Q] Prove that all harmonic are present in the vibrations of the air column in a pipe open at both ends.
Answers
Answer:
In the case of pipe open at both the ends, the air molecules near the open end are free to vibrate , hence they vibrate with maximum amplitude.
Explanation:
ʟᴇᴛꜱ ᴄᴏɴꜱɪᴅᴇʀ ᴀ ᴏʀɢᴀɴ ᴏꜰ ᴡɪɴᴅ ᴘɪᴘᴇ ᴄʟᴏꜱᴇᴅ ᴀᴛ ᴏɴᴇ ᴇɴᴅ ᴀɴᴅ ᴏᴘᴇɴ ᴀᴛ ᴏᴛʜᴇʀ ᴇɴᴅ. ɪꜰ ᴀ ᴠɪʙʀᴀᴛɪɴɢ ᴏꜰ ᴛᴜɴɴɪɴɢ ꜰᴏʀᴋ ᴏꜰ ꜰʀᴇQᴜᴇɴᴄʏ ʜᴇʟᴅ ɴᴇᴀʀ ɪᴛꜱ ᴏᴘᴇɴ ᴇɴᴅ, ɪᴛ ꜱᴇɴᴅꜱ ʟᴏɴɢɪᴛᴜᴅɪɴᴀʟ ᴡᴀᴠᴇꜱ ɪɴ ᴛʜᴇ ᴘɪᴘᴇ,ᴛʜᴇʏ ᴀʀᴇ ʀᴇꜰʟᴇᴄᴛᴇᴅ ꜰʀᴏᴍ ᴄʟᴏꜱᴇ ᴇɴᴅ. ʜᴇʀᴇ ᴛʜᴇ ɪɴᴄɪᴅᴇɴᴛꜱ ᴀɴᴅ ʀᴇꜰʟᴇᴄᴛᴇᴅ ꜰʀᴏᴍ ᴄʟᴏꜱᴇᴅ ᴡᴀᴠᴇꜱ ɪɴᴛᴇʀꜰᴇʀᴇ ᴀɴᴅ ꜰᴏʀᴍᴇᴅ ꜱᴛᴀᴛɪᴏɴᴀʀʏ ᴡᴀᴠᴇ.
ʟᴇɴɢᴛʜ ᴏꜰ ᴏʀɢᴀɴ ᴘɪᴘᴇ, ɪ=1/4x ᴡᴀᴠᴇʟᴇɴɢᴛʜ=ꜱᴏ ꜰᴜɴᴅᴀᴍᴇɴᴛᴀʟ ꜰʀᴇQᴜᴇɴᴄʏ =ᴠ/4ɪ,ᴡʜᴇʀᴇ ᴠ ɪꜱ ꜱᴘᴇᴇᴅ ᴏꜰ ꜱᴏᴜɴᴅ.
ꜰɪʀꜱᴛ ᴏᴠᴇʀᴛᴏɴᴇꜱ ʟᴇɴɢᴛʜ ᴏꜰ ᴏʀɢᴀɴ ᴘɪᴘᴇ, 1=1=
ɴᴏᴡ ꜰʀᴇQᴜᴇɴᴄʏ ᴏꜰ ꜰɪʀꜱᴛ ᴏᴠᴇʀᴛᴏɴᴇ=3ᴠ/4ʟ
ᴇɢ.1ꜱᴛ ᴏᴠᴇʀᴛᴏɴᴇꜱ=3ʀᴅ ʜᴀʀᴍᴏɴɪᴄ.
ᴛʜᴇʀᴇꜰᴏʀᴇ ɪᴛ ɪꜱ ᴄʟᴇᴀʀ ᴛʜᴀᴛ ᴀʟʟ ʜᴀʀᴍᴏɴɪᴄꜱ ᴀʀᴇ ᴘʀᴇꜱᴇɴᴛ ɪɴ ᴛʜᴇ ᴠɪʙʀᴀᴛɪᴏɴꜱ ᴏꜰ ᴀɪʀ ᴄᴏʟᴜᴍɴ ɪɴ ᴀ ᴘɪᴘᴇ ᴏᴘᴇɴ ᴀᴛ ʙᴏᴛʜ ᴇɴᴅꜱ..