Question

In Fig. 16-24, wave 1 consists of a rectangular peak of height 4 units and width d, and a rectangular valley of depth 2 units and width. The wave travels rightward along an xaxis. Choices 2, 3, and 4 are similar waves, with the same heights, depths and widths, that will travel leftward along that axis and through wave 1. Right-going wave 1 and one of the left-going waves will interfere as they pass through each other. With which left-going wave will the interference give, for an instant, (a) the deepest valley, (b) a flat line, and (c) a flat peak 2dwide?

Short answer

a) The left-going wave (4) will give interference for the deepest valley.

b) The left-going wave (4) will give interference for a flat line.

c) The left-going wave (3) will give interference for a flat peak $2d$wide.

Step-by-step solution

Step 1: Given

The height of the rectangular peak of the wave 1 is$h=4\mathrm{unit}$

The width of the rectangular peak of the wave 1 is$d$

The depth of the rectangular peak of the wave 1 is$d=2unit$

Determining the concept

Use the concept of standing waves. Apply the superposition principle for interference of the waves.

According to the superposition principle, if two sinusoidal waves of the same amplitude and same frequency travel in opposite directions along a stretched string, they interfere to produce a resultant sinusoidal wave.

Formulae are as follow:

$y^{\text{'}}\left(x,+t\right)=y_1\left(x,t\right)+y_2\left(x,t\right)$

Where, t is time , x-y axis.

Determining the interference for the deepest valley

a)

The deepest valley :

The wave (1) has depth and wave (4) has depth. When they interfere with each other, then their resultant has the deepest valley as according to the superposition principle.

Therefore, wave (1) can interfere with wave (4) that gives the deepest valley than other waves.

Hence, the left-going wave (4) will give interference for the deepest valley.

Determining the interference for a flat line

b)

A flat line:

When wave (1) interferes with wave (4), according to the superposition principle, they cancel each other. Hence, their resultant as a flat line is produced.

Hence, the left-going wave (4) will give interference for a flat line.

Determining the interference for a flat peak 2d wide

c)

A flat peak $2d$wide:

According to the superposition principle, when wave (1) interferes with wave (3), then their resultant has a level peak $2d$wide.

Hence, the left-going wave (3) will give interference for a flat peak$2d$wide.

Therefore, the deepest valley, a flat line, and a level peak $2d$wide can be found by using the superposition principle.