A wave is a physical disturbance that propagates through a medium or space without the actual physical transfer of matter as a whole. Waves transport both energy and information from one point to another.
Historical Chronicle: Christiaan Huygens, Robert Hooke, and Isaac Newton
Waves are classified based on the direction of oscillation of the medium particles relative to the direction of wave propagation.
In a transverse wave , the particles of the medium oscillate perpendicular to the direction of wave propagation. These waves can only propagate in media that can sustain shearing stress, such as solids.
In a longitudinal wave , the particles of the medium oscillate parallel to the direction of wave propagation. These waves can propagate in all elastic media (solids, liquids, and gases) because all states of matter can sustain compressive strain.
Waves on the surface of water are a combination of both longitudinal and transverse motions. They are categorized into:
A progressive wave is a wave that travels continuously through a medium. Mathematically, a sinusoidal progressive wave travelling along the positive x-axis is represented by:
$$y(x, t) = a \sin(kx - \omega t + \phi)$$Where:
The wavelength ($\lambda$) is the minimum distance between two points having the same phase. The angular wave number ($k$) is defined as:
$$k = \frac{2\pi}{\lambda}$$The period ($T$) is the time taken for a particle to complete one full oscillation. The frequency ($\nu$) and angular frequency ($\omega$) are given by:
$$\omega = \frac{2\pi}{T} = 2\pi\nu$$The speed of propagation of a wave is determined by the properties of the medium through which it travels.
The speed ($v$) depends on the tension ($T$) in the string and its linear mass density ($\mu$):
$$v = \sqrt{\frac{T}{\mu}}$$Mathematical Problem Statement: A steel wire 0.72 m long has a mass of 5.0 &× 10 -3 kg . If the wire is under a tension of 60 N , what is the speed of transverse waves on the wire?
Step-by-Step Algorithmic Solution Trace:
- Calculate the linear mass density ($\mu$):$$\mu = \frac{m}{L} = \frac{5.0 \times 10^{-3}\text{ kg}}{0.72\text{ m}} \approx 6.9 \times 10^{-3}\text{ kg m}^{-1}$$
- Substitute $T = 60\text{ N}$ and $\mu$ into the speed formula:$$v = \sqrt{\frac{60}{6.9 \times 10^{-3}}} \approx 93\text{ m s}^{-1}$$
Final Evaluation Output: $v = 93\text{ m s}^{-1}$
The speed of a longitudinal wave in a medium depends on its bulk modulus ($B$) and mass density ($\rho$):
$$v = \sqrt{\frac{B}{\rho}}$$For a solid bar, the bulk modulus is replaced by Young's modulus ($Y$):
$$v = \sqrt{\frac{Y}{\rho}}$$For an ideal gas, Newton assumed the process to be isothermal ($B = P$), leading to Newton's formula:
$$v = \sqrt{\frac{P}{\rho}}$$Laplace corrected this by noting that pressure variations occur too rapidly for heat exchange, making the process adiabatic ($B = \gamma P$):
$$v = \sqrt{\frac{\gamma P}{\rho}}$$Mathematical Problem Statement: Estimate the speed of sound in air at standard temperature and pressure (STP) using both Newton's formula and Laplace's correction. (Mass of 1 mole of air = 29.0 &× 10 -3 kg , $\gamma = 1.4$).
Step-by-Step Algorithmic Solution Trace:
- Calculate the density of air ($\rho$) at STP:$$\rho = \frac{\text{mass of 1 mole}}{\text{volume of 1 mole}} = \frac{29.0 \times 10^{-3}\text{ kg}}{22.4 \times 10^{-3}\text{ m}^3} = 1.29\text{ kg m}^{-3}$$
- Calculate speed using Newton's formula ($P = 1.01 \times 10^5\text{ N m}^{-2}$):$$v = \sqrt{\frac{1.01 \times 10^5}{1.29}} \approx 280\text{ m s}^{-1}$$
- Calculate speed using Laplace's correction:$$v = \sqrt{\frac{1.4 \times 1.01 \times 10^5}{1.29}} \approx 331.3\text{ m s}^{-1}$$
Final Evaluation Output: Newton's Speed = $280\text{ m s}^{-1}$, Laplace's Speed = $331.3\text{ m s}^{-1}$
The principle of superposition states that when two or more wave disturbances overlap in a medium, the resultant displacement is the algebraic sum of the displacements of the individual waves:
$$y(x, t) = y_1(x, t) + y_2(x, t)$$When two sinusoidal waves of equal amplitude and frequency but differing in phase by $\phi$ overlap, the resultant wave is:
$$y(x, t) = \left[2a \cos\left(\frac{\phi}{2}\right)\right] \sin\left(kx - \omega t + \frac{\phi}{2} ight)$$When a wave encounters a boundary, it undergoes reflection.
The superposition of two identical waves travelling in opposite directions produces a standing wave :
$$y(x, t) = [2a \sin(kx)] \cos(\omega t)$$For a string of length $L$ fixed at both ends, the boundary conditions require nodes at both ends, leading to quantized wavelengths and frequencies:
$$\lambda = \frac{2L}{n}, \quad \nu = \frac{nv}{2L} \quad (n = 1, 2, 3, \dots)$$The phenomenon of beats occurs when two harmonic sound waves of nearly equal frequencies $\nu_1$ and $\nu_2$ propagate in the same direction, causing a periodic waxing and waning of sound intensity.
The number of beats heard per second is equal to the difference between the two frequencies:
$$\nu_{\text{beat}} = |\nu_1 - \nu_2|$$