Plane waves

We consider a disturbance which depends on only one position variable $x$. For example $x$ could be along the length of the beam. We have the wave equation

$$\frac{\partial^2 \xi}{\partial x^2} - \frac{1}{c^2 } \ \frac{\partial^2 \xi}{\partial t^2} = 0 \,.$$ (15.30)

In solving the wave equation it is convenient to introduce two new variables

$$\begin{eqnarray*} w_1 &=& x + ct \ \mbox{and} \ w_2 = x -ct \\ x &=& \frac{w_1 + w_2}{2} \ \mbox{and} \ t= \frac{w_1 - w_2}{2c} \end{eqnarray*}$$

We can represent $\xi(x,t)$ as a function of $w_1$ and $w_2$ i.e. $\xi ( w_1, w_2)$
Also

$$\begin{eqnarray*} \frac{\partial \xi}{\partial x} & =& \frac{\partial w_1}{\part... ...{\partial \xi}{\partial w_1} + \frac{\partial \xi}{\partial w_2} \end{eqnarray*}$$

Similarly

$$\frac{1}{c} \ \frac{\partial \xi}{\partial t} = \frac{\partial \xi}{\partial w_1} - \frac{\partial \xi}{\partial w_2}$$ (15.31)

Using these the wave equation becomes

$$\frac{\partial^2 \xi}{\partial x^2}-\frac{1}{c^2} \ \frac{\partia... ... \frac{\partial}{\partial w_1} - \frac{\partial}{\partial w_2} \right)^2 \xi =0$$ (15.32)

which gives us the condition

$$\frac{\partial^2}{\partial w_1 \partial w_2} \xi (w_1, w_2 ) = 0$$ (15.33)

Possible solutions:-

1. $\xi ( w_1, w_2)$ = Constant ( not of interest )
2. $\xi (w_1,w_2) = f_1 (w_1)$ (function of $w_1$ alone.)
3. $\xi (w_1,w_2) = f_2 (w_2)$ (function of $w_2$ alone.)

Any linear superposition of 2 and 3 above is also a solution.

$$\xi ( w_1, w_2 ) = c_1 f_1( w_1 ) + c_2 f_2(w_2)$$ (15.34)

To physically interpret these solutions we revert back to $(x,t)$.

Let us first consider solution 2 i.e. any arbitrary function of $w_1 = x+ct.$

$$\xi ( x,t ) = f_1 ( x+ ct)$$ (15.35)

At $t=0$ we have

(15.36)

at $t= 1$ we have $\xi(x,1) = f_1( x+c)$ i.e the origin x =0 has now shifted to $x= -c$. and at $t= \tau,$ the origin shift to . The form of the disturbance remains unchanged and the disturbance propagates to the left i.e. along the $-x$ direction with speed $c$ as shown in the left panel of Figure 15.6.

Figure 15.6: A left moving wave and a right moving wave

Similarly, the solution 3 corresponds to a right travelling wave

$$\xi ( x, t) = f_2 (x-ct)$$ (15.37)

which propagates along with speed $c$ as shown in the right panel of Figure 15.6

$$\xi(x,t)=a_1 f_1(x+c t) + a_2 f_2(x-c t)$$ (15.38)

Any arbitrary combination of a left travelling solution and a right travelling solution is also a solution to the wave equation

The value of $\xi$ is constant on planes perpendicular to the $x$ axis and these solutions are plane wave solutions. The sinusoidal plane wave

$$\xi(x,t)=a \cos[k (x-c t) ]$$ (15.39)

that we have studied earlier is a special case of the more general plane wave solution.