Energy can travel along a rope in the form of a wave. The energy is transferred along the rope in the form of disturbances – a piece of the rope gets tugged upward by its left neighbour, moves up a bit and then in turn tugs upward on its right neighbour. That is a wave.

The energy can travel along the rope in any shape.

You are cleaning your apartment with a vacuum cleaner and you have strayed from one room into the other and suddenly find yourself out of power cord.

You can either (A) walk back to the first room to unplug the cord or (B) unplug it from right here. If you send a wave pulse along the power cord, it will travel all the way to the first room. Suppose that at certain time $t=0$, the height of the displacement you created along the wire looks like the function: \[ y(x) = Ae^{-x^2}. \] where the $x$ axis measures the distance from your hand along the wire.

Assuming the shape of the “disturbance” along the wire stays the same, we can write down the equation which describes its shape at a later time simply by translating the function in the time domain. The shape of the disturbance when $t=\tau$ will be given by: \[ y(x;t=\tau) = Ae^{-(x-v\tau)^2}, \] where $v$ is velocity of “disturbance propagation” along the power cord.

If you get the size of the pulse $A$ to be big enough, the “tug” by the last piece of power cord on the power socket will pull it out of the wall outlet.

• $t$: The time.
• $x$: The distance along the wire.
• $y(x,t)$: The displacement of the wire as a function of time and distance.
• $v$: The propagation velocity of waves along the wire or rope.

Define $x_c(t)$ to be centre position of the pulse in the above example as a function of time. The equation which describes a disturbance travelling with velocity $v$ is given by the formula: \[ y(x,t) = f( x - x_c(t) ), \qquad x_c(t) = vt. \] where $f(x)=y(x;t=0)$ is the function which describes the displacement along the wire at $t=0$. This is the equivalent of uniform velocity motion in kinematics $x(t)=x_i + vt$ (UVM). In the case of the wire however, the energy travels as a pulse along the wire instead of as the kinetic energy of an object.

In the next couple of sections we will see a lot of expression of the form: \[ g(x,t) = f( \alpha x - \beta t ), \] where $\alpha$ and $\beta$ are some constants. This corresponds to a travelling wave $g(x,t)$ which has the shape of $f(\alpha x)$ at $t=0$ and travels along the $x$ axis with velocity $v=\beta/\alpha$.

\[ y(x,t) = A \cos\left[ \frac{2\pi}{\lambda}( x - vt) \right]. \]

If you continuously “wiggle” the end of the wire using a simple harmonic motion $y_{SHM}(t)=A\cos(\omega t )$ then a travelling wave will form along the wire. The equation for this travelling wave is: \[ y(x,t) = A\cos( k x - \omega t ), \] where $v = \omega / k$ is the velocity of propagation for this wave along the wire.

The quantity $k$ is not directly observable in the real world, but we have the equation \[ k = \frac{ 2 \pi }{ \lambda }, \] which relates $k$ to the wavelength $\lambda$[m]. This relationship is analogous to the relationship between the angular frequency $\omega$ and the period $T$.

Note that the propagation velocity of this wave is given by: \[ v = \omega/k = \frac{2\pi f}{ 2\pi / \lambda} = \lambda f. \] Remember this equation because it applies to ALL WAVES.

In the next couple of sections we will look at different kinds of waves.

Sound vibrations travel through air in the form of a pressure wave. The speed of sound in air is: \[ v_{snd} = . \]

Thus

http://en.wikipedia.org/wiki/Wave