Ngā Tātai mō te Roa o te Aho, te Papatipu Aho, te Auautanga, me te Taumahatanga Aho
Pengantar
He wāhanga nui ngā aho i roto i ngā taonga puoro maha, mai i ngā kitā ki ngā vaiorine. He mea nui te roa, te papatipu, te auau, me te kukū o te aho ki te whakatau i te teitei me te kounga o te oro e puta mai ana. I roto i tēnei tuhinga, ka matapakihia e tātou te hononga o ēnei tawhā ki a rātou anō mā te ahupūngao me te whakamahinga o aua tawhā i roto i te mahi, inā koa i roto i te horopaki o ngā taonga puoro.
Roa o te aho
Ko te roa o te aho (\(L\)) ko te tawhiti i waenganui i ngā pūwāhi e rua e piri ana, e puritia ana rānei te aho. I roto i ngā taonga puoro, ka inehia tēnei roa mai i te pūwāhi ka tīmata te wiri o te aho ki te pūwāhi ka mutu te wiri.
Ka pā te roa o te aho ki tōna auau taketake. I te nuinga o te wā, ka roa ake te aho, ka iti ake te auau e puta mai ana. Ka kitea te whanaungatanga i waenga i te roa o te aho me te auau i roto i te whārite taketake mō te ngaru tū i runga i te aho:
\[ f = \frac{n}{2L} \sqrt{\frac{T}{\mu}} \]
Kei hea:
– Ko te auau (Hz) te \( f \),
– Ko te tau orooro \( n \) (1 mō te auau taketake, 2 mō te orooro tuarua, me ētahi atu),
– Ko te roa o te aho (mita) ko \( L \),
– Ko te kukū aho (Newton) te \( T \),
– Ko te \( \mu \) te taumaha mō ia wae roa o te aho (kg/m).
Papatipu Aho
Ko te papatipu o te aho (\(m\)) ko te papatipu katoa o te aho. Ko te papatipu mō ia wae roa (\(\mu\)) ko te papatipu o te aho mō ia mita kotahi o te roa o te aho, ā, ka whakaaturia penei:
\[ \mu = \frac{m}{L} \]
Kei hea:
– Ko te taumaha katoa o te aho (kg) ko \( m),
– Ko te \( L \) te roa o te aho (mita).
He mea nui tēnei papatipu mō ia wae roa hei whakatau i te auau o te wiri o te aho. Ko te nui ake o te \(\mu\), ko te iti iho o te auau e whakaputaina ana e te aho. Nā te mea he nui ake te pūngao e hiahiatia ana e ngā aho taumaha hei wiri i tētahi auau kua whakaritea.
Auautanga Aho
Ko te auau (\(f\)) te maha o ngā wiri ia hekona e puta mai ana i tētahi aho. Ko te auau taketake (\(f_1\)) o tētahi aho wiri te auau iti rawa, ā, ka taea te tatau mā te whakamahi i te tātai:
\[ f_1 = \frac{1}{2L} \sqrt{\frac{T}{\mu}} \]
Ka taea hoki te tatau i te auautanga orooro mā te whakamahi i tēnei tātai, mā te whakakapi i te tau orooro \(n\) ki te whārite:
\[ f_n = \frac{n}{2L} \sqrt{\frac{T}{\mu}} \]
Kei hea:
– Ko te auau o te haurore nth (Hz) te \( f_n \),
– He tau ōrohiko a \( n \).
Te Taumahatanga o te Aho
Ko te kume (\(T\)) te kaha e pā ana ki te aho hei whakakume. Ka inehia tēnei kume i roto i ngā Newton (N) ā, he mea nui e whakatau ana i te auau o te wiri o te aho.
Ki te piki ake te taumahatanga o te aho, ka piki ake anō hoki te auau o te wiri. Ka taea te whakamārama i tēnei whanaungatanga mā te whakamahi i te tātai auau taketake kua whakahuatia i runga ake nei:
\[ f = \frac{1}{2L} \sqrt{\frac{T}{\mu}} \]
Mai i tēnei tātai, ka kitea e tātou he rite tonu te auau ki te pūtake tapawhā o te ngaohiko. Ko te tikanga, ki te whakareatia te ngaohiko, ka piki ake te auau mā te tauwehenga o te pūtake tapawhā o te rua.
