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Chapter 14: Waves

Unit X – Oscillations and Waves

14.1 Introduction to Waves

A wave is a disturbance that propagates through a medium (or vacuum), transferring energy without the net transfer of matter.

Types of waves by medium:

  • Mechanical waves: Require a medium (sound, water waves, seismic waves)
  • Electromagnetic waves: Do not require a medium (light, radio waves, X-rays)

14.2 Transverse and Longitudinal Waves

Transverse Waves

  • Particle displacement is perpendicular to the direction of wave propagation.
  • Example: Waves on a string, light waves.

Longitudinal Waves

  • Particle displacement is parallel to the direction of wave propagation.
  • Consist of compressions (C) and rarefactions (R).
  • Example: Sound waves.
Transverse vs Longitudinal Waves Transverse Wave particle → wave direction Longitudinal Wave C R C R → wave direction

14.3 Characteristics of a Wave

  • Wavelength (\(\lambda\)): Distance between two consecutive points in the same phase (m)
  • Amplitude (A): Maximum displacement from equilibrium
  • Frequency (f): Number of oscillations per second (Hz)
  • Time period (T): \(T = 1/f\)
  • Wave speed (v): \(v = \lambda f = \lambda/T\)
  • Wave number (k): \(k = 2\pi/\lambda\)
  • Angular frequency: \(\omega = 2\pi f\)

14.4 Displacement Relation for a Progressive Wave

A plane progressive (travelling) wave moving in +x direction:

\[ y(x, t) = A\sin(kx - \omega t + \phi) \]

where:

  • \(k = \dfrac{2\pi}{\lambda}\) (wave number)
  • \(\omega = 2\pi f\) (angular frequency)
  • \(\phi\) = initial phase

Wave speed:

\[ v = \frac{\omega}{k} = \frac{\lambda}{T} = f\lambda \]

14.5 Speed of Different Waves

Speed of Transverse Wave in a String

\[ v = \sqrt{\frac{T}{\mu}} \]

where \(T\) = tension, \(\mu\) = linear mass density (kg/m)

Speed of Sound in a Medium (Newton-Laplace)

\[ v = \sqrt{\frac{B}{\rho}} \]

For ideal gas (\(v_{sound}\)):

\[ v = \sqrt{\frac{\gamma P}{\rho}} = \sqrt{\frac{\gamma RT}{M}} \]

At 0°C, \(v_{sound} \approx 332 \text{ m/s}\) in air; at 27°C, \(\approx 347 \text{ m/s}\)

14.6 Principle of Superposition

When two or more waves overlap, the resultant displacement at any point is the algebraic sum of their individual displacements:

\[ y = y_1 + y_2 \]

14.7 Reflection of Waves

  • At a rigid boundary (closed end): Wave is reflected with a phase change of π (inversion).
  • At a free end (open end): Wave is reflected without phase change (crest reflects as crest).

14.8 Standing (Stationary) Waves

When two identical waves travel in opposite directions, they interfere to form standing waves.

\[ y = 2A\cos(kx)\sin(\omega t) \]

  • Nodes: Points of zero displacement: \(kx = n\pi\) (\(n\) = integer)
  • Antinodes: Points of maximum displacement: \(kx = (2n+1)\frac{\pi}{2}\)

Adjacent nodes are separated by \(\lambda/2\).

14.9 Standing Waves in Strings (Stretched String Fixed at Both Ends)

Harmonics in a Stretched String n=1 (f₁) n=2 (f₂=2f₁) n=3 (f₃=3f₁) N A N L

Harmonics for string (both ends fixed):

\[ \lambda_n = \frac{2L}{n}, \quad f_n = n\frac{v}{2L} = nf_1 \quad (n = 1, 2, 3, …) \]

Fundamental frequency:

\[ f_1 = \frac{1}{2L}\sqrt{\frac{T}{\mu}} \]

14.10 Standing Waves in Air Columns

Open Pipe (Open at Both Ends)

Both ends are antinodes:

\[ f_n = n\frac{v}{2L} \quad (n = 1, 2, 3, …) \]

All harmonics are present.

Closed Pipe (Closed at One End)

Closed end → node, open end → antinode:

\[ f_n = (2n-1)\frac{v}{4L} \quad (n = 1, 2, 3, …) \]

Only odd harmonics are present.

