CBSE Class 11 Physics Chapter 15 Waves Notes

Key Definitions in Waves

Wave Motion

A disturbance that propagates through a medium (or vacuum, for electromagnetic waves) by transferring energy from one point to another without any net transport of matter.

Wavelength (λ)

The distance between two consecutive crests or troughs (or any two corresponding points) in a wave. SI unit: meter (m).

Frequency (f)

The number of complete oscillations or cycles made by a particle of the medium in one second. SI unit: hertz (Hz).

Time Period (T)

The time taken for one complete oscillation or cycle of a wave. It is the reciprocal of frequency (T = 1/f). SI unit: second (s).

Amplitude (A)

The maximum displacement of the particles of the medium from their mean (equilibrium) position. SI unit: meter (m).

Wave Velocity (v)

The speed at which the wave disturbance propagates through the medium. It is given by v = fλ. SI unit: m/s.

Phase

The state of oscillation of a particle in a wave, describing its position and direction of motion at a particular instant. Represented by (kx - ωt + φ).

Superposition Principle

When two or more waves overlap, the resultant displacement at any point and at any instant is the vector sum of the displacements due to individual waves.

Standing Waves

Waves formed by the superposition of two identical progressive waves travelling in opposite directions, resulting in fixed positions (nodes) of zero displacement and maximum displacement (antinodes).

Understanding Wave Motion and Types

Wave motion is a fundamental phenomenon describing the propagation of disturbances through a medium or space. It is characterized by the transfer of energy without any net transport of the medium's particles. There are broadly two types of waves based on the requirement of a medium:

1. Mechanical Waves: These waves require a material medium (solid, liquid, or gas) for their propagation. They are caused by the oscillations of particles within the medium. Examples include sound waves, water waves, and seismic waves. The speed of mechanical waves depends on the elastic and inertial properties of the medium.

1. Electromagnetic Waves: These waves do not require any material medium for their propagation and can travel through a vacuum. They consist of oscillating electric and magnetic fields perpendicular to each other and to the direction of wave propagation. Examples include light, radio waves, X-rays, and gamma rays. All electromagnetic waves travel at the speed of light in a vacuum ($c = 3 imes 10^8$ m/s).

Waves can also be classified based on the direction of particle oscillation relative to the direction of wave propagation:

• Transverse Waves: In these waves, the particles of the medium oscillate perpendicular to the direction of wave propagation. For instance, in a wave on a string, the string particles move up and down while the wave travels horizontally. Light is an electromagnetic transverse wave. They exhibit crests (maximum upward displacement) and troughs (maximum downward displacement).

• Longitudinal Waves: In these waves, the particles of the medium oscillate parallel to the direction of wave propagation. Sound waves are classic examples of longitudinal waves, where compressions (regions of high density and pressure) and rarefactions (regions of low density and pressure) propagate through the medium. The oscillations are along the line of energy transfer. Understanding these distinctions is crucial for solving problems related to wave characteristics and their behavior in different media.

Key Formulas and Must-Remember Concepts

• Wave Equation: The general equation for a progressive harmonic wave travelling in the positive x-direction is $y(x,t) = A \sin(kx - \omega t + \phi)$, where $k = 2\pi/\lambda$ (angular wave number) and $\omega = 2\pi f$ (angular frequency).
• Wave Speed: $v = f\lambda = \omega/k$. This fundamental relation connects wave speed, frequency, and wavelength.
• Speed of Transverse Wave on a Stretched String: $v = \sqrt{T/\mu}$, where T is the tension in the string and $\mu$ is the linear mass density (mass per unit length).
• Speed of Longitudinal Wave in a Fluid: $v = \sqrt{B/\rho}$, where B is the bulk modulus and $\rho$ is the density of the fluid. For an ideal gas, $v = \sqrt{\gamma RT/M}$ (Newton-Laplace formula).
• Principle of Superposition: Net displacement = Vector sum of individual displacements ($y = y_1 + y_2 + ...$). This leads to interference and beat phenomena.
• Standing Waves in a String (fixed ends): Wavelengths allowed are $\lambda_n = 2L/n$, and frequencies are $f_n = n(v/2L)$, where $n=1, 2, 3...$. $n=1$ is fundamental, $n=2$ is first overtone (second harmonic).
• Standing Waves in an Open Organ Pipe: Wavelengths allowed are $\lambda_n = 2L/n$, and frequencies are $f_n = n(v/2L)$, where $n=1, 2, 3...$. Only integral multiples of fundamental frequency are present.
• Standing Waves in a Closed Organ Pipe: Wavelengths allowed are $\lambda_n = 4L/(2n-1)$, and frequencies are $f_n = (2n-1)(v/4L)$, where $n=1, 2, 3...$. Only odd harmonics are present.
• Beats: Formed by the superposition of two waves of slightly different frequencies. Beat frequency $f_{beat} = |f_1 - f_2|$.
• Doppler Effect: Apparent change in frequency of sound (or light) due to relative motion between source and observer. For sound: $f' = f \frac{v \pm v_o}{v \mp v_s}$ (use top signs for approach, bottom for recession).
• Energy carried by a wave is proportional to the square of its amplitude and the square of its frequency ($E \propto A^2 f^2$).

