Section 5.5 · Problems · 1 of 6

Problem 5.5.1: Velocity and direction of propagation

Interactive: The Wave Equation

Problem 5.5.1

Velocity and direction of propagation

Find the velocity and the direction of propagation of the phenomena denoted by the below functions:

$u = sin (2 x - 3 t)$
$u = sin^{2} (x + 3 t)$
$u = sin (\frac{3}{2} x - 3 t) cos (\frac{1}{2} x - 9 t)$
$u = sin 2 x cos 6 t$

Try it yourself first.

Strategy

By d'Alembert, Eq. (5.5.20), a function of $x - v t$ alone travels rigidly in the $+ x$ direction with speed $v$ , and a function of $x + v t$ alone travels in the $- x$ direction. So factor the coefficient of $x$ out of the argument. A function that is neither may still be a sum of such functions: try to write it as one with a trigonometric identity.

Take the coefficient of $x$ out:

u = sin (2 x - 3 t) = sin [2 (x - 1.5 t)],

a function of $x - 1.5 t$ : speed $v = 1.5$ , towards $+ x$ .

$u = sin^{2} (x + 3 t)$ is a function of $x + 3 t$ . It is not a sinusoid, but d'Alembert allows any twice differentiable shape: speed $v = 3$ , towards $- x$ .

The two factors alone would suggest the speeds $3/ (3/2) = 2$ and $9/ (1/2) = 18$ , and neither is the answer. Use $sin A cos B = \frac{1}{2} [sin (A + B) + sin (A - B)]$ with $A = \frac{3}{2} x - 3 t$ and $B = \frac{1}{2} x - 9 t$ :

A + B u = 2 x - 12 t = 2 (x - 6 t), A - B = x + 6 t, = \frac{1}{2} sin [2 (x - 6 t)] + \frac{1}{2} sin (x + 6 t) .

These are two waves of the same speed $v = 6$ , one towards $+ x$ with the wavelength $π$ and one towards $- x$ with the wavelength $2 π$ .

The same identity with $A = 2 x$ and $B = 6 t$ :

u = sin 2 x cos 6 t = \frac{1}{2} sin [2 (x - 3 t)] + \frac{1}{2} sin [2 (x + 3 t)],

two equal waves of speed $v = 3$ in opposite directions: a standing wave. Nothing propagates on balance, and the zeros $x = nπ /2$ stay where they are.

Answer

(a) v = 1.5 towards + x, (b) v = 3 towards - x, (c) v = 6 in both directions (two waves), (d) v = 3 in both directions (standing wave)

Pick (a), (b), (c) or (d) in the menu below: (c) changes its shape because its two parts run in opposite directions at $v = 6$ , and (d) keeps its zeros $x = nπ /2$ fixed.

InteractiveLoading simulation…

Check

Each function must satisfy $u_{xx} = u_{tt} / v^{2}$ . For (d), $u_{xx} = - 4 u$ and $u_{tt} = - 36 u$ , so $v^{2} = 36/4 = 9$ and $v = 3$ . For (c) the two parts have $u_{xx} / u_{tt} = 4/144$ and $1/36$ , both giving $v^{2} = 36$ , so the sum obeys one wave equation with $v = 6$ . In (a) $u_{xx} / u_{tt} = 4/9$ and $v = 1.5$ . Figure 5.5.13 shows the four functions at three instants.

(a) \sin(2x-3t) shifts right by 1.5t. — (a) $sin (2 x - 3 t)$ shifts right by $1.5 t$ .

(b) \sin^2(x+3t) shifts left by 3t. — (a) $sin (2 x - 3 t)$ shifts right by $1.5 t$ .

Remark

A product of a function of $x$ and a function of $t$ , as in (d), is never a single travelling wave. Snapshots tell the cases apart at a glance, which makes them a useful check on the algebra.

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