Question

Figure 16-25agives a snapshot of a wave traveling in the direction of positive xalong a string under tension. Four string elements are indicated by the lettered points. For each of those elements, determine whether, at the instant of the snapshot, the element is moving upward or downward or is momentarily at rest. (Hint:Imagine the wave as it moves through the four string elements, as if you were watching a video of the wave as it traveled rightward.) Figure 16-25bgives the displacement of a string element located at, say, x=0as a function of time. At the lettered times, is the element moving upward or downward or is it momentarily at rest?

Short answer

a) In figure a), the string elements $a$and $b$are moving upwards and the string elements $c$and $d$are moving downwards.

b) In figure b), the string elements $e$and $f$are moving downwards and the string elements $g$and $h$are moving upwards.

Step-by-step solution

Step 1: Given

i) The figure a) shows a snapshot of a wave traveling in the positive x direction along a string under tension.

ii) The figure b) shows the displacement of a string element located at x=0 as a function of time.

iii) Hint: Imagine the wave as it moves through the four string elements, as if you were watching a video of the wave as it traveled rightward.

Determining the concept

Use the concept of wave motion.

According to the concept of wave motion, the position of the string elements can be determined. Also the transverse velocity of the string elements and whether it’s positive and negative can be found.

Formulae are as follow:

$u=\frac{dy\left(x=0,t\right)}{dt}$

Where, U is transverse velocity, t is time.

 Determining at the instant of the snapshot, the position of the element is moving upward, downward or is it momentarily at rest.

a)

The position of the element in figure (a):

The wave motion at time $t$ is shown by a continuous line and its position $t+dt$is shown by a dotted line.

During the time , the string elements and are moving upwards and $c$and $d$are moving downwards.

Hence, at the instant of the snapshot, elements $a$and $b$are moving upward while elements $c$and $d$ are moving downwards.

Determining during the time dt, the element is moving upward, downward or is it momentarily at rest.

b)

The position of the element in figure (a)

obtain the plot of the transverse velocity$u$ of the wave at the position,
$x=0,u=\frac{dy\left(x=0,t\right)}{dt}$

Find the function, that is, whether the function $u$is positive or negative at that specific position.

At the instant e, the position of the string element is below the amplitude position,

Hence, the string elementmoves downward.

At the instant f, the position of the string element is below the equilibrium position and is moving toward the lowest point.

Hence, at the instant, the string element moves downward.

At the instant g, the position of the string element is above the lowest position and is moving towards the equilibrium position.

Hence, at the instant, the string element moves upward.

At instant h, the position of the string element is below the highest point that is amplitude. It is moving towards the highest position.

Hence, at the moment$h$,the string element moves upward.

Hence, in figure b), the string elements $e$and $f$are moving downwards and the string elements $g$and $h$are moving upwards.

Therefore, the position of the string elements along the plot can be found by using the concept of the wave speed on a stretched string.