# Why do dispersive waves get wider?

Consider the two waves $$y_1=Acos(\omega_1 t+k_1 x), \tag{1}$$ $$y_2=Acos(\omega_2 t+k_2 x), \tag{2}$$

where $\omega_i=k_iv(k_i)$ for $i=1,2$ so we have a dispersive medium. Then if we take their superposition to get the wave:

$$y=2Acos(\bar {\omega}t + \bar {k}x)cos\left(\frac{\Delta \omega \ t}{2} + \frac{\Delta k \ x}{2}\right), \tag{3}$$

where

$$\bar {\omega} = \frac {\omega _1 + \omega _2}{2}, \ \ \ \bar {k} = \frac {k_1 + k_2}{2}, \tag{4}$$

$$\Delta \omega = \omega _1 - \omega _2, \ \ \ \Delta k = k_1 - k_2. \tag{5}$$

Then the waves in wave packets are ment to spread out. Why? In this case at any time $t=t_0$ the wavelength of the wavepacket is given by $\lambda=4 \pi /(k_1+k_2)$ which is a constant and therefore the wavepackets should stay the same length, so why do they spread out?

• This isn't a wave packet, it's just a superposition of two plane-waves. A wave packet needs some spacial envelope (which is the thing that spreads out) – Shep Feb 14 '15 at 19:50
• @Shep why is $2Acos((\omega_1-\omega_2)t/2+(k_1-k_2)x/2)$ not my envelope? – Quantum spaghettification Feb 14 '15 at 19:53
• I suppose that's a fine envelope. But more importantly, this wave packet is already as "spread out" as it possibly could be: it's periodic out to $\pm$ infinity. For an example of a function with a (finite) envelope, see here: en.wikipedia.org/wiki/Wave_packet#Dispersive – Shep Feb 14 '15 at 19:59
• @Shep do you mind looking at my other question physics.stackexchange.com/q/165060, along with the coments. – Quantum spaghettification Feb 14 '15 at 20:03
• – Qmechanic Feb 14 '15 at 20:03

By definition, we say that waves are dispersive if their phase velocity if a function of wavenumber. That is, $$c\equiv \frac{\omega}{k}=c(k).$$
What you've sketched above is an argument related to the group velocity. The key property of dispersion is the functional form of the angular frequency $\omega$. The relationship $\omega$ has with $k$, called a dispersion relationship, is related to the properties of the medium. For deep water surface gravity waves $\omega=\sqrt{gk}$, hence $c$ is dispersive.
For shallow water linear waves $\omega = \sqrt{gh}k$ with $h$ the depth of the water. The phase velocity of these waves is independent of $k$, hence all of these waves travel at the same speed, and a packet of waves will not disperse.