Introduction

Waves are disturbances that propagate through a medium, transferring energy without the bulk movement of matter. Examples include ripples on water, sound waves, and light waves. While mechanical waves (like those on a string, water waves, sound waves) require a medium, electromagnetic waves (like light, radio waves) can travel through a vacuum. Matter waves are associated with particles like electrons and are part of quantum mechanics. This chapter focuses on mechanical waves, illustrating their connection to harmonic oscillations.

Transverse And Longitudinal Waves

Waves are classified based on the direction of oscillation of the medium's constituents relative to the direction of wave propagation:

• Transverse Waves: The particles of the medium oscillate perpendicular to the direction of wave propagation. Examples include waves on a stretched string and light waves. Transverse waves require a medium with a shear modulus of elasticity (like solids).
• Longitudinal Waves: The particles of the medium oscillate parallel to the direction of wave propagation. Examples include sound waves in air and pressure waves in solids. Longitudinal waves require a medium with a bulk modulus of elasticity, making them possible in solids, liquids, and gases.

Some waves, like those on the surface of water, can be a combination of both transverse and longitudinal motions.

Displacement Relation In A Progressive Wave

A travelling wave can be mathematically described by a function of position (x) and time (t). A sinusoidal travelling wave propagating in the positive x-direction is given by:

$$ y(x, t) = a \sin(kx - \omega t + \phi) $$

or equivalently, as a linear combination of sine and cosine functions.

Amplitude And Phase

• Amplitude (a): The maximum displacement of a particle of the medium from its equilibrium position. It is always a positive quantity.
• Phase (kx - ωt + φ): The argument of the sine function, which determines the displacement of the wave at a given position and time.
• Phase constant (φ): The phase at x=0 and t=0, determined by initial conditions.

Wavelength And Angular Wave Number

• Wavelength (λ): The minimum distance between two points on the wave that have the same phase (e.g., two consecutive crests or troughs). It is related to the angular wave number (k) by:

$$ \lambda = \frac{2\pi}{k} $$

• Angular Wave Number (k): Represents the spatial rate of change of the wave's phase. Its SI unit is rad m^-1.

Period, Angular Frequency And Frequency

• Period (T): The time it takes for one complete oscillation of a particle in the medium.
• Angular Frequency (ω): Related to the period by:

$$ \omega = \frac{2\pi}{T} $$

Its SI unit is rad s^-1.

• Frequency (ν): The number of oscillations per unit time, related to angular frequency by:

$$ \nu = \frac{1}{T} = \frac{\omega}{2\pi} $$

Its SI unit is Hertz (Hz).

For a wave travelling in the negative x-direction, the displacement function is $y(x, t) = a \sin(kx + \omega t + \phi)$.

The Speed Of A Travelling Wave

The speed of a travelling wave (v) is the speed at which a point of fixed phase (like a crest) moves. It is related to wavelength (λ) and frequency (ν) by:

$$ v = \lambda \nu = \frac{\lambda}{T} $$

This relationship holds for all progressive waves.

Speed Of A Transverse Wave On Stretched String

The speed of a transverse wave on a stretched string depends on the tension (T) in the string and its linear mass density (μ, mass per unit length):

$$ v = \sqrt{\frac{T}{\mu}} $$

The speed is determined by the properties of the medium, not the wave's frequency or wavelength.

Speed Of A Longitudinal Wave (Speed Of Sound)

The speed of a longitudinal wave in a medium depends on its elastic properties (bulk modulus B for fluids, Young's modulus Y for solids) and its mass density (ρ):

$$ v = \sqrt{\frac{B}{\rho}} \quad \text{(for fluids)} $$

$$ v = \sqrt{\frac{Y}{\rho}} \quad \text{(for solids)} $$

For an ideal gas, Newton's formula initially suggested $v = \sqrt{P/\rho}$, where P is the pressure. However, Laplace corrected this, realizing that sound propagation involves adiabatic processes, not isothermal ones. The correct formula, considering the adiabatic index γ ($= C_p/C_v$), is:

$$ v = \sqrt{\frac{\gamma P}{\rho}} $$

This corrected formula agrees well with experimental values.

The Principle Of Superposition Of Waves

The principle of superposition states that when two or more waves overlap in the same medium, the resultant displacement at any point is the algebraic sum of the displacements due to each individual wave.

Mathematically, if $y_1(x, t)$ and $y_2(x, t)$ are the displacements due to two waves, the net displacement is $y(x, t) = y_1(x, t) + y_2(x, t)$. This principle explains phenomena like interference.

When two sinusoidal waves with the same amplitude (a) and frequency, but with a phase difference (φ), overlap, the resultant wave has the same frequency but its amplitude depends on the phase difference: $A(\phi) = 2a \cos(\phi/2)$.

• Constructive Interference: Occurs when waves are in phase (φ = 0 or multiple of 2π), resulting in maximum amplitude (2a).
• Destructive Interference: Occurs when waves are out of phase (φ = π or odd multiple of π), resulting in minimum or zero amplitude.

Reflection Of Waves

When a wave encounters a boundary, it can be reflected. The nature of reflection depends on the boundary condition:

• Reflection at a Rigid Boundary: If a wave reflects from a rigid boundary (e.g., a string tied to a fixed wall), the reflected wave undergoes a phase reversal (180° or π phase shift).
• Reflection at a Free Boundary: If a wave reflects from a free or open boundary (e.g., a string tied to a freely moving ring), the reflected wave has no phase change.

Standing Waves And Normal Modes

When waves are reflected from both ends of a medium (like a string fixed at both ends or an air column in a pipe), they interfere to form standing waves (or stationary waves). In standing waves, the wave pattern does not propagate; instead, the medium oscillates in segments. The amplitude varies along the medium, with points of zero amplitude called nodes and points of maximum amplitude called antinodes.

The possible frequencies at which a system can sustain standing waves are called its normal modes or natural frequencies. These are determined by the boundary conditions:

• String fixed at both ends (length L): Wavelengths are $\lambda_n = 2L/n$, and frequencies are $\nu_n = nv/2L$, where $n = 1, 2, 3, ...$. The fundamental frequency is for n=1.
• Pipe open at both ends (length L): All harmonics are possible. Frequencies are $\nu_n = nv/2L$, $n = 1, 2, 3, ...$.
• Pipe open at one end, closed at the other (length L): Only odd harmonics are possible. Frequencies are $\nu_n = (n+1/2)v/L$, $n = 0, 1, 2, 3, ...$.

When the frequency of an external driving force matches a natural frequency of the system, resonance occurs, leading to large amplitude oscillations.

Beats

Beats occur when two sound waves of slightly different frequencies ($\nu_1$ and $\nu_2$) and comparable amplitudes are superposed. The resulting sound has an average frequency ($\nu_{avg} = (\nu_1 + \nu_2)/2$) but its intensity varies periodically with time. This periodic variation in intensity is called beats, and its frequency is equal to the difference between the two frequencies:

$$ \nu_{beat} = |\nu_1 - \nu_2| $$

This phenomenon is used in tuning musical instruments.

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