(2L)=120/(4)=30 Hz.

Worked Numerical 6 — Beat Frequency

Question: Two tuning forks have frequencies 256 Hz and 260 Hz. Find the beat frequency.

f_beat=|260−256|=4 Hz.

Worked Numerical 7 — Closed Pipe

Question: An ideal closed pipe is 0.85 m long. Taking sound speed as 340 m/s, find its fundamental frequency.

f₁=v/(4L)=340/(4×0.85)=100 Hz.

Worked Numerical 8 — Phase Difference

Question: Two points on a sinusoidal wave are separated by λ/4. Find their phase difference.

Δφ=2π(Δx/λ)=2π(1/4)=π/2 rad, or 90°.

Important Derivation — Wave Speed on a String

Consider a small element of a stretched string. The tension provides the restoring force when the string is slightly curved. Applying Newton’s second law to the element and taking the small-slope limit gives the wave equation. Comparing the resulting equation with the standard wave equation yields v=√(T/μ). Thus the speed depends on tension and mass per unit length, not directly on the amplitude of a small linear wave.

Important Derivation — Standing Waves on a Fixed String

At each fixed end, displacement must be zero, so both ends are nodes. The allowed patterns require the string length to contain an integral number of half wavelengths: L=nλ/2. Therefore λ_n=2L/n and f_n=nv/(2L).

Explained MCQs

A wave has frequency 10 Hz and wavelength 3 m. Its speed is:
Answer: 30 m/s. Use v=fλ.
In a longitudinal sound wave, particles oscillate:
Answer: Parallel to the direction of propagation. This produces compressions and rarefactions.
Distance between two consecutive nodes in a standing wave is:
Answer: λ/2.
A closed pipe supports which harmonics in the ideal model?
Answer: Odd harmonics. Its boundary conditions permit frequencies (2n−1)v/(4L).
Two sound sources of 500 Hz and 506 Hz produce:
Answer: 6 beats per second. Beat frequency is the absolute difference of frequencies.
If tension in a stretched string becomes four times, wave speed becomes:
Answer: Twice. Since v∝√T.

Competency Question

A guitar string is tightened while its length and mass per unit length remain essentially unchanged. Explain how and why the frequency of its fundamental vibration changes, using the relationship between tension and wave speed.

HOTS

Two waves have equal amplitudes and frequencies and travel in opposite directions along the same string. Explain how a standing wave forms, identify nodes and antinodes, and explain why the pattern does not transport energy along the string in the same way as a progressive wave.

Common Mistakes

Confusing wave speed with the oscillation speed of individual particles.
Using wavelength in centimetres while frequency and speed are in SI units without conversion.
Forgetting that f=1/T and ω=2πf.
Mixing the boundary conditions of open and closed organ pipes.
Thinking a standing wave is produced by only one travelling wave.
Using f_beat=f₁+f₂ instead of the absolute difference.
Assuming every wave needs a material medium; electromagnetic waves can propagate through vacuum.

Quick Revision

v=fλ
f=1/T
ω=2πf
k=2π/λ
y=A sin(kx−ωt+φ)
Δφ=2πΔx/λ
String: v=√(T/μ)
Fixed string: f_n=nv/(2L)
Open pipe: f_n=nv/(2L)
Closed pipe: f_n=(2n−1)v/(4L)
Beat frequency=|f₁−f₂|
Sound in gas: v=√(γP/ρ)

Class 11 Physics – Waves