Formation of Standing Waves (Stationary Waves)

1. — Standing waves are formed when two identical progressive waves (same amplitude, frequency, and wavelength) travelling in opposite directions superimpose upon each other in a confined medium. This often happens when a wave reflects off a boundary and interferes with the incident wave.
2. — At certain fixed positions in the medium, the two waves always interfere destructively. This results in zero displacement at all times. These points are called nodes. The particles at nodes remain permanently at their equilibrium positions.
3. — Midway between any two consecutive nodes, the two waves always interfere constructively. This results in maximum displacement (amplitude equal to twice the amplitude of individual waves). These points are called antinodes. Particles at antinodes oscillate with maximum amplitude.
4. — Unlike progressive waves, standing waves do not transfer energy from one point to another. The positions of nodes and antinodes are fixed. All particles between two consecutive nodes oscillate in phase, but out of phase with particles in the next segment. The distance between two consecutive nodes or antinodes is $\lambda/2$, and between a node and an adjacent antinode is $\lambda/4$.
5. — Standing waves are crucial for musical instruments. In a string fixed at both ends, nodes form at the ends. In an open organ pipe, antinodes form at both open ends. In a closed organ pipe, a node forms at the closed end and an antinode at the open end. This dictates the allowed frequencies (harmonics and overtones) produced by these instruments.

Worked Examples

• {"heading":"Example 1: Wave Speed Calculation","bodyMarkdown":"Problem: A sound wave has a frequency of 250 Hz and a wavelength of 1.36 m. Calculate its speed.\n\nSolution: Using the wave speed formula $v = f\\lambda$.\nGiven $f = 250 \\text{ Hz}$, $\\lambda = 1.36 \\text{ m}$.\n$v = 250 \\text{ Hz} \\times 1.36 \\text{ m} = 340 \\text{ m/s}$.\n"}
• {"heading":"Example 2: Fundamental Frequency of a String","bodyMarkdown":"Problem: A string of length 0.5 m and mass 5 g is under a tension of 80 N. Find its fundamental frequency.\n\nSolution: First, calculate linear mass density $\\mu = \\text{mass}/\\text{length} = 0.005 \\text{ kg} / 0.5 \\text{ m} = 0.01 \\text{ kg/m}$.\nSpeed of wave on string $v = \\sqrt{T/\\mu} = \\sqrt{80 \\text{ N} / 0.01 \\text{ kg/m}} = \\sqrt{8000} \\approx 89.44 \\text{ m/s}$.\nFor fundamental frequency ($n=1$) in a string fixed at both ends, $f_1 = v/(2L)$.\n$f_1 = 89.44 \\text{ m/s} / (2 \\times 0.5 \\text{ m}) = 89.44 \\text{ Hz}$.\n"}
• {"heading":"Example 3: Doppler Effect (Approaching Source)","bodyMarkdown":"Problem: A car approaches a stationary observer at 20 m/s, blowing a horn of frequency 400 Hz. If the speed of sound in air is 340 m/s, what is the apparent frequency heard by the observer?\n\nSolution: Using the Doppler effect formula for source approaching stationary observer: $f' = f \\frac{v}{v - v_s}$.\nGiven $f = 400 \\text{ Hz}$, $v = 340 \\text{ m/s}$, $v_s = 20 \\text{ m/s}$, $v_o = 0$.\n$f' = 400 \\text{ Hz} \\times \\frac{340 \\text{ m/s}}{(340 - 20) \\text{ m/s}} = 400 \\times \\frac{340}{320} = 400 \\times \\frac{17}{16} = 25 \\times 17 = 425 \\text{ Hz}$.\n"}

Exam Tip: Doppler Effect Sign Convention

A common mistake in Doppler Effect problems is getting the signs wrong. Remember the simple rule: **if the source and/or observer are moving towards each other, the apparent frequency increases (numerator gets larger, denominator gets smaller). If they are moving away from each other, the apparent frequency decreases (numerator gets smaller, denominator gets larger).** Always ensure consistent application of this logic rather than rote memorization of four formulas. Pay attention to whether the observer or source is moving. Also, remember that for light, the formula is slightly different and only depends on relative velocity.

Practice Questions with Solutions

• Q: What is the main difference between transverse and longitudinal waves based on particle motion? A: In transverse waves, particles oscillate perpendicular to wave propagation. In longitudinal waves, particles oscillate parallel to wave propagation.
• Q: If the tension in a stretched string is quadrupled, how does the speed of a transverse wave on it change? A: The speed ($v = \sqrt{T/\mu}$) will double, as $v \propto \sqrt{T}$. $\sqrt{4T} = 2\sqrt{T}$.
• Q: What is the condition for two waves to produce constructive interference? A: Constructive interference occurs when the phase difference between the two waves is an even multiple of $\pi$ (i.e., $0, 2\pi, 4\pi, ...$) or path difference is an integral multiple of wavelength ($0, \lambda, 2\lambda, ...$). The crest of one wave falls on the crest of another.
• Q: An organ pipe is open at both ends. If its fundamental frequency is $f$, what are the frequencies of its first two overtones? A: For an open organ pipe, all harmonics are present. The first overtone is the second harmonic ($2f$), and the second overtone is the third harmonic ($3f$).

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