16.1 Traveling Waves

A wave is a disturbance that moves from the point of origin with a wave velocity v.
A wave has a wavelength $λ$ , which is the distance between adjacent identical parts of the wave. Wave velocity and wavelength are related to the wave’s frequency and period by $v = \frac{λ}{T} = λ f .$
Mechanical waves are disturbances that move through a medium and are governed by Newton’s laws.
Electromagnetic waves are disturbances in the electric and magnetic fields, and do not require a medium.
Matter waves are a central part of quantum mechanics and are associated with protons, electrons, neutrons, and other fundamental particles found in nature.
A transverse wave has a disturbance perpendicular to the wave’s direction of propagation, whereas a longitudinal wave has a disturbance parallel to its direction of propagation.

16.2 Mathematics of Waves

A wave is an oscillation (of a physical quantity) that travels through a medium, accompanied by a transfer of energy. Energy transfers from one point to another in the direction of the wave motion. The particles of the medium oscillate up and down, back and forth, or both up and down and back and forth, around an equilibrium position.
A snapshot of a sinusoidal wave at time $t = 0.00 s$ can be modeled as a function of position. Two examples of such functions are $y (x) = A sin (k x + ϕ)$ and $y (x) = A cos (k x + ϕ) .$
Given a function of a wave that is a snapshot of the wave, and is only a function of the position x, the motion of the pulse or wave moving at a constant velocity can be modeled with the function, replacing x with $x \mp v t$ . The minus sign is for motion in the positive direction and the plus sign for the negative direction.
The wave function is given by $y (x, t) = A sin (k x - ω t + ϕ)$ where $k = 2 π / λ$ is defined as the wave number, $ω = 2 π / T$ is the angular frequency, and $ϕ$ is the phase shift.
The wave moves with a constant velocity $v_{w}$ , where the particles of the medium oscillate about an equilibrium position. The constant velocity of a wave can be found by $v = \frac{λ}{T} = \frac{ω}{k} .$

The speed of a wave on a string depends on the linear density of the string and the tension in the string. The linear density is mass per unit length of the string.
In general, the speed of a wave depends on the square root of the ratio of the elastic property to the inertial property of the medium.
The speed of a wave through a fluid is equal to the square root of the ratio of the bulk modulus of the fluid to the density of the fluid.
The speed of sound through air at $T = 20 ° C$ is approximately $v_{s} = 343.00 m/s .$

The energy and power of a wave are proportional to the square of the amplitude of the wave and the square of the angular frequency of the wave.
The time-averaged power of a sinusoidal wave on a string is found by $P_{ave} = \frac{1}{2} μ A^{2} ω^{2} v,$ where $μ$ is the linear mass density of the string, A is the amplitude of the wave, $ω$ is the angular frequency of the wave, and v is the speed of the wave.
Intensity is defined as the power divided by the area. In a spherical wave, the area is $A = 4 π r^{2}$ and the intensity is $I = \frac{P}{4 π r^{2}} .$ As the wave moves out from a source, the energy is conserved, but the intensity decreases as the area increases.

Superposition is the combination of two waves at the same location.
Constructive interference occurs from the superposition of two identical waves that are in phase.
Destructive interference occurs from the superposition of two identical waves that are $180 ° (π radians)$ out of phase.
The wave that results from the superposition of two sine waves that differ only by a phase shift is a wave with an amplitude that depends on the value of the phase difference.

A standing wave is the superposition of two waves which produces a wave that varies in amplitude but does not propagate.
Nodes are points of no motion in standing waves.
An antinode is the location of maximum amplitude of a standing wave.
Normal modes of a wave on a string are the possible standing wave patterns. The lowest frequency that will produce a standing wave is known as the fundamental frequency. The higher frequencies which produce standing waves are called overtones.