14.11 Beats

When two sound waves of slightly different frequencies \(f_1\) and \(f_2\) interfere:

\[ \text{Beat frequency} = |f_1 - f_2| \]

14.12 Doppler Effect

The Doppler effect is the change in observed frequency due to relative motion between source and observer.

\[ f_{obs} = f_0\left(\frac{v + v_o}{v - v_s}\right) \]

where:

  • \(v\) = speed of sound in medium
  • \(v_o\) = speed of observer (+ if moving towards source)
  • \(v_s\) = speed of source (+ if moving towards observer)

Key Formulas Summary

FormulaQuantity
\(v = f\lambda\)Wave speed
\(y = A\sin(kx - \omega t)\)Progressive wave
\(v = \sqrt{T/\mu}\)Speed in string
\(v = \sqrt{\gamma RT/M}\)Speed of sound in gas
\(f_n = nv/(2L)\)Harmonics in string / open pipe
\(f_n = (2n-1)v/(4L)\)Harmonics in closed pipe
Beat freq = \(|f_1 - f_2|\)Beats

Practice Questions

Section A – MCQ (1 mark each)

Q1. A wave of frequency 500 Hz travels at 340 m/s. Its wavelength is:

(a) 0.68 m   (b) 0.34 m   (c) 1.7 m   (d) 170 m

Answer

(a) 0.68 m — \(\lambda = v/f = 340/500 = 0.68 \text{ m}\)

Q2. The closed organ pipe produces harmonics in the ratio:

(a) 1:2:3   (b) 1:3:5   (c) 2:4:6   (d) 1:2:4

Answer

(b) 1:3:5 — only odd harmonics (\(f, 3f, 5f, …\))

Q3. Two sound waves of frequencies 256 Hz and 260 Hz are sounded together. The number of beats heard per second is:

(a) 2   (b) 4   (c) 8   (d) 516

Answer

(b) 4 — Beat frequency = \(|260 - 256| = 4\) Hz.

Section B – Short Answer (2–3 marks)

Q4. A string of length 0.5 m is fixed at both ends and vibrates in its fundamental mode. If the speed of the wave in the string is 120 m/s, find the frequency.

Answer

\(f_1 = \dfrac{v}{2L} = \dfrac{120}{2 \times 0.5} = \dfrac{120}{1} = \mathbf{120 \text{ Hz}}\)

Q5. Explain why sound travels faster in summer than in winter.

Answer

Speed of sound in air: \(v = \sqrt{\dfrac{\gamma RT}{M}}\)

In summer, temperature \(T\) is higher → \(v\) is greater.

In winter, temperature is lower → \(v\) is smaller.

Hence sound travels faster in summer.

Quantitatively: \(v \propto \sqrt{T}\); for every 1°C rise in temperature, speed increases by approximately 0.61 m/s.

Section D – Competency-Based Questions

Q6. (Case Study) An ambulance is approaching a stationary observer at 20 m/s with its siren blowing at 1000 Hz. Speed of sound = 340 m/s.

(i) What frequency does the observer hear as the ambulance approaches?

(ii) What frequency does the observer hear as the ambulance moves away (at the same speed)?

(iii) Explain the Doppler effect using the compression and stretching of wave fronts.

(iv) State one application of the Doppler effect in medicine.

Answer

(i) Source approaches, observer stationary (\(v_o = 0\), \(v_s = 20\)):

\[ f_{obs} = f_0\frac{v}{v - v_s} = 1000 \times \frac{340}{340 - 20} = 1000 \times \frac{340}{320} \approx \mathbf{1063 \text{ Hz}} \]

(ii) Source moves away:

\[ f_{obs} = f_0\frac{v}{v + v_s} = 1000 \times \frac{340}{360} \approx \mathbf{944 \text{ Hz}} \]

(iii) As the source approaches, it “catches up” with previously emitted waves, compressing them → wavelength decreases → frequency increases. As it moves away, waves are stretched → frequency decreases.

(iv) Echocardiography / Doppler ultrasound — measures blood flow velocity by detecting the Doppler shift in reflected ultrasound waves. Used to detect blockages, measure heart function.

Q7. (Assertion-Reason) Assertion (A): Standing waves do not transport energy from one place to another.

Reason (R): In standing waves, the nodes are stationary and there is no net flow of energy.

(a) Both A and R true; R explains A

(b) Both A and R true; R does not explain A

(c) A true; R false

(d) A false; R true

Answer

(a) Both are true and R correctly explains A. In standing waves, energy oscillates between KE (at antinodes) and PE (at nodes) but is not transported from one point to another.