Tauira Tātaitanga
Me titiro tātou ki ētahi tauira tātaitanga kia pai ake ai te mārama ki te whanaungatanga i waenga i te roa o te aho, te papatipu, te auau, me te kume.
Tauira 1: Te Whakatau i te Auautanga o te Aho
Me kī he aho kotahi mita te roa, he papatipu katoa he 0.01 kg, ā, he 100 N te kume. Me whakatau te auau taketake o tēnei aho.
1. Tātaihia te papatipu mō ia wae roa:
\[ \mu = \frac{m}{L} = \frac{0.01}{1} = 0.01 \, \text{kg/m} \]
2. Whakamahia te tātai auau taketake:
\[ f_1 = \frac{1}{2L} \sqrt{\frac{T}{\mu}} \]
\[ f_1 = \frac{1}{2 \times 1} \sqrt{\frac{100}{0.01}} \]
\[ f_1 = \frac{1}{2} \sqrt{10000} \]
\[ f_1 = \frac{1}{2} \whakareatia ki te 100 \]
\[ f_1 = 50 \, \kuputuhi{Hz} \]
Nō reira, ko te auau taketake o tēnei aho ko 50 Hz.
Tauira 2: Te Whakatau i te Ngaohiko e Hiahiatia ana
Me kī tātou e hiahia ana ki te whakatau i te kume e hiahiatia ana mō tētahi aho he 0.5 mita te roa, ā, he 0.02 kg te taumaha katoa hei whakaputa i tētahi auau taketake o te 440 Hz (te auau paerewa o te nota A).
1. Tātaihia te papatipu mō ia wae roa:
\[ \mu = \frac{m}{L} = \frac{0.02}{0.5} = 0.04 \, \text{kg/m} \]
2. Whakamahia te tātai auau taketake, ka whakaoti mō \(T\):
\[ f_1 = \frac{1}{2L} \sqrt{\frac{T}{\mu}} \]
\[ 440 = \frac{1}{2 \times 0.5} \sqrt{\frac{T}{0.04}} \]
\[ 440 = \frac{1}{1} \sqrt{\frac{T}{0.04}} \]
\[ 440 = \sqrt{\frac{T}{0.04}} \]
\[ 440^2 = \frac{T}{0.04} \]
\[ 193600 = \frac{T}{0.04} \]
\[ T = 193600 \whakareatia ki te 0.04 \]
\[ T = 7744 \, \kuputuhi{N} \]
Nō reira, ko te ngaohiko e hiahiatia ana hei whakaputa i te auau taketake o te 440 Hz ko 7744 N.
Ngā Whakamahinga Whai Hua
1. Ngā Taonga Puoro
E whakamahia ana ngā aho i roto i ngā momo taonga puoro, tae atu ki ngā kitā, ngā vaiorini, me ngā piano. He mea nui te mārama ki te whanaungatanga i waenga i te roa o ngā aho, te papatipu, te auau, me te kukū hei whakahāngai tika i tētahi taonga puoro me te whakaputa i te oro e hiahiatia ana.
2. Hangarau
E whakamahia ana hoki ngā aho i roto i ngā hangarau hou, pērā i ngā pūoko me ngā taputapu ine. Hei tauira, ka whakamahia e te ine taumahatanga ngā panonitanga o te kume i roto i te aho hei ine i te whakarerekētanga me te ahotea i roto i tētahi rauemi.
3. Rangahau me te Mātauranga
I roto i te mātauranga me te rangahau ahupūngao, ka whakamahia ngā whakamātautau aho hei ako i ngā ngaru me ngā wiri. Mā tēnei ka āwhina i ngā ākonga ki te mārama ki ngā ariā taketake o te ahupūngao ngaru me te oro.
Whakamutunga
He mea nui te mārama ki ngā tātai me ngā whanaungatanga i waenga i te roa o te aho, te papatipu, te auau, me te kume i roto i ngā mara maha, inā koa te puoro me te hangarau. Mā te whakamahi i ngā tātai ahupūngao taketake, ka taea e tātou te tatau me te matapae i te whanonga o ngā aho i raro i ngā āhuatanga rerekē, hei āwhina i a tātou i roto i te whānuitanga o ngā tono mahi, mai i te whakahāngai i ngā taonga puoro ki ngā inenga tika i roto i te hangarau hou.
