Physical waves, or mechanical waves, form through the vibration of a medium, be it a string, the Earth's crust, or particles of gases and fluids. Waves have mathematical properties that can be analyzed to understand the motion of the wave. This article introduces these wave properties, but will not address how to apply them in specific situations in physics.
Transverse & Longitudinal Waves
There are two types of mechanical waves.A transverse wave is such that the displacements of the medium are perpendicular (transverse) to the direction of travel of the wave along the medium. Vibrating a string in periodic motion, so the waves move along it, is a transverse wave, as are waves in the ocean.
A longitudinal wave is such that the displacements of the medium are back and forth along the same direction as the wave itself. Sound waves, where the air particles are pushed along in the direction of travel, is an example of a longitudinal wave.
Even though the waves discussed in this article will refer to travel in a medium, the mathematics introduced here can be used to analyze properties of nonmechanical waves. Electromagnetic radiation, for example, is able to travel through empty space, but still has the same mathematical properties as other waves. For example, the Doppler effect for sound waves is well known, but there exists a similar Doppler effect for light waves, and they are based around the same mathematical principles.
What Causes Waves?
 Waves can be viewed as a disturbance in the medium around an equilibrium state, which is generally at rest. The energy of this disturbance is what causes the wave motion. A pool of water is at equilibrium when there are no waves, but as soon as a stone is thrown in it, the equilibrium of the particles is disturbed and the wave motion begins.
 The disturbance of the wave travels, or propogates, with a definite speed, called the wave speed (v).
 Waves transport energy, but not matter. The medium itself doesn't travel; the individual particles undergo backandforth or upanddown motion around the equilibrium position.
The Wave Function
To mathematically describe wave motion, we refer to the concept of a wave function, which describes the position of a particle in the medium at any time. The most basic of wave functions is the sine wave (shown to the right, or sinusoidal wave, which is a periodic wave (i.e. a wave with repetitive motion).It is important to note that the wave function doesn't depict the physical wave, but rather it's a graph of the displacement about the equilibrium position. This can be a confusing concept, but the useful thing is that we can use a sinusoidal wave to depict most periodic motions, such as moving in a circle or swinging a pendulum, which don't necessarily look wavelike when you view the actual motion.
Properties of the Wave Function

wave speed (v)  the speed of the wave's propagation

amplitude (A)  the maximum magnitude of the displacement from equilibrium, in SI units of meters. The amplitude is indicated in blue in the picture.

period (T)  is the time for one wave cycle (two pulses, or from crest to crest or trough to trough), in SI units of seconds (though it may be referred to as "seconds per cycle").

frequency (f)  the number of cycles in a unit of time. The SI unit of frequncy is the hertz (Hz) and
1 Hz = 1 cycle/s = 1 s^{1}

angular frequency (omega)  is 2pi times the frequency, in SI units of radians per second.

wavelength (lambda)  the distance between any two points at corresponding positions on successive repetitions in the wave, so (for example) from one crest or trough to the next, in SI units of meters. The wavelength is indicated in red in the picture.

wave number (k)  also called the propagation constant, this useful quantity is defined as 2 pi divided by the wavelength, so the SI units are radians per meter.
 pulse  one halfwavelength, from equilibrium back
v = lambda / T = lambda fThe vertical position of a point on the wave, y, can be found as a function of the horizontal position, x, and the time, t, when we look at it. We thank the kind mathematicians for doing this work for us, and obtain the following useful equations to describe the wave motion:omega = 2 pi f = 2 pi/T
T = 1 / f = 2 pi/omega
k = 2pi/omega
omega = vk
y(x, t) = A sin omega(t  x/v) = A sin 2pi f(t  x/v)y(x, t) = A sin 2pi(t/T  x/v)
y(x, t) = A sin (omega t  kx)
The Wave Equation
One final feature of the wave function is that applying calculus to take the second derivative yields the wave equation, which is an intriguing and sometimes useful product (which, once again, we will thank the mathematicians for and accept without proving it):d^{2}y / dx^{2} = (1 / v^{2}) d^{2}y / dt^{2}The second derivative of y with respect to x is equivalent to the second derivative of y with respect to t divided by the wave speed squared. The key usefulness of this equation is that whenever it occurs, we know that the function y acts as a wave with wave speed v and, therefore, the situation can be described using the wave function.