"Wave Phenomena"

About

Waves carry energy and information without transporting matter. From sound waves to light waves, from ripples on a pond to seismic waves — wave phenomena are fundamental to physics. This chapter covers the types, properties, and behaviour of waves including reflection, superposition, standing waves, beats, and the Doppler effect.

Key Concepts

14.1 Wave Motion

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

Key wave parameters:

14.2 Types of Waves

14.3 Mathematical Description

A wave travelling along +x direction: $y=A\sin (\omega t-kx)$

A wave travelling along −x direction: $y=A\sin (\omega t+kx)$

Phase and path difference: $\text{Phase difference }\theta =\frac{2\pi }{\lambda }\times \text{path difference}$

14.4 Speed of Waves

On a stretched string: $v=\sqrt{\frac{T}{\mu }}$

Where $T$ = tension, $\mu$ = mass per unit length.

In gases (sound): $v=\sqrt{\frac{\gamma P}{\rho }}=\sqrt{\frac{\gamma RT}{M}}$

14.5 Superposition Principle

When two or more waves overlap, the resultant displacement is the algebraic sum of individual displacements: $y=y_1+y_2+y_3+\dots$

14.6 Standing Waves (Stationary Waves)

Formed when two identical waves travel in opposite directions: $y=2A\cos kx\cdot \sin \omega t$

Nodes: Points of zero displacement ($\cos kx=0$). Node spacing = $\lambda /2$.

Antinodes: Points of maximum displacement ($\cos kx=\pm 1$). Antinode spacing = $\lambda /2$.

Standing waves on a string fixed at both ends: $L=n\frac{\lambda }{2},{\nu }_n=\frac{n}{2L}\sqrt{\frac{T}{\mu }}$

Where $n=1,2,3,\dots$ (harmonics).

14.7 Beats

When two waves of slightly different frequencies superpose, the resultant amplitude varies periodically — producing beats.

Beat frequency: ${\nu }_{\text{beat}}=|{\nu }_1-{\nu }_2|$

14.8 Doppler Effect

The apparent change in frequency of a wave due to relative motion between source and observer.

Source moving toward stationary observer: ${\nu }^{\prime }=\nu \frac{v}{v-v_s}$

Source moving away from stationary observer: ${\nu }^{\prime }=\nu \frac{v}{v+v_s}$

Observer moving toward stationary source: ${\nu }^{\prime }=\nu \frac{v+v_o}{v}$

Applications: Radar speed guns, astronomical redshift, medical ultrasound.

INTEXT QUESTIONS 14.1

Q1. State the differences between longitudinal and transverse waves.

Ans:

Q2. Write the relation between phase difference and path difference.

Ans: $\theta =\frac{2\pi }{\lambda }\times p$

Where $\theta$ = phase difference, $p$ = path difference, $\lambda$ = wavelength.

Q3. Two simple harmonic waves are represented by equations $y_1=a\sin (\omega t-kx)$ and $y_2=a\sin [(\omega t-kx)+\varphi ]$. What is the phase difference between these two waves?

Ans: The phase difference between the two waves is $\varphi$.

Terminal Exercise

1. Define wave motion. Distinguish between transverse and longitudinal waves with examples.

2. Derive the relation $v=\nu \lambda$ for a progressive wave.

3. State and explain the principle of superposition of waves.

4. Derive the equation of a standing wave: $y=2A\cos kx\cdot \sin \omega t$. Define nodes and antinodes.

5. A string of length 1 m and mass 2 g is stretched with a tension of 80 N. Find the fundamental frequency.

6. Explain the formation of beats. Two tuning forks of frequencies 256 Hz and 260 Hz are sounded together. Find the beat frequency.

7. State and explain the Doppler effect. Derive the expression for apparent frequency when: (a) source moves toward a stationary observer, (b) observer moves toward a stationary source.

8. A train moving at 72 km/h sounds its whistle of frequency 500 Hz. What frequency does a stationary observer hear as the train (a) approaches, (b) recedes? (Speed of sound = 340 m/s)

9. The fundamental frequency of a stretched string is 200 Hz. What will be the frequency of the (a) second harmonic, (b) third harmonic?

10. Explain why sound travels faster in solids than in gases.

11. A wave is represented by $y=0.05\sin (100t-2x)$ where $x$ and $y$ are in metres and $t$ in seconds. Find: (a) amplitude, (b) wavelength, (c) frequency, (d) wave speed.

12. Derive the expression for the speed of a transverse wave on a stretched string: $v=\sqrt{T/\mu }$.

Worked Examples

Example 1: Wave Speed

Problem: A wave has wavelength 2 m and frequency 170 Hz. Find its speed.

Solution: $v=\nu \lambda =170\times 2=340\text{ m/s}$

Example 2: String Frequency

Problem: A string of length 0.5 m, mass 5 g, tension 100 N. Find fundamental frequency.

Solution: $\mu =\frac{0.005}{0.5}=0.01\text{ kg/m}$ ${\nu }_1=\frac{1}{2L}\sqrt{\frac{T}{\mu }}=\frac{1}{1}\sqrt{\frac{100}{0.01}}=100\text{ Hz}$

Example 3: Doppler Effect

Problem: An ambulance siren at 800 Hz approaches at 30 m/s. Find the frequency heard. ($v_{\text{sound}}=340$ m/s)

Solution: ${\nu }^{\prime }=\nu \frac{v}{v-v_s}=800\times \frac{340}{340-30}=800\times \frac{340}{310}\approx 877\text{ Hz}$

Common Mistakes

1. Confusing wave speed with particle speed: Wave speed ($v=\nu \lambda$) is constant in a medium; particle speed varies during oscillation.
2. Thinking nodes move in standing waves: Nodes are fixed points — they don't move.
3. Using the wrong sign in Doppler effect: Source approaching → denominator decreases → frequency increases.
4. Forgetting that beats need nearly equal frequencies: If frequencies are too different, beats are not perceptible.
5. Confusing longitudinal with transverse using "parallel/perpendicular": In longitudinal, particles vibrate ALONG the direction of wave propagation.

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