Waves/Sine Waves

Sine Waves
A particularly simple kind of wave, the sine wave, is illustrated in figure 1.2. This has the mathematical form:
 * $$h(x) = h_0 \sin ( 2 \pi x / \lambda ) ,$$ (2.1)

where
 * h is the displacement (which can be either longitudinal or transverse),
 * $$h_0$$ is the maximum displacement, sometimes called the amplitude of the wave,
 * &lambda; is the wavelength.tory Physics fig 1.2.png|Figure 1.2: Sine wave]] -->

Figure 1.2: Definition sketch for a sine wave, showing the wavelength &lambda; and the amplitude $$h_0$$ and the phase &phi; at various points.

So far we have only considered a sine wave as it appears at a particular time. All interesting waves move with time. The movement of a sine wave to the right, a distance $$d$$ may be accounted for by replacing $$x$$ in the above formula by $$x - d$$. If this movement occurs in time $$t$$, then the wave moves at velocity $$c = d/t$$. Solving this for $$d$$ and substituting yields a formula for the displacement of a sine wave as a function of both distance $$x$$ and time $$t$$:
 * $$ h(x) = h_0 \sin [ 2 \pi (x - ct) / \lambda ] . $$ (2.2)

The time for a wave to move one wavelength is called the period of the wave: $$T = \lambda / c$$. Thus, we can also write
 * $$h(x) = h_0 \sin [ 2 \pi ( x / \lambda - t / T ) ] . $$(2.3)

Physicists actually like to write the equation for a sine wave in a slightly simpler form. Defining the wavenumber as $$k = 2 \pi / \lambda$$ and the angular frequency as $$\omega = 2 \pi / T$$, we write
 * $$ h(x) = h_0 \sin (kx - \omega t) . $$(2.4)

We normally think of the frequency of oscillatory motion as the number of cycles completed per second and is given by $$f = 1/T$$. It is related to the angular frequency omega by $$\omega = 2 \pi f$$. The angular frequency is used because it is directly analogous to the wavenumber, see above. Converting between the two is not difficult. Frequency is measured in units of hertz, abbreviated Hz; $$1 \mbox{Hz} = 1 \mbox{ s}^{-1}$$ and angular frequency $$\omega$$ is in units of radians per second. (Not converting between frequency and angular frequency is a common mistake in many calculations)

The argument of the sine function is by definition an angle. We refer to this angle as the phase of the wave, $$\phi = kx - \omega t$$. The difference in the phase of a wave at fixed time over a distance of one wavelength is $$2 \pi$$, as is the difference in phase at fixed position over a time interval of one wave period.

As previously noted, we call $$h_0$$, the maximum displacement of the wave, the amplitude. Often we are interested in the intensity of a wave, which is defined as the square of the amplitude, $$I = h_0^2$$.

The wave speed we have defined above, $$c = \lambda /T$$, is actually called the phase speed. Since $$\lambda = 2 \pi /k$$ and $$T = 2 \pi / \omega$$, we can write the phase speed in terms of the angular frequency and the wavenumber:
 * $$c = \frac{\omega}{k} \mbox{(phase speed)} . $$ (2.5)