Question

The amplitudes and phase differences for four pairs of waves of equal wavelengths are (a) 2 mm, 6 mm, and $\mathbf{\pi rad}$, (b) 3 mm, 5 mm, and$\mathbf{rad}$ (c) 7 mm, 9 mm, and (d) 2 mm, 2 mm, and 0 rad. Each pair travels in the same direction along the same string. Without written calculation, rank the four pairs according to the amplitude of their resultant wave, greatest first.

(Hint:Construct phasor diagrams.)

Short answer

The four pairs can be ranked according to the amplitude of their resultant wave, taking the greatest first as,$y_a=y_d>y_b=y_c$

Step-by-step solution

Given

2 mm, 6mm and $\tau \tau \mathrm{rad}$

3 mm, 5mm and $\tau \tau \mathrm{rad}$

2 mm, 2mm and $0rad$

Hint: Construct a phasor diagram.

Determining the concept

Use the concept of phasors and draw the circuit diagram.

Formulae are as follow:

$\overrightarrow{y_m}=\overrightarrow{y_{m1}}+\overrightarrow{y_{m2}}$

Where, y is vector.

Determining the rank the four pairs according to the amplitude of their resultant wave, taking the greatest first.

a) The amplitude of their resultant wave:

$y_{m1}=2mm,y_{m2}=6mmand\varnothing =\tau \tau rad$

Draw the phasor diagram for the given amplitudes by using a vector diagram.

b) The amplitude of their resultant wave:

$y_{m1}=3mm,y_{m2}=5mmand\varnothing =\tau \tau rad$

Draw the phasor diagram for the given amplitudes by using a vector diagram.

c) The amplitude of their resultant wave:

$y_{m1}=7mm,y_{m2}=9mmand\varnothing =\tau \tau rad$

Draw the phasor diagram for the given amplitudes by using a vector diagram.

d) The amplitude of their resultant wave:

Draw the phasor diagram for the given amplitudes, by using a vector diagram.

Therefore, from the phasor diagram, rank of the four pairs according to the amplitude of their resultant wave, taking the greatest first is,

$y_a=y_d>y_b=y_c$

Therefore construct the phasor diagram by using the vector diagram concept. find their resultant wave by using vector addition law and rank them, taking the greatest first.