Chapter 14: Waves | Summary

A sinusoidal travelling wave propagating in the positive x-direction is mathematically described by y(x, t) = a sin(kx − ωt + φ), where a is the amplitude (maximum displacement), k is the angular wave number (k = 2π/λ), ω is the angular frequency (ω = 2π/T), and φ is the initial phase angle. The argument (kx − ωt + φ) is called the phase of the wave. For a wave travelling in the negative x-direction, the displacement relation is y(x, t) = a sin(kx + ωt + φ).

The amplitude a represents the maximum displacement of medium particles from equilibrium. The wavelength λ is the minimum distance between two points with the same phase. The period T is the time taken by any element to complete one full oscillation. The frequency ν = 1/T gives the number of oscillations per second, measured in hertz. The angular frequency ω = 2π/T, and the angular wave number k = 2π/λ. These quantities are related through the wave speed: v = ω/k = λ/T = λν.

The speed of a mechanical wave is determined by the inertial and elastic properties of the medium, not by the wavelength or frequency of the wave itself. For a transverse wave on a stretched string with tension T and linear mass density μ, the wave speed is v = √(T/μ). This formula is obtained by dimensional analysis; the exact derivation using Newton's laws confirms the dimensionless constant to be unity.

For longitudinal waves (sound), the speed in a fluid medium with bulk modulus B and density ρ is v = √(B/ρ). In a solid bar, the relevant modulus is Young's modulus Y, giving v = √(Y/ρ). Newton's original formula for sound in a gas assumed isothermal changes and gave v = √(P/ρ), which underestimates the measured speed by about 15%. Laplace corrected this by noting that sound propagation is adiabatic, giving v = √(γP/ρ), where γ = Cp/Cv. This corrected formula agrees with the experimental value of 331 m/s for air at STP.

When two or more waves travel simultaneously through the same medium, the net displacement of any element of the medium is the algebraic sum of the individual displacements due to each wave. This is the principle of superposition. Mathematically, if y1(x, t), y2(x, t), ..., yn(x, t) are the individual wave functions, the resultant displacement is y(x, t) = y1 + y2 + ... + yn. Each pulse or wave retains its identity after crossing others.

When two harmonic waves of the same frequency and amplitude but differing in phase by φ are superposed, the resultant amplitude is A(φ) = 2a cos(φ/2). If φ = 0 (waves in phase), the amplitude is 2a — this is constructive interference. If φ = π (waves completely out of phase), the amplitude is zero — this is destructive interference. The principle of superposition is the foundation for understanding interference, standing waves, and beats.

When a wave encounters a boundary, it gets reflected. At a rigid boundary (such as a fixed end of a string or a closed end of a pipe), the reflected wave undergoes a phase change of π (180°). This is because the displacement at a rigid boundary must always be zero, which is satisfied only if the incident and reflected waves differ in phase by π. If yi = a sin(kx − ωt) is the incident wave, the reflected wave at a rigid boundary is yr = −a sin(kx + ωt) = a sin(kx + ωt + π).

At an open (non-rigid) boundary, such as the open end of an organ pipe or a string tied to a freely moving ring, the reflected wave has no phase change. The net maximum displacement at an open boundary is twice the amplitude. When reflection occurs at two boundaries simultaneously, as in a string fixed at both ends or an air column in a closed pipe, steady wave patterns called standing waves or stationary waves are set up through repeated reflections.

When two identical waves travel in opposite directions, their superposition produces standing waves. For waves y1 = a sin(kx − ωt) and y2 = a sin(kx + ωt), the resultant is y(x, t) = 2a sin(kx) cos(ωt). In this pattern, x and t appear separately rather than in the combination (kx − ωt), so the wave pattern does not travel. Points of zero displacement are called nodes, and points of maximum amplitude are antinodes. Adjacent nodes (or antinodes) are separated by λ/2.

For a string of length L fixed at both ends, the allowed wavelengths are λ = 2L/n and the corresponding frequencies are νn = nv/2L for n = 1, 2, 3, ... The lowest frequency (n = 1) is the fundamental mode or first harmonic. For a pipe closed at one end and open at the other, only odd harmonics are possible: νn = (n + 1/2)v/2L for n = 0, 1, 2, ... with fundamental v/4L. A pipe open at both ends supports all harmonics, similar to a string fixed at both ends. These natural frequencies are called normal modes, and each is a resonant frequency of the system.

Beats are produced when two sound waves of slightly different frequencies ν1 and ν2 (with comparable amplitudes) are superposed. The resultant wave oscillates at the average frequency (ν1 + ν2)/2, but its amplitude varies periodically. The intensity alternately waxes and wanes with a beat frequency equal to the difference of the two frequencies: νbeat = ν1 − ν2. This phenomenon is audible when the beat frequency is in the range of about 1 to 10 Hz.

Mathematically, superposing s1 = a cos ω1t and s2 = a cos ω2t gives a resultant s = [2a cos ωb t] cos ωa t, where ωb = (ω1 − ω2)/2 and ωa = (ω1 + ω2)/2. The amplitude 2a cos ωb t varies with frequency ωb, but since the amplitude reaches maximum twice per cycle, the beat frequency is 2ωb = ω1 − ω2. Musicians use beats to tune their instruments: they adjust the tension until no beats are heard, indicating identical frequencies.

Chapter 14 on Waves provides a comprehensive foundation for understanding how mechanical disturbances propagate through elastic media. The key insight is that waves transport energy and information without any net transfer of matter. The mathematical description y(x, t) = a sin(kx − ωt + φ) encapsulates all wave parameters — amplitude, wavelength, frequency, speed, and phase — in a single compact equation. The wave speed depends entirely on medium properties (tension and linear mass density for strings; bulk modulus and density for sound) and not on the wave's frequency or amplitude. The principle of superposition elegantly explains interference, reflection, standing waves, and beats as natural consequences of waves coexisting in a medium.

From a board examination perspective, this chapter is highly significant and carries substantial weightage. Students must be comfortable deriving and applying the formulas for wave speed on a string (v = √T/μ) and in a gas (v = √γP/ρ, Laplace correction), setting up normal mode frequencies for strings and pipes (both open and closed), identifying nodes and antinodes in standing wave patterns, and calculating beat frequency. Conceptual questions on phase change during reflection, difference between travelling and standing waves, and why transverse waves cannot propagate in fluids are frequently asked. Numerical problems based on Examples 14.2 through 14.6 in the chapter are direct models for CBSE board questions.