Question

The displacements of two sinusoidal waves propagating through a string are given by the following equations:-
$y_{1} = 4 \sin(20x - 30t)$
$y_{2} = 4 \sin(25x - 40t)$
where x and y are in centimeter and t is in second.
a) calculate the phase difference between these two waves at the points x= 5 cm and t = 2s.
b) when these two waves interfere , what are the maximum and minimum values of the intensity?​

Answer 1

ɢɪᴠᴇɴ :-

$y_{1} = 4 \sin(20x - 30t)$

ᴀɴᴅ,

$y_{2} = 4 \sin(25x - 40t)$

➻ a) Tᴏ ғɪɴᴅ ᴘʜᴀsᴇ ᴅɪғғᴇʀᴇɴᴄᴇ ᴡʜᴇɴ x=5cm and t= 2 s:

$➙ y_{1} = 4 \sin(20 \times 5 - 30 \times 2)$

$= 4 \sin(100 - 60)$

$= 4 \sin40$

$➙ y_{2} = 4 \sin(25 \times 5 - 40 \times 2)$

$= 4 \sin(125 - 80)$

$= 4 \sin45$

➪Pʜᴀsᴇ ᴅɪғғᴇʀᴇɴᴄᴇ ɪs 5 ʀᴀᴅɪᴀɴ ʙᴇᴄᴀᴜsᴇ

= |45-40| = 5 ʀᴀᴅɪᴀɴ .

b) Tᴏ ғɪɴᴅ ᴛʜᴇ ᴍᴀxɪᴍᴜᴍ ᴀɴᴅ ᴍɪɴɪᴍᴜᴍ ᴠᴀʟᴜᴇs ᴏғ ᴛʜᴇ ɪɴᴛᴇɴsɪᴛʏ :

➪ Aᴍᴘʟɪᴛᴜᴅᴇs ᴏғ ᴛʜᴇ ᴛᴡᴏ ᴡᴀᴠᴇs ᴀʀᴇ :-

$A_{1} = 4cm \: and \: A_{2} = 4cm$

$I_{max} = {( A_{1} + A_{2}) }^{2}$

$I_{max} = (4 + 4 {)}^{2}$

$➪ I_{max} = 64$

➙ ᴡʜᴇɴ ᴛʜᴇ ᴘʜᴀsᴇ ᴅɪғғᴇʀᴇɴᴄᴇ ɪs ᴢᴇʀᴏ ᴀɴᴅ -

$I_{min} = {( A_{1} - A_{2}) }^{2}$

$I_{min} = (4 - 4 {)}^{2}$

$➙ \: I_{min} = 0$

ᴡʜᴇɴ ᴛʜᴇ ᴘʜᴀsᴇ ᴅɪғғᴇʀᴇɴᴄᴇ ɪs :-$\pi$

Hᴏᴘᴇ ɪᴛ ʜᴇʟᴘs ᴜ !❤︎