Waves/Types of Waves

Types of Waves
In order to make the above material more concrete, we now examine the characteristics of various types of waves which may be observed in the real world.

Ocean Surface Waves
Figure 1.3: Wave on an ocean of depth $$H$$. The wave is moving to the right and the particles of water at the surface move up and down as shown by the small vertical arrows. The particles move up, and gravity is the 'restoring' force. Water waves are also longitudinal, the water particles moving forward, then back. The restoring forces are more complex, but involve the inertia of the mass of water surrounding. Think of water waves as superimposed transverse and longitudinal waves. The net particle paths are nearly circular. This is typical of waves that travel along the boundary (interface) between two substances, in this case water and air.

These waves are manifested as undulations of the ocean surface as seen in figure 1.3. The speed of ocean waves is given by the formula
 * $$v = \left( \frac{g \tanh (kH)}{k} \right)^{1/2}, $$ (2.6)

where $$g = 9.8 \mbox{ m} \mbox{ s}^{-2}$$ is a constant related to the strength of the Earth's gravity, $$H$$ is the depth of the ocean, and the hyperbolic tangent is defined as 2.7
 * $$\tanh (x) = \frac{e^x - e^{-x}}{e^x + e^{-x}} . $$ (2.7)

Figure 1.4: Plot of the function $$\tanh (x)$$. The dashed line shows our approximation $$\tanh (x) \approx x$$ for $$\vert x\vert << 1$$.

As figure 1.4 shows, for very small x, we can approximate the hyperbolic tangent by $$\tanh (x) \approx x$$, while for very large x it is positive 1 for positive x and negative 1 for negative x. This leads to two limits: Since $$x = k H$$, the shallow water limit, which occurs when $$kH << 1$$, yields a wave speed of
 * $$v \approx (gH)^{1/2}, \mbox{(shallow water waves)} , $$ (2.8)

while the deep water limit, which occurs when $$kH >> 1$$, yields
 * $$v \approx (g/k)^{1/2}, \mbox{(deep water waves)} . $$ (2.9)

Notice that the speed of shallow water waves depends only on the depth of the water and on $$g$$. In other words, all shallow water waves move at the same speed.

On the other hand, deep water waves of longer wavelength (and hence smaller wavenumber) move more rapidly than those with shorter wavelength. Waves for which the wave speed varies with wavelength are called dispersive. Thus, deep water waves are dispersive, while shallow water waves are non-dispersive.

For water waves with wavelengths of a few centimeters or less, surface tension becomes important to the dynamics of the waves. In the deep water case the wave speed at short wavelengths is actually given by the formula
 * $$ c = (g/k + A k )^{1/2} $$ (2.10)

where the constant $$A$$ is related to surface tension and depends on the surfaces involved. For an air-water interface near room temperature, $$A \approx 74 \mbox{ cm}^3 \mbox{ s}^{-2}